A. Mg–{Ce, Nd, Zn} binary systems

Backscattered electron images of the Mg–Ce diffusion couples, annealed at 400, 450, and 490 °C for 120, 96, and 48 h, respectively, are shown in Figs. 1(a)–1(c). Four diffusion layers were observed in all the diffusion couples. WDS spot-analysis confirmed the presence of four compounds, namely, MgCe, Mg₃Ce, Mg₄₁Ce₅, and Mg₁₂Ce. This is consistent with the Mg–Ce phase diagram, ^{Reference Okamoto20} shown in Fig. 1(d), in the temperature range between 400 and 490 °C. The interface between diffusion layers indicates the local equilibria between the corresponding compounds. The occurrence of the correct number of layers and associated compounds is necessary for sound interdiffusion analysis using diffusion couples.

FIG. 1.

Backscattered electron images of Mg–Ce diffusion couples annealed at (a) 400 °C for 120 h, (b) 450 °C for 96 h, (c) 490 °C for 48 h, and (d) Ce–Mg binary phase diagram ^{Reference Okamoto20} with the annealing temperatures marked as dashed lines.

WDS line-scans were performed across the diffusion zones between the end-members. The composition profile of Mg–Ce diffusion couple, annealed at 490 °C and 48 h, is presented in Fig. 2. The composition profiles within the Mg₁₂Ce and Mg₄₁Ce₅ layers show constant compositions of 92 and 88.6 at.% Mg, respectively. This means that these compounds are stoichiometric. The diffusion layers of MgCe and Mg₃Ce show deviation from constant composition. Thus, Mg₃Ce is a solid solution with a homogeneity range of 1.0, 1.3, and 2.3 at.% at 400, 450, and 490 °C, respectively. Although the Mg and Ce terminal far ends of the diffusion couples were used as standard metals for WDS calibration, the stoichiometry in the MgCe phase was not 1:1 in the three diffusion couples. However, this deviation is not expected to give significant errors in the diffusion coefficients calculations.

FIG. 2.

Composition profiles of the Mg–Ce diffusion couples annealed at 490 °C and 48 h.

The dark spots in the Ce end-member are residues from the hard constituents of the polishing paste as inferred from the WDS analysis. The same residues were seen in the pure Ce end-member in the work of Kevorkov and Pekguleryuz. ^{Reference Kevorkov and Pekguleryuz21} This is because pure Ce is very soft. Despite ultrasonic cleaning in alcohol bath, alumina particles were embedded in Ce and were difficult to remove.

Backscattered electron images of Mg–Nd diffusion couples, annealed at 400, 450, and 490 °C for 120, 96, and 4 h, respectively, are shown in Figs. 3(a)–3(c). Three diffusion layers appeared in all diffusion couples. These three layers are MgNd, Mg₃Nd, and Mg₄₁Nd₅. This agrees with the Mg–Nd phase diagram ^{Reference Okamoto22} shown in Fig. 3(d). WDS line-scans were used to determine the homogeneity ranges of the terminal members and intermediate compounds. The terminal solubility of Mg in Nd was determined from the three diffusion couples as 6.3, 7.6, and 8.2 at.% Mg, at 400, 450, and 490 °C, respectively.

FIG. 3.

Backscattered electron images of Mg–Nd diffusion couples annealed at (a) 400 °C for 120 h, (b) 450 °C for 96 h, (c) 490 °C for 48 h, and (d) the Mg–Nd phase diagram ^{Reference Okamoto22} with the annealing temperatures marked as dashed lines.

The plot of composition profile of the diffusion couple annealed at 490 °C and 48 h is presented in Fig. 4. The composition profiles within the Mg₄₁Nd₅ layer show constant composition of ∼88.6 at.% Mg; whereas, the diffusion layers of Mg₃Nd appeared with a slope, which indicates the presence of a homogeneity range at 400, 450, and 490 °C as 3.35, 3.61, and 3.85 at.% Mg, respectively. MgNd occurred in all diffusion couples with very low thicknesses of 1.5, 1.9, and 2.1 µm at 400, 450, and 490 °C, respectively. The spatial displacement of WDS line-scans was reduced to ∼0.5 µm across the MgNd diffusion layer to acquire more readings from its thin layer.

FIG. 4.

Composition profile of the Mg–Nd diffusion couple annealed at 490 °C and 48 h.

The Mg₄₁Nd₅ diffusion layer, in the couple annealed at 400 °C for 120 h, shown in Fig. 3(a), appeared with remaining traces of Mg₃Nd. Similar to this was observed in the Mg–Ce diffusion couple annealed under the same conditions, Fig. 1(a). It seems that the growth rate of the Mg₄₁Ce₅ and Mg₄₁Nd₅ compounds at the Mg₃RE/Mg₄₁RE₅ interface is relatively high at 400 °C. This will be discussed in more details with experimental evidence in the interdiffusion coefficient calculations of this section.

The activation energy and preexponential factor of the growth of the Mg–Ce compounds were found higher than those of Mg–Nd compounds, as can be concluded from Table IX. This means that higher energy is required for the Ce–Mg compounds to grow at the same rate as those of Nd–Mg. Thus, the total thickness of Ce–Mg diffusion layers is higher than those of Mg–Nd, in spite of the fact that the Mg–Ce and Mg–Nd diffusion couples were annealed at the same time and temperature as demonstrated in Fig. 5. The increase in the activation energy of Mg–Ce compounds can be correlated to the larger atomic radius of Ce. This was proved using density-functional theory (DFT), where all RE elements exhibit significant increase in the activation energy with increasing the solute volume. ^{Reference Huber, Elfimov, Rottler and Militzer23}

TABLE IX.

Interdiffusion coefficients at Boltzmann–Matano planes at the interfaces of the Mg–{Ce, Nd, Zn} diffusion layers.

FIG. 5.

The total thickness of Mg–Ce and Mg–Nd diffusion zones with temperature regardless the annealing time.

Huber et al. ^{Reference Huber, Elfimov, Rottler and Militzer23} used DFT to calculate activation energies for diffusion of aluminum (Al), calcium (Ca), and several RE elements in Mg. The predicted solute-vacancy binding and exchange energies increase and decrease, respectively, with increasing solute size. Using the eight-frequency model for hcp lattices, they ^{Reference Huber, Elfimov, Rottler and Militzer23} determined correlation factors for each solute.

Backscattered electron images of Mg–Zn diffusion couples, annealed at 250, 275, and 300 °C for 552, 408, and 504 h, are shown in Figs. 6(a)–6(c). The 300 °C diffusion couple was first annealed for 350 h to follow the growth trend, i.e., higher temperature would require less annealing time. All the phases at 300 °C were observed; nevertheless, the cracks in the Mg₂Zn₁₁ diffusion layer did not allow for determining the interfaces on both sides clearly. This could be due to brittleness of this compound as well as the presence of the original diffusion interface. Thus, the same diffusion couple was annealed for 504 h to overcome the cracking problem by obtaining thicker Mg₂Zn₁₁ layer. Four intermetallic compounds, namely, Mg₁₂Zn₁₃, Mg₂Zn₃, MgZn₂, and Mg₂Zn₁₁, were observed. Figure 6(d) shows the Mg₁₂Zn₁₃ as a thin layer growing between Mg₂Zn₃ and pure Mg. The Mg₅₁Zn₂₀ was not observed in the Mg–Zn diffusion couples, because the annealing temperatures were intentionally selected below its stability temperature. This provides more practical diffusion data regarding the low temperature compounds that are present in Mg alloys at service temperature.

FIG. 6.

Backscattered electron images of Mg–Zn diffusion couples annealed at (a) 250 °C for 552 h, (b) 275 °C for 408 h, (c) 300 °C for 504 h, and (d) magnified image from (c).

The composition profile across Mg–Zn diffusion couple, annealed at 300 °C for 504 h, is shown in Figs. 7(a) and 7(b). The composition profiles confirmed the existence of three stoichiometric compounds (Mg₁₂Zn₁₃, Mg₂Zn₃, and Mg₂Zn₁₁) and one intermediate solid solution, MgZn₂. The solubility range of MgZn₂ was determined at 250, 275, and 300 °C as ∼1, 1.3, and 2.1 at.%, respectively. Figure 7(b) represents part of the composition profile shown in Fig. 7(a) to emphasis on the solubility of Zn in Mg. Accordingly, the solubility of Zn in Mg was determined as 1.5 at.% Zn at 300 °C.

FIG. 7.

(a) and (b) The composition profile across Mg–Zn diffusion couple annealed at 300 °C for 504 h.

The experimental composition profiles determined by WDS line-scans for all diffusion couples were analyzed for interdiffusion coefficients using Boltzmann–Matano method, ^{Reference Matano7,Reference Shewmon8} in which interdiffusion fluxes can be measured at any location of a diffusion couple. ^{Reference Matano7} In this work, the position of Boltzmann–Matano interface was determined from the composition profiles when the integrated areas above and under the curve are equal [Eq. (4)]. Using this method, Boltzmann–Matano plane position was determined at every intermetallic interface within the Mg–{Ce, Nd, Zn} and Zn–{Ce, Nd} diffusion couples. Once the plane position is known, the composition-dependent diffusion coefficient can be calculated for any given composition based on Eq. (1). The following example illustrates the calculation of composition-dependent interdiffusion coefficient at the Mg₃Nd/Mg₄₁Nd₅ interface at 450 °C. The composition of Nd falls across the interface from 25 at.% until it reaches 10.8 at.% as shown in Fig. 8. Then, the integrated areas above and below the curve can be calculated while changing the location of the vertical line until the areas are equal. The point at which the areas are equal represents Boltzmann–Matano plane location. The same procedure is followed to determine Boltzmann–Matano plane at the other interfaces.

FIG. 8.

Demonstration of Boltzmann–Matano plane determination at the Mg₃Nd/Mg₄₁Nd₅ interface.

The value of $\tilde D$ can be calculated at any given composition C by measuring the area $\int_{{C^0}}^C {\left( {x - {x_0}} \right){\rm{d}}C}$ and the slope ${{{\rm{d}}x} \over {{\rm{d}}C}}$ . For Mg₃Nd/Mg₄₁Nd₅ interface, $\int_{{C^0}}^C {\left( {x - {x_0}} \right){\rm{d}}C} = 5.173 \times {10^{ - 4}}$ , ${{{\rm{d}}x} \over {{\rm{d}}C}} = - 19,145$ . Thus according to Eq. (1), $\tilde D = 3.0 \times 10^{ - 13} \,{\rm{cm}}^2 /$ s. Similarly calculations at other interfaces in all the studied systems were performed. Table IX summarizes the Boltzmann–Matano plane positions and the calculated interdiffusion coefficients at every interface and temperature of Mg–{Ce, Nd, Zn} diffusion couples.

Figures 1 and 3(a) show that apparent two phase regions exist in the Mg–Ce and Mg–Nd diffusion couples annealed at 400 °C for 120 h. The interdiffusion coefficients of Mg₃Ce/Mg₄₁Ce₅ and Mg₃Nd/Mg₄₁Nd₅ interfaces, at 400 °C, show higher values than for the other adjacent interfaces, Table IX. This means that the interatomic diffusion of Mg atoms through Mg₄₁{Ce,Nd}₅ compound layers is faster than those of neighboring compounds at 400 °C resulting in a very wavy interface that appears as two phase region. Also, this could be the reason of having traces of Mg₃{Ce,Nd} compounds in the Mg₄₁{Ce,Nd}₅ diffusion layers.

Lal and Levy ^{Reference Lal and Levy10} measured the diffusion coefficient of Ce–Mg system considering a solid solution having a continuous composition profile assuming complete miscibility between the two components. However, the Ce–Mg should be treated as multiple-phase system, since four stable intermetallic compounds exist in the temperature range of 550–600 °C.

Xu et al. ^{Reference Xu, Chumbley and Laabs11} measured the diffusion coefficients of Mg–Nd system as a result of solid–liquid diffusion reaction. Therefore their measured coefficients were much higher than those obtained by Xu et al., ^{Reference Xu, Chumbley, Weigelt and Laabs12} who used solid–solid diffusion reaction. Because Mg₃Nd/Mg₄₁Nd₅ and Mg₄₁Nd₅/Mg interfaces were missing in the work of Xu et al., ^{Reference Xu, Chumbley, Weigelt and Laabs12} the diffusion coefficient at the Mg₃Nd/Mg interface could not be compared with their work. Comparing the interdiffusion coefficients of Mg–Nd intermetallics calculated in this work, Table IX, with the values reported by Xu et al., ^{Reference Xu, Chumbley, Weigelt and Laabs12} Table III, the diffusion coefficients at the Nd/MgNd and MgNd/Mg₃Nd interfaces show the same order of magnitude. However, the magnitude of the current values is much higher than those calculated by Brennan et al. ^{Reference Brennan, Bermudez, Sohn, Pool and Kainer13} Same observation was noticed regarding the magnitude of interdiffusion coefficients of the growth of the Mg–Zn compounds. There is an agreement between Sakakura and Sugino ^{Reference Sakakura and Sugino15} and Brennan et al., ^{Reference Brennan, Bermudez, Kulkarni and Sohn16} whereas the values provided in this work are much lower. This is due to the fact that the interdiffusion coefficients were measured within each individual intermetallic layer ^{Reference Brennan, Bermudez, Sohn, Pool and Kainer13,Reference Brennan, Bermudez, Kulkarni and Sohn16} using the parabolic growth constant formula ^{Reference Philibert24} instead of the diffusion at the interface. Boltzmann–Matano interface is defined as being the plane across which an equal number of atoms have crossed in both directions. ^{Reference Green, Humphreys, Mackenzie and Jones25} The current work focuses on providing these coefficients at the interface of the intermetallic compounds. Thus, our results are in good agreement with Xu et al. ^{Reference Xu, Chumbley, Weigelt and Laabs12} who also obtained the diffusion coefficients at the interfaces.

Matano ^{Reference Matano7} considered the value of $\tilde D$ as a function of composition where the coefficient of diffusion decreases rapidly with composition; except at some point, where the values retained nearly the same even though the composition is changing. The point where the diffusion coefficient becomes almost constant represents the location of Matano plane. This implies that, once we move away from the interface, the diffusion coefficient value changes. This is illustrated in Fig. 9, which represents the drop in the $\tilde D$ value with composition and then the value remains almost constant where Matano plane exists. Thus, the calculated coefficients here could be more or less close to the values given in Refs. Reference Brennan, Bermudez, Sohn, Pool and Kainer13, Reference Sakakura and Sugino15, and Reference Brennan, Bermudez, Kulkarni and Sohn16, if one considers the transient diffusion values and not those at the interface. The $\tilde D$ value near Mg₃Nd diffusion layer at 450 °C, in Fig. 9, reads as 1.63 × 10⁻¹² cm²/s, which is still close to 1.7 × 10⁻¹¹ cm²/s that was reported by Brennan et al. ^{Reference Brennan, Bermudez, Sohn, Pool and Kainer13} at the same temperature. This indicates that farther from the interface toward the center of the Mg₃Nd layer, the diffusion coefficient becomes closer to the values measured by Brennan et al. ^{Reference Brennan, Bermudez, Sohn, Pool and Kainer13} The same applies to other intermetallic compounds.

FIG. 9.

(a) The composition profile at the Mg₃Nd/Mg₄₁Nd₅ interface, and (b) corresponding interdiffusion coefficient versus at.% Nd composition at the Mg₃Nd/Mg₄₁Nd₅ interface.

The composition-dependent interdiffusion coefficients were then used to calculate the activation energies and the preexponential factors at the interfaces of the Mg–{Ce, Nd, Zn} diffusion couples based on Eq. (5). The plots of $\log {\tilde D_0}$ against 1/T are presented in Figs. 10(a)–10(c).

FIG. 10.

${\rm{Log}}{\tilde D_0}$ against 1/T plots at the intermetallic interfaces of (a) Mg–Ce, (b) Mg–Nd, and (c) Mg–Zn diffusion layers.

These plots were used to estimate the activation energy and preexponential factor, for the growth of the Mg–Ce, Mg–Nd, and Mg–Zn diffusion layers at every interface, based on the data provided in Table IX and Arrhenius equation. The activation energy and preexponential factor of the growth of the Mg–Ce, Mg–Nd, and Mg–Zn layers are listed in Table X.

TABLE X.

Activation energy and preexponential factor of the growth of the Mg–Ce, Mg–Nd, and Mg–Zn diffusion layers.

The compound formation and growth take place at the interface where two substances of dissimilar chemical natures react with one another. So, the growth rate depends on the chemical composition of the initial phases. The layer of the same intermetallic compound may grow at the interface between pure elements, between another compound and pure element, or between two other compounds. ^{Reference Dybkov26} Hence, it is concluded that the compound itself does not diffuse and the interdiffusion coefficient, activation energy, and preexponential factor are characteristics of the diffusing atoms at certain location in the diffusion couple.

According to the results obtained from Table X, the activation energy values of the growth of the Mg–Zn intermetallics, obtained from this work, were comparable to those calculated by Sakakura and Sugino. ^{Reference Sakakura and Sugino15} On the other hand, the preexponential factor of the same compounds is much lower, because of the difference in the interdiffusion coefficients values. The activation energy of the growth of the Mg–Nd compounds obtained from this work was lower than the values reported by Brennan et al. ^{Reference Brennan, Bermudez, Sohn, Pool and Kainer13} This might be due to imposing a parabolic diffusion-controlled growth equation to calculate the interdiffusion coefficients of the corresponding phases without taking the diffusion reaction at the interfaces into account. Parabolic diffusion equations can only be used to examine whether the phases grow by a diffusion-controlled process, because the growth constant is calculated based on the total layer thickness. This may be correct for isomorphous systems but for systems that form intermediate compounds, such as the systems tackled in this article, Boltzmann–Matano method is more appropriate.

B. Zn–{Ce–Nd} binary systems

Many attempts were carried out to obtain successful Zn–{Ce, Nd} diffusion couples by the current authors. Formation of all expected diffusion layers while preventing Zn evaporation is a real challenge. In the following two systems, pure Zn disappeared from the contacting interface of the two end-members after annealing at 400 °C for 240 h. However, the diffusion reaction took place and all the expected compounds formed. It was observed that the diffusion reaction took place due to the migration of Zn atoms toward RE end-members. No diffusion reaction took place upon annealing at 300 °C for up to 504 h. Annealing at 300 °C for longer time might result in diffusion. However, this could lead to increasing the gap between the two end-members due to Zn volatility and Kirkendall effect. ^{Reference Kirkendall27}

Backscattered electron image of Ce–Zn diffusion couple, annealed at 400 °C for 240 h, is shown in Fig. 11(a). Figure 11(b) represents the diffusion layers adjacent to Ce end-member. Seven diffusion layers, representing CeZn, CeZn₂, CeZn₃, Ce₃Zn₁₁, Ce₁₃Zn₅₈, Ce₂Zn₁₇, and CeZn₁₁, were observed and determined using WDS spot analysis. Two sublayers were also observed at the Ce₁₃Zn₅₈/Ce₂Zn₁₇ interface. WDS line-scan was performed to measure the composition profile across the diffusion layers as shown in Fig. 11(c). For showing the composition of all the observed phases clearly, the data ranged from 60 to 300 µm, located in the very wide layer of Ce₁₃Zn₅₈, were cut from the graph. Two intermetallic compounds were not detected by the WDS line-scan, because they formed in sublayers with nonuniform interfaces. These compounds are: CeZn₅ and Ce₃Zn₂₂, which are located between Ce₁₃Zn₅₈ and Ce₂Zn₁₇. However, these compounds were observed in other Ce–Zn diffusion couple annealed at 400 °C for 48 h, which means that they are stable at this temperature. The couple annealed at 400 °C for 48 h was not used for interdiffusion calculations, because the diffusion layers were very thin.

FIG. 11.

(a) and (b) backscattered electron images of Ce–Zn diffusion couples annealed at 400 °C for 240 h, (c) WDS composition profile, and (d) the Ce–Zn phase diagram ^{Reference Dybkov30} with the annealing temperature marked as dashed line.

In a binary diffusion couple, the number of the intermetallic compounds in every diffusion layer cannot exceed one except at the interface. However, the diffusion layers of Ce₁₃Zn₅₈ and Ce₂Zn₁₇ show the growth of sublayers at the interface, as shown in Fig. 11(a). As a result, two intermetallic compounds, CeZn₅ and Ce₃Zn₂₂, were presumed missing. It is highly probable that all the intermetallic compounds of the Ce–Zn system were formed in the diffusion couple, because of the presence of the Ce-rich CeZn and the Zn-rich Ce₂Zn₁₁ compounds in the sublayers. This was based on the fact that the layer-growth kinetics in a diffusion couple is sequential and not simultaneous. ^{Reference Okamoto28} This means that the two far-end compound layers grow initially, at the interface with the pure elements, due to the participation of diffusing atoms of one of two components and surface atoms of the other component. Once these layers reach their maximum thickness, they are consumed partially to form the next layers. ^{Reference Dybkov29} From the Ce–Zn diffusion couple annealed at 400 °C for 48 h, Fig. 12(a), it was noticed that the compounds CeZn₅ and Ce₃Zn₂₂ were imbedded in the Ce₁₃Zn₅₈ and Ce₂Zn₁₇ diffusion layers, representing Ce₁₃Zn₅₈ + CeZn₅ and Ce₃Zn₂₂ + Ce₂Zn₁₇ sublayers as shown in Fig. 12(b). In total, nine intermetallic compounds were found in this diffusion couple, which are equal to the number of the stable compounds at 400 °C of the Ce–Zn binary phase diagram ^{Reference Dybkov30} shown in Fig. 11(d). Figure 12(c) shows the wavy growth of Ce, CeZn₂, CeZn₃, Ce₃Zn₁₁, and Ce₁₃Zn₅₈ diffusion layers. This is attributed to the anisotropic diffusion of Ce and Zn resulting in uneven growth of these layers as will be discussed in the Nd–Zn diffusion couple part of this work.

FIG. 12.

(a) Backscattered electron image of the Ce–Zn diffusion couple annealed at 400 °C for 48 h, (b) diffusion sublayer formation at the Ce₁₃Zn₅/Ce₃Zn₂₂ interface, and (c) anisotropic diffusion and irregular growth of the CeZn, CeZn₂, CeZn₃, Ce₃Zn₁₁, and Ce₁₃Zn₅₈ diffusion layers.

Thus, CeZn₅ and Ce₃Zn₂₂ compounds might have been consumed upon the formation of Ce₁₃Zn₅₈ and Ce₂Zn₁₇ in the Ce–Zn diffusion couple annealed for 240 h. In addition, the fast formation kinetics of Ce₁₃Zn₅₈, which has obviously larger thickness (266.4 µm) than other layers, led to the consumption of the diffusing Zn atoms contributing further to the consumption of the CeZn₅ and Ce₃Zn₂₂ compounds. Similarly, Loo and Rieck ^{Reference van-Loo and Rieck31} and Wöhlert and Bormann ^{Reference Wöhlert and Bormann32} reported the absence of phases other than TiAl₃ in the Al–Ti diffusion couples. Wöhlert and Bormann ^{Reference Wöhlert and Bormann32} made their diffusion couples using α₂-Ti₃Al and γ-TiAl sandwiched between Ti and Al layers. The results indicated that the phase absence in favor of Ti₃Al was caused by different growth velocities due to different diffusion fluxes. The kinetics of layers consumption was tackled thoroughly by Dybkov. ^{Reference Okamoto28,Reference Dybkov29}

Backscattered electron image of Nd–Zn diffusion couple, annealed at 400 °C for 20 h, is shown in Fig. 13(a). Figure 13(b) represents an enlarged part near Nd end-member. Eight diffusion layers, representing NdZn, NdZn₂, NdZn₃, Nd₃Zn₁₁, Nd₁₃Zn₅₈, Nd₃Zn₂₂, Nd₂Zn₁₇, and NdZn₁₁ intermetallic compounds, were observed and determined using WDS spot analysis. WDS line-scan was performed across these layers to measure the composition profile shown in Fig. 13(c).

FIG. 13.

(a) and (b) Backscattered electron images of Nd–Zn diffusion couples annealed at 400 °C and 240 h, (c) WDS composition profile, and (d) the Nd–Zn phase diagram ^{Reference Okamoto33} with the annealing temperature marked as dashed line.

Although the diffusion layer of NdZn could be observed in the micrograph [Fig. 13(b)], it could not be detected in the composition profile because it is very thin. Similar to Fig. 11(c), the data ranged from 140 to 500 µm were cut from the graph for clarity. Based on the Nd–Zn phase diagram, ^{Reference Okamoto33} shown in Fig. 13(d), all the expected stable phases at 400 °C were detected.

The composition profile in Fig. 13(c) shows that a few values deviated from the stoichiometric composition of Nd₁₃Zn₅₈ (18.3 at.% Nd and 81.6 at.% Zn) near the interface with Nd₃Zn₁₁. This can be explained from the backscattered electron image in Fig. 13(b), which shows a sublayer containing Mg₁₃Zn₅₈ dendrites extending in Nd₃Zn₁₁ layer. Therefore, the WDS line-scan provided the composition of Nd₃Zn₁₁ for some points and an average composition of the two phases for other points.

Initially, it was difficult to explain the existence of the sublayers near the Nd₁₃Zn₅₈/Nd₃Zn₁₁ interface without experimental proof. Also, the interfaces between the layers are wavy indicating an isotropic diffusion between Mg and Nd. The same applies to Mg and Ce diffusion. In this work, an explanation is provided based on extensive experimental studies of the Nd–Zn diffusion couple annealed at 300 °C for 576 h. The evolution of the diffusion process is discussed in Fig. 14. Backscattered electron images of this diffusion couple are shown in Figs. 14(a)–14(c). These micrographs were taken along the diffusion interface, where the diffusion process has just started at one location at the interface but progressed in another location. The contrast of these micrographs was modified to show details of the diffusion layers. The colored-schematic, shown in Fig. 14(d), was developed from Fig. 14(c) using image processing software to show the dendritic growth mechanism of Nd₁₃Zn₅₈.

FIG. 14.

Evolution of phase growth in the Nd–Zn diffusion couple annealed at 300 °C for 576 h. (a) Initial diffusion stage, (b) advanced diffusion stage, (c) dendritic growth representation, and (d) colored-schematic showing the dendritic growth of Nd₁₃Zn₅₈. Reader is encouraged to refer to the online version for colored image.

Figure 14(a) is an initial diffusion stage showing that the thickness of NdZn₂ layer is the largest among other layers. At this stage, Nd₁₃Zn₅₈ dendrites were initially growing in the NdZn₃ layer. According to the Nd–Zn phase diagram, Fig. 13(d), NdZn₃ cannot be in equilibrium with Nd₁₃Zn₅₈. There must be an intermediate layer representing Nd₃Zn₁₁ compound, equilibrating with both NdZn₃ and Nd₁₃Zn₅₈ compounds. Also, if the diffusion process is interrupted at this stage, a two-compound layer, NdZn₃ + Nd₁₃Zn₅₈, can appear. Figure 14(b) shows an advanced diffusion stage, where Nd₁₃Zn₅₈ dendrites are continuously growing, displacing both interfaces, NdZn₃/Nd₁₃Zn₅₈ and NdZn₃/NdZn₂, forward at the expense of NdZn₂ layer. In other words, part of NdZn₂ is consumed to form NdZn₃ and part of NdZn₃ is consumed to form Nd₁₃Zn₅₈. The consumption means that more Zn atoms are transferred within NdZn₃ layer toward NdZn₂. This was demonstrated with dotted arrows in Fig. 14(c). At the bottom of Fig. 14(b), Nd₃Zn₁₁ started to form, which means that Nd₁₃Zn₅₈ reached its maximum thickness. In Fig. 14(c), the NdZn₂/NdZn₃ interface seems not to be straight; dotted curves were drawn to show that the wavy interface was due to the growth of Nd₁₃Zn₅₈ dendrites. With the aid of Fig. 14(d), it is concluded that the Nd₁₃Zn₅₈ dendrites were growing longitudinally across the NdZn₃ layer. The dendrites, then, became wider due to the side expansion resulted from the formation of secondary dendrites branched from the primary ones. The expansion continues with time, and the gaps between these branches, representing NdZn₃, become smaller until they disappear. This explanation might give an approach to understand that the growth mechanism of every diffusion layer is unique.

The location of Boltzmann–Matano plane and the composition-dependent interdiffusion coefficients of the Zn–{Ce, Nd} intermetallics are listed in Table XI. Zn–{Ce, Nd} systems are considered among the difficult systems to be studied by diffusion couple experiments, because they contain large number of intermetallic compounds and the compositions of the Zn-rich compounds are very close to each other. In this work, two successful diffusion couples, i.e., Ce–Zn and Nd–Zn, were obtained and the diffusion data of the 15 observed compounds are provided. The interdiffusion coefficients of the components of the Ce–Zn system were not calculated at six metallic interfaces, because three phases, namely, CeZn₅, Ce₃Zn₂₂, and Zn, were formed in the sublayers which were not parallel to the original interface. These compounds might be partially consumed during the diffusion process, as discussed earlier in the Zn–{Ce, Nd} diffusion couples section. These interfaces were marked with letter (a). Also, the experimentally observed interfaces, determined despite the missing compounds, were marked with bold font and letter (b), as indicated in Table XI. The same procedure was applied to the interfaces in the Nd–Zn diffusion couple. Although NdZn diffusion layer was discovered by WDS spot analysis to be very thin, the interdiffusion calculations were not performed at the corresponding interfaces for the accuracy.

TABLE XI.

Boltzmann–Matano plane locations and composition-dependent interdiffusion coefficients at the interface of the Zn–{Ce, Nd} diffusion layers.

^a Indicates the missing interfaces based on the WDS composition profiles.

^b Indicates the location of the experimentally available interfaces.