Modeling of Intermetallic Compounds Growth Between Dissimilar Metals

In many important binary metal couples, the binary system is characterized by limited solid solubility and the formation of intermetallic compounds at intermediate compositions. During reactive interdiffusion, IMC will form once sufficient diffusion across the interface has occurred, and it will then grow. The nucleation stage is often ignored since it usually occurs at very short times with respect to total annealing time. Therefore, the final thickness of the IMC layer is controlled mainly by growth. The growth kinetics of IMC phases are mainly controlled by the interdiffusion coefficient. Kajihara[1] developed a model to calculate the effective interdiffusion coefficient of an IMC phase from a binary system where only one compound will be formed. Since the thickness of an IMC phase is usually quite small with respect to the total specimen size, the two metal substrates can be considered as semi-infinite boundaries. The composition of an element at the two sides of an interface can be calculated using thermodynamic calculation software if both sides of the moving interface are assumed to be in a local equilibrium state. In Kajihara’s model,[1,2] the moving rate of the interface is assumed to be controlled by the volume diffusion in the neighboring phases, and all other effects caused by impurities or defects are ignored. It is also assumed that the volume diffusion coefficient is independent of phase composition.

To extend this approach to a system with two IMC phases, consider a hypothetical situation: a binary alloy system A-B forms two IMC phases A₂B (γ-phase) and AB (β-phase) at the interface between A and B, respectively, as schematically illustrated in Figure 1. The initial boundary conditions are

During growth of intermetallic phases, the boundary conditions of each phase can be described by

Since there are two intermetallic phase, two more boundary conditions are added compared to Kajihara’s model for a single IMC. In Eqs. [2c] through [2h], z _ab , z _bc , and z _cd are the distance of each interface to the original interface between A and B, as shown in Figure 1, and their values can be negative, zero, or positive depending on their position to the original A-B interface. According to Kajihara[1] and invoking Kidson’s model[11] for the growth kinetics of multiple intermetallic phases in a binary system, the distance values can be calculated by Eqs. [3a] through [3c]: where t is time, D _A, D _γ , D _β , and D _B are the interdiffusion coefficient of the A substrate, the γ-A₂B phase, the β-AB phase, and the B substrate, respectively. K _Aγ , K _γA, K _γβ , K _βγ , K _βB, and K _Bβ are dimensionless proportionality coefficients, which are related to the initial boundary conditions. According to Kajihara’s model, incorporating the initial and boundary conditions shown in Eqs. [2a] through [2h] into Fick’s 2nd law, the proportionality coefficients have the following relationships with the concentration profile[1,2]: where C _A0 , C _Aγ , C _γA, C _γβ , C _βγ , C _βB, C _Bβ, and C _B0 represent concentrations of the element A at one side of an interface as shown in Figure 1. All the K values can be calculated by these equations when the concentration profile data have been obtained.

The thickness of the two IMC phases can be simply calculated by where l _γ and l _β represent the thickness of the γ-phase and the β-phase, respectively.

If the Al-Mg binary system is taken as an example, the IMC phases to be considered are Mg₁₇Al₁₂ and Al₃Mg₂.[5,9,10,12,17‐19] These two IMC phases have different growth kinetics during long time annealing.[9,10,12,13,17] The Mg concentration at each side of every interface was obtained by thermodynamic calculations using the thermodynamic software package Pandat with the database PanAl2012. In order to calculate the values of the positions for each interface, only the Mg concentration profile, the interdiffusion coefficient of each phase, and the time are required as inputs.

It has been widely reported that grain boundaries can act as fast diffusion paths.[14,16,21] According to Heitjans,[14] the pre-exponent factor D ₀ should be the same for lattice diffusion and grain boundary diffusion, but the activation energy for grain boundary diffusion is about half of that for lattice diffusion. Diffusion coefficients for lattice diffusion and grain boundary diffusion can be expressed using the classic Arrhenius relationship as shown in Eqs. [6a] and [6b], respectively[14]: where D _l is the lattice diffusion coefficient, D _gb is the grain boundary diffusion coefficient, D _0l and D _0gb are pre-exponent factors for lattice diffusion and grain boundary diffusion, Q _l and Q _gb are the activation energies for lattice diffusion and grain boundary diffusion, respectively.

The contribution that grain boundaries can make to the overall diffusion rate also depends on the area of grain boundary per unit volume (area fraction) and grain boundary width. These factors can be combined to calculate an effective volume fraction available for grain boundary diffusion, as given by Eq. [7][14]:where q is a numerical factor depending on the grain shape, q = 1 for columnar grains, and q = 3 for equiaxed grains;[14] δ is grain boundary width, which can be approximately assumed as three times of the atomic diameter,[4] and L is grain size. Therefore, the overall effective diffusion coefficient can be calculated as weighted average of grain boundary diffusion and lattice diffusion contributions as[14]where D _eff is the overall effective diffusion coefficient.

The grain size within the IMC is not usually constant but evolves due to grain growth processes as the layer thickens. This must be accounted for to accurately determine the contribution made by grain boundary diffusion. Grain growth kinetics generally follows the typical grain growth equation[20]:where L ₁ is the average grain size at the time t ₁ , L ₂ is the average grain size at the time t ₂ , n is the exponent factor, and k is a constant.

Measured values of the exponent (n) range from 2 to 4 for different metals.[20] This classic grain growth model is valid for equiaxed grains. For columnar grains, which are often observed in the IMC layer formed in a diffusion couple, if only the widening of grains is considered, then Eq. [9] can also be applied, but the grain size L should be the average width of the columnar grains measured from 2-D section.[20] It is difficult to determine the appropriate values for k and n a priori, and so in the present work, these are derived by fitting to experimental measurements of the IMC grain size made at different times. A further complication arises because the grain size is often not uniform through the IMC layer. In this case, it is diffusion through the region of the largest grain size that will be rate limiting since there will be fewer fast diffusion pathways. Therefore, it is the grain size of the largest grain region (which usually makes up most of the layer thickness) that is used as the effective grain size in the model. This will be demonstrated for the case of the Al-Mg couple in the results section.

Once the grain size is known, the effective diffusion coefficient for the IMC phase at each temperature at each time step can be predicted using the model described in Eqs. [6] through [8]. The IMC growth kinetics can then be predicted using the multi-IMC-phase diffusion model, which is expressed in Eqs. [3] through [5]. The model structure is schematically illustrated in the flow chart shown in Figure 2. If the interdiffusion coefficients of the IMC phases are unknown, the present model can also be applied to calculate the pre-exponential factor D ₀ and the activation energy Q of the IMC phases if the thickness of each IMC phase has been determined experimentally.

The Mg₁₇Al₁₂ phase and the Al₃Mg₂ phase are the two IMC phases expected to form in Al-Mg reactive diffusion couples. In order to clearly distinguish the two IMC phases and analyze the IMC grain size, EBSD was applied to study the IMC layers. Figure 3 shows an example EBSD phase identification map after 10 minutes annealing at 673 K (400 °C). It can be seen that at short time (Figure 3(a)), the Mg₁₇Al₁₂ phase is thicker than the Al₃Mg₂ phase. After a long annealing time (3 hour or even 72 hour), the Al₃Mg₂ phase becomes much thicker than the Mg₁₇Al₁₂ phase [Figures 3(b) and (c)]. Further details are given elsewhere.[5] Mg₁₇Al₁₂ phase is the first formed IMC phase, but the Al₃Mg₂ phase has a faster growth rate. Therefore, as the annealing time is increased, the Al₃Mg₂ phase eventually becomes thicker than the Mg₁₇Al₁₂ phase. In each sample, the grain size of the Mg₁₇Al₁₂ phase is quite uniform, and thus applying the average grain size for the modeling calculation is acceptable. However, the grain size for Al₃Mg₂ phase varies with the distance to the Al substrate. New small Al₃Mg₂ grains always nucleate at the interface between Al₃Mg₂/Al with the thickening of the Al₃Mg₂ phase, and they increase in size toward the interface between Al₃Mg₂/Mg₁₇Al₁₂. Near the interface to the Mg₁₇Al₁₂ phase, the Al₃Mg₂ grains can be as large as the Mg₁₇Al₁₂ grains. This change in grain size can clearly be seen in Figure 3. As discussed previously, the rate-limiting step will be the diffusion through material with the largest grain size. Therefore, in the model, the grain size at the largest grain region is applied in Eq. [8] to calculate the effective diffusion coefficient D _eff of the IMC layer.

From Figure 3(c), a tertiary IMC phase is present at the interface between the Mg₁₇Al₁₂ phase and the Al₃Mg₂ phase after a long time heat treatment at 673 K (400 °C). There are a number of candidate tertiary IMC phases that can form after long time annealing of Al-Mg couples.[22‐24] Possible phases include ε-phase, R-phase, λ-phase, or ζ-phase. EDX analysis of the phase composition shows it contains ~56 at. pct Al, which best matches the expected chemistry for ε-phase. This phase only formed after very extended (e.g., 72 h) isothermal heat treatment, and does not normally arise in the timescale used for industrial dissimilar metal joining processes. Therefore, it is not considered in the present modeling calculation.

The thickness data and the grain size data obtained from the present experimental results are shown in Figure 4 at different temperatures and times. The grain size data are measured from 2-D section to get the true grain size. The growth behaviors of each IMC layer at various temperatures are plotted in Figures 4(a) and (b). The growth kinetics of the Al₃Mg₂ phase is always faster than the Mg₁₇Al₁₂ phase at the temperature range, and the kinetics curves of both the IMC phases generally follow the parabolic relationship with time. Figure 4(c) shows the average grain width measured from a 2-D section of Mg₁₇Al₁₂, and Figure 4(d) shows the average grain width measured from a 2-D section of Al₃Mg₂ in the region close to the interface where the grain size is largest. From the grain size data in Figures 4(c) and (d), the grain growth behavior at these temperatures generally follows the parabolic relationship in Eq. [9] with the exponent factor n = 2. In Figures 4(a) and (b), the error bars are standard deviations of IMC thickness measured at five different positions. In Figures 4(c) and (d), the error bars are standard deviations of the width of all measured grains in a sample.

The model to predict the IMC thickness evolution for both Mg₁₇Al₁₂ and Al₃Mg₂ phases was implemented in MATLAB based on the theory already presented. Equilibrium compositions of the phases at the interfaces were calculated as a function of temperature using Pandat as already described.

To predict the IMC growth kinetics, the model requires diffusion data for both grain boundary and lattice diffusion. The diffusion data available in the literature extracted from measuring IMC thickening rates provides only an effective value for activation energy that combines both grain boundary and lattice diffusion.[12,13] If these values are used as inputs to the model (assuming they apply to lattice diffusion only), then as shown in Figure 5 the thickening rates are always overestimated by the model. This is because the contribution from grain boundary and lattice diffusion is not properly considered separately as required by the model. These calculations demonstrate that values of D ₀ and Q from the literature cannot be directly applied in the present model, but the new accurate values of diffusion coefficient parameters including D _0l , D _0gb, Q _l , and Q _gb are necessary.

The present model can also be utilized to calculate the interdiffusion coefficients of IMC phases. In order to calculate the interdiffusion coefficients of the two IMC phases, the grain growth kinetics of each IMC phase and the thickness of each IMC layer after a heat treatment are required. The self-diffusion coefficient of Mg and Al are obtained from Brennan.[6] The effective interdiffusion coefficients of the Mg₁₇Al₁₂ phase and the Al₃Mg₂ phase can be calculated by Eqs. [3] through [5]. The individual contributions from lattice and grain boundary diffusion can then be calculated using the measured grain size data at each isothermal holding time as shown in Figure 4(b). The calculated D ₀ and Q values for both mechanisms are as shown in Table I.

With the diffusion coefficient data from Table I, the present model was also utilized to predict the thickness of IMC layers at different temperatures. Modeling predictions are compared with experimental measured IMC growth for Al-Mg couples with different initial conditions taken from the literature[10,25] as shown in Figure 6. The experimental data in Figure 6(a) are from Panteli.[25] In that study, couples between Al alloy (AA6111) and Mg alloy (AZ31) were lightly welded before the heat treatment, and thus the sample conditions should be quite similar to those in the present work. The experimental data in Figure 6(b) were from a different technique, that of diffusion bonding of pure Al-Mg couples with no pre-welding, only contact under pressure.[10] The IMC grain size data were not measured in these studies.[10,25] Therefore, it was assumed that the initial grain size in the IMC layer was the same as that measured in the present work, but subsequent grain growth was allowed to vary as a function of temperature according to the grain growth model already outlined. The good agreement between predictions and experiments suggests that an accurate a priori knowledge of the initial grain size in the IMC is not needed for the present model to be used successfully, because the grain size during most of the IMC thickening time is determined more strongly by the temperature at which grain growth occurs.