Modeling of Intermetallic Compounds Growth Between Dissimilar Metals

A model has been developed to predict growth kinetics of the intermetallic phases (IMCs) formed in a reactive diffusion couple between two metals for the case where multiple IMC phases are observed. The model explicitly accounts for the effect of grain boundary diffusion through the IMC layer, and can thus be used to explore the effect of IMC grain size on the thickening of the reaction layer. The model has been applied to the industrially important case of aluminum to magnesium alloy diffusion couples in which several different IMC phases are possible. It is demonstrated that there is a transition from grain boundary-dominated diffusion to lattice-dominated diffusion at a critical grain size, which is different for each IMC phase. The varying contribution of grain boundary diffusion to the overall thickening kinetics with changing grain size helps explain the large scatter in thickening kinetics reported for diffusion couples produced under different conditions.


Introduction
Intermetallic compounds (IMCs) are often formed at the interface between dissimilar metals during diffusion bonding, welding, and mechanical milling etc. [1][2][3][4][5]. The IMCs usually have properties that are very different from the base metals and thus have a critical effect in controlling overall joint performance.
Diffusion bonding is a widely used technique to study the growth kinetics of IMC phases between dissimilar metals. In a diffusion bonded couple, two different metals are brought into intimate contact and then annealed at an appropriate temperature below the eutectic melting temperature. Solid state diffusion occurs between the two metals and leads to the formation of one or more IMC phases at the interface between the two metal substrates once a critical level of enrichment is reached. Once the IMC has formed a continuous layer at the interface, further growth is only possible by diffusion through the IMC itself, and an abrupt change in kinetics is often observed. To produce a bond with good mechanical properties it is usually essential to maintain the IMC layer thickness below a critical value, since IMC phases are typically brittle and can lead to a marked loss in joint toughness if allowed to become too thick [5][6][7][8]. Therefore, many researchers have studied reactive diffusion in a large number of binary alloy diffusion couples and there have also been several attempts to model the evolution of the IMC layer [1][2][3][4][5]. It is widely accepted that reactive diffusion is mainly governed by volume diffusion [1][2][3]9,10], and the thickness of an IMC phase generally follows a parabolic relationship with annealing time [3,5,9,10].
It is common in analyzing reactive interdiffusion to measure the thickness of the IMC layer as a function of time under isothermal conditions and then fit the data to a parabolic law of the form: where l is the layer thickness, k the parabolic coefficient and t is time. By performing such experiments at a range of temperatures, an effective activation energy Q and the pre-exponent factor k 0 for the thickening kinetics are obtained from the Arrhenius equation [1][2][3]10]. According to Kidson [11], the parabolic coefficient is a mixture of many different parameters, and the simple Arrhenius dependence on temperature is not strictly correct. It is found that the values of activation energy that are derived from such experiments show a large variation even for the same system between studies, e.g. the activation energy for thickening of the Al 3 Mg 2 phase, which is the main IMC phase in the Al-Mg binary diffusion system, is reported in a range from 65 kJ/mol to 86 kJ/mol, which is a very significant difference given the exponential dependence of kinetic processes on Q [5,10,12,13].
The diffusion coefficient is the key parameter to describe the diffusivity of an element [14]. Since the IMC growth is mainly controlled by volume diffusion, the interdiffusion coefficient can be used to compare the growth rates between IMC phases. However, the interdiffusion coefficient of an IMC phase cannot be simply obtained, and must usually be extracted from experimental measurements using a number of modelling techniques [1,2,12,15]. The Boltzmann-Matano method [15] is one of the most successful models to 4 calculate the interdiffusion coefficient of IMC phases . This method requires measuring the composition profile along the IMC layers, based on the calculation of interdiffusion fluxes of individual components. Kajihara [1,2] also developed a model to calculate the interdiffusion coefficient of the IMC phase in a binary alloy system for the case where only one IMC forms. In Kajihara's model, the composition profile is not required, but the compositions at each side of each interface are necessary [1,2]. Another advantage of this model is that it can also be utilized in reverse (once the interdiffusion coefficient in the IMC is known) to predict the IMC growth kinetics. However, this model cannot be applied for many practically important binary alloy systems where two or more IMC phases form.
In order to predict the IMC growth kinetics, interdiffusion coefficients of all the phases are required. However, a wide range of interdiffusion coefficient values for each IMC phase is often reported, making it difficult to accurately predict the thickening kinetics [10,12,13]. A possible reason for this is that the interdiffusion values are derived from experiments that do not explicitly consider the different pathways that contribute to diffusion, e.g. grain boundary and lattice diffusion.
Grain boundaries provide fast diffusion pathways, so the interdiffusion coefficient depends on the availability of such pathways, i.e. on grain size and shape [14,16]. Few researchers consider the effect of grain boundary diffusion on the effective interdiffusion coefficient of IMC phases [1][2][3]12,13]. However, this can be important, since there are significant differences in grain size depending on the process used to form the IMC phase. Part of the scatter in the 5 effective interdiffusion coefficients reported for IMC phases is undoubtedly due to variations in grain size between specimens, which is often neglected. This paper focuses on the industrially important case of aluminium to magnesium alloy couples. Such couples are of importance in lightweight vehicle structures produced by dissimilar metal joining of magnesium and aluminium alloys. In Al-Mg couples, two different IMC phases Mg 17 Al 12 and Al 3 Mg 2 are typically observed [5,9,10,12,[17][18][19] and have different growth kinetics [9,10,12].
Once the IMC layers exceed a critical thickness they are known to produce a sharp reduction in the strength and ductility of the dissimilar metal joint.
Understanding how to control IMC growth in such a system containing multiple IMCs is thus of great practical as well as scientific importance.
The aim of the present work is to develop a model to predict growth kinetics of each IMC phase in a binary alloy system where multiple phases can form. The effect of grain size on diffusion is also explicitly considered so that results from conditions that lead to a difference in grain size can be compared. The basic approach is from Kajihara's model [1,2] extended to include multiple phases. A classical grain growth model [20] is applied to predict IMC grain size. The model is demonstrated by application to the case of Al-Mg alloy couples produced by ultrasonic welding and diffusion annealing.

Dual IMC phase growth model
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 A 2 B (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: 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.

Grain boundary diffusion
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 same for lattice diffusion and grain boundary diffusion, but the activation energy 11 for grain boundary diffusion is about half of that for lattice diffusion. Diffusion coefficients for lattice diffusion and grain bo undary diffusion can be expressed using the classic Arrhenius relationship as shown in equations 6a and 6b, respectively [14]: where is the lattice diffusion coefficient, is the grain boundary diffusion coefficient, and are pre-exponent factors for lattice diffusion and grain boundary diffusion, and 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 equation 7 [14]: where is a numerical factor depending on the grain shape, =1 for columnar grains, and =3 for equiaxed grains [14], is grain boundary width, which can be approximately assumed as 3 times of the atomic diameter [4], and 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 is the overall effective diffusion coefficient.

Grain growth
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 is the average grain size at the time t 1 , 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 equation 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 13 through the IMC layer. In this case, it is diffusion through the region of 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 equations 6-8. The IMC growth kinetics can then be predicted using the multi-IMC-phase diffusion model, which is expressed in equations 3-5.
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.

Grain growth of IMC layers
The Mg 17 Al 12 phase and the Al 3 Mg 2 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 Mg 17 Al 12 phase is thicker than the Al 3 Mg 2 phase. After a long annealing time (3 hours or even 72 hours) annealing, the Al 3 Mg 2 phase becomes much thicker than the Mg 17 Al 12 phase (Figure 3 (b) and (c)). Further details are given elsewhere [5]. Mg 17 Al 12 phase is the first formed IMC phase, but the Al 3 Mg 2 phase has a faster growth rate. Therefore, as the annealing time 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 si ze.
The growth behaviors of each IMC layer at various temperatures are plotted in

IMC thickness prediction
The model to predict the IMC thickness evolution for both Mg 17 Al 12 and Al 3 Mg 2 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

Interdiffusion coefficients calculation
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 ].  Table 1. 20 With the diffusion coefficient data from Table 1, the present model was also utilized to predict the thickness of IMC layers at different temperatures.
Modelling 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  The reason for the different controlling mechanisms for the two IMC phases in the Al-Mg couple is the difference in activation energy of diffusion in each phase.
According to the present results, and consistent with data from the literature [10,12,13], the Mg 17 Al 12 phase has much higher activation energy for lattice diffusion than the Al 3 Mg 2 phase. The higher activation energy for diffusion in the Mg 17 Al 12 phase makes lattice diffusion more difficult and therefore the reduction in activation energy associated with grain boundaries has a stronger e ffect on D.
This effect can be seen by considering the influence of changing activation energy for a fixed temperature and grain size.

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The effect of activation energy on the coefficients for grain boundary and lattice diffusion at 673 K (400°C) with a fixed grain width of 2 µm is plotted in Figure 8.
This shows the expected activation energy for lattice diffusion at which a transition from lattice to grain boundary diffusion will dominate.
In this paper, the model is applied for the Al-Mg couple where two typical IMC phases could be formed. However, the model can also be utilized to predict IMC growth behavior or calculate diffusion coefficients for other couples of dissimilar metals where one or more IMC phases could be formed.