**1. Introduction**

Dissimilar joining of aluminum alloy sheets to steel sheets is an indispensable key process for producing multimaterial car bodies, offering both high crash safety and low vehicle weight. Fusion welding processes in particular have marked advantages regarding the efficient joining of hybrid parts of complex shapes. However, thermal joining of aluminum- (Al) to iron- (Fe) based materials is known to be associated with the formation of intermetallic (IM) Al*x*Fe*<sup>y</sup>* phases at the bonding interface [1–3]. The formation of these phases is mandatory for bonding of the dissimilar materials, however, excessive formation results in brittleness and therefore in poor mechanical properties of the joints. Thus, controlling the thermodynamically unavoidable interfacial reaction between iron and aluminum is a critical issue regarding the performance of dissimilar joints.

Laboratory experiments have identified that in most cases two main IM phases form at the interface between solid iron or steel and liquid aluminum or its alloys: Al5Fe2 as the major η-phase [4–8], together with Al3Fe (also referred as Al13Fe4) as the minor θ-phase [9–34]. Some researchers have found additional Al*x*Fe*<sup>y</sup>* phases, e.g., AlFe3 or AlFe [7,14–16]. Dybkov [12] reported 'paralinear' growth of the IM phases, meaning that with increasing time the thickness of the Al5Fe2 phase tends to grow towards a certain limit, while the thickness of the Al3Fe phase grows almost linearly after a non-linear initial period. Bouayad et al. [17] also observed that growth of the Al5Fe2 phase follows a parabolic relationship, but the growth of the Al3Fe phase follows a linear relationship. However, according to Bouché et al. [18], both the Al5Fe2 phase as well as the Al3Fe phase exhibit parabolic growth after an initial non-parabolic transient period. The Al5Fe2 crystals are assumed to have a much higher growth rate than the Al3Fe crystals, since the tongue-like Al5Fe2 sub-layer is generally observed to be markedly thicker than the serrated Al3Fe sub-layer.

At least one, or even both, of these two phases was also found to form in dissimilar cold metal transfer (CMT) welding/brazing of aluminum alloys to steel [35–48]. In comparison to conventional gas metal arc (GMA) welding processes, the CMT process is operated with significantly reduced heat input [47,48], which restricts the growth of the AlxFey phases and therefore limits the thickness of the IM layer [42,43]. The process temperature is high enough to melt the aluminum base material and the aluminum-based filler, but the steel base material remains solid. Thus, dissimilar joining is achieved by a combination of aluminum welding and steel brazing. The single-sided CMT process in particular offers high potential regarding flexible and efficient butt-welding of aluminum alloy sheets to zinc-coated steel sheets, which is of utmost interest in the automotive industry. However, note that the steel sheet can be used in the as-cut condition, i.e., the cutting edge of the steel sheet is uncoated [49].

The growth of the thickness of the IM layer, *xIM* (m), can be expressed as a function of time, *t* (s). Diffusion-controlled layer growth, which is assumed as dominant in low temperature and solid state welding processes (e.g., CMT), is commonly expressed using a power-law function:

$$\mathbf{x}\_{IM} = \left(kt\right)^{n} \tag{1}$$

For parabolic growth, *n* = 0.5. The temperature-dependent growth rate coefficient *k* (m2/s) is expected to follow an Arrhenius relationship, where *k*<sup>0</sup> (m2/s) is the growth constant, *Q* (J/mol) is the activation energy, *T* (K) is the absolute temperature, and *R* = 8.314 J/molK is the gas constant:

$$k = k\_0 \exp\left(-\frac{Q}{RT}\right) \tag{2}$$

Table 1 contains values of *Q* and *k*<sup>0</sup> as reported in the literature for calculation of the timedependent parabolic growth of the major η-phase or of the IM layer, respectively. Both of these constants are usually determined by fitting experimental data captured at different temperatures. Obviously, considerable variations exist between the reported values, which can be attributed to differences in the materials investigated, the experimental conditions, and in formulating the growth equation. Note that most of the experiments have been conducted at laboratory conditions within comparatively narrow temperature ranges. Therefore, the influence of transient or non-uniform temperature fields—as occur, for instance, in most industrial welding processes—on the formation of the IM layer is not considered. Furthermore, if iron or aluminum of technical pureness are used for experiments, the growth constants do not consider the influence of alloying elements, which are known to influence the growth of the IM layer and which are normally present in industrial processes. In particular, increasing the silicon content of aluminum alloys retards IM layer growth [4,10,27–33,50,51], but increasing the zinc content promotes IM layer growth [4,34,51]. Increasing the carbon content of steels also retards the growth [52,53]. Note that the constants given in References [54–59] were determined in experiments where both iron and aluminum were solid (s).

During recent years, different methods have been applied by researchers to model the IM layer, since this layer represents a critical feature influencing the mechanical properties of aluminum-steel joints. Rong et al. [60] conducted thermophysical simulations to clarify reaction mechanisms and growth kinetics at the interface between solid steel and liquid aluminum, and to predict the average thickness of the IM layer. Das et al. [61] proposed a combined theoretical–experimental method, including a numerical model and a set of measured results, to estimate the thickness of the IM layer as a function of key process parameters in a lap joint configuration. Zhang et al. [62] used the Monte Carlo

(MC) method to model the growth of IM compounds, and validated their results with bead-on-plate welding of aluminum alloy onto galvanized mild steel.


**Table 1.** Activation energies and growth constants, as reported in the literature.

\* maximum thickness of the IM layer, \*\* mean thickness of the IM layer, (s) solid, (l) liquid.

This work presents a numerical method, which allows fast estimation and three-dimensional (3D) visualization of the IM layer, because the highly irregular microscopic interface between the IM layer and the weld seam is approximated as a smooth surface. 3D visualization of the layer enables the identification of critical weld seam regions where comparatively thin (possibility of insufficient bonding) or thick (brittleness of the joint) IM layers occur. The presented method, which includes (i) calculation of the temperature at the bonding interface between the steel sheet and the aluminum weld by means of finite element (FE) simulation, (ii) validation of the obtained numerical results with temperature curves measured in CMT welding, (iii) prediction of the thickness of the IM layer based on the calculated temperature field, and finally (iv) validation of the predicted thickness with micrographs of weld cross-sections, has already been presented by the authors of this article [63]. This method was also successfully applied by Borrisutthekul et al. [64] for estimating the effect of different heat flow control measures on the thickness of the IM layer in laser welding, and by Mezrag et al. [65] for the indirect determination of the process efficiency in CMT welding.
