*3.4. Calculation of the IM Layer Thickness*

The thickness of the IM layer was calculated using the software Matlab (R 2013, MathWorks, Natick, MA, USA). During the non-isothermal welding process, the temperature inside the weld seam and inside the heat affected zone varied strongly with time, *T* = *T*(*t*). Accordingly, the growth rate coefficient varied with time too, *k* = *k*(*t*). Thus, the simple power law function of Equation (1), which is valid for isothermal conditions, cannot describe the growth of the IM layer over the entire welding cycle. However, since assuming *T* and therefore *k* as constant is feasible within small time increments, *dt*, the corresponding growth increment, *dxIM*, can be calculated based on the first derivation of Equation (1):

$$d\mathbf{x}\_{IM} = n \, k^n t^{n-1} dt \tag{7}$$

For discrete steps, and if parabolic growth (*n* = 0.5) of the IM layer is assumed, Equation (7) can be transformed into Equation (8). For each of these steps, *i* = 1, 2, 3...*m*, one can calculate the growth increment, Δ*xIM*,*i*, based on the actual time of growth, *ti*, the time increment, Δ*ti*, and the growth rate coefficient, *ki*, as follows:

$$
\Delta x\_{IM,i} = \sqrt{\frac{k\_i}{4t\_i}} \,\Delta t\_i \tag{8}
$$

Figure 5 illustrates schematically the relationships between the thickness of the IM layer, *xIM*, and the time of growth, *t*, as well as between the growth increment, Δ*xIM*,*i*, and the time increment, Δ*ti*, as expressed by Equation (8).

**Figure 5.** Schematic illustration of the relationship between IM layer thickness and growth time.

Using Equation (9), *ki* was calculated with the constants *Q* = 190 kJ/mol and *k*<sup>0</sup> = 1.5 m2/s, which lay within the range of the values given in Table 1 for the combination of liquid technically pure (i.e., particularly silicon-free) aluminum and solid low-carbon steel:

$$k\_i = k\_0 \exp\left(-\frac{Q}{RT\_i}\right) \tag{9}$$

Keep in mind that *k*<sup>0</sup> is the general growth constant, which is not related to *t*<sup>0</sup> = 0. By contrast, the growth rate coefficient *ki* is certainly related to the time *ti*. The temperature *Ti* was obtained from the finite element simulation, utilizing the constant time increment Δ*ti*. Note that *t*<sup>0</sup> = *tw*0, because *tw*<sup>0</sup> = 0 when the welding process starts, but *t*<sup>0</sup> = 0 when the local temperature exceeds the limit of *T*<sup>0</sup> = 400 ◦C for the growth of the IM layer. At temperatures below this limit diffusion is quite slow and therefore reactions between aluminum and steel are more or less negligible [5]. According to Equation (10) the total thickness of the IM layer, *xIM*, can be finally calculated by adding all growth increments, Δ*xIM*,*i*:

$$
\Delta \mathbf{x}\_{IM} = \sum\_{i=1}^{m} \Delta \mathbf{x}\_{IM,i} \tag{10}
$$
