*4.3. Microstructural Evaluation: Primary Dendrite Arm Spacing (PDAS)*

Understanding the relationship between the laser parameters and the microstructure can require an understanding of the microstructural evaluation of the joints via the numerical and analytical studies. By doing so, the mechanical characteristics of laser welded components can be estimated using this invaluable knowledge. Primary dendrite arm spacing (PDAS) can be an appropriate microstructural instance with which the mechanical features of welded materials can be elaborated upon [9]. As a result, the correlation between the solidification process and primary dendrite arm spacing has become the attention of various researches [5,33,34]. The Kurz and Fisher (KF) model is one of the most traditional models with which the calculation of PDAS becomes available [34]. It should be mentioned that this model (Equation (13)) has been utilized in Al-Mg systems in which PDAS has reported to be accurate with regards to experimental results [5].

$$\delta = 4.3 \left( \frac{\Delta T\_0.D.\Gamma}{k\_0} \right)^{1/4} . G^{-1/2}. \text{R}^{-1/4} \tag{13}$$

in this equation, *G*, which is known as temperature gradient, is the variation of temperature around a particular location at a given time; *R*, which is known as solidification rate or growth rate, is the travel speed of the solid-liquid interface at a given temperature; δ is the primary dendrite arm spacing (*m*). The values of Δ*T*0, *D*, *k*0, and Γ are intrinsic characteristics of the material which are provided in Table 5 [5,20].


**Table 5.** Material characteristics of the AA5456 used in the Kurz and Fisher (KF) model.

As shown in Figure 10, the variation of primary dendrite arm spacing (PDAS) obtained from the numerical modeling and the analytical method is illustrated versus the heat input of the laser beam, and a comparison between these results and experimental one has been made. Obviously, the G (temperature gradient) and R (solidification rate) values obtained from the numerical modeling and the analytical Rosenthal equation are different so that Equation (13) results in different values for PDAS attained from the numerical modeling and the analytical method. Wang et al. [35] have observed that both calculation approaches illustrate the same trend in which the PDAS is increased by enhancing heat input. It should be taken into account that the PDAS, which is predicted by the analytical method, is attained to be slightly larger than that from the numerical modeling. To observe the validation of the results obtained from the numerical modeling, an experiment was conducted using the same laser parameters as for the numerical method, and these results are compared with each other. Figure 10 shows the transverse-section of the laser welded material in which the laser parameters are selected from A2 in Table 2. As shown in Figure 11, the PDAS value is 1.029 μm at the heat input of 30 <sup>J</sup>·mm<sup>−</sup>1. It is apparent that the results obtained from the numerical modeling and the analytical method are consistent with the result from an experiment which was done at the heat input of 30 <sup>J</sup>·mm<sup>−</sup>1. Apparently, both of the numerical and analytical methods are capable to produce the primary dendrite arm spacing (PDAS), but it needs to have more experimental results to validate the numerical and analytical methods efficiently and compare their results to see which one is more accurate that the other one. Thus, there should be more experiments, specifically at various heat inputs, to verify the numerical modeling and the analytical method. Furthermore, with more available data, the determination of better methods will be attainable as well.

**Figure 10.** Variation of primary dendrite arm spacing (PDAS) obtained from the experiment, the numerical modeling, and the analytical method versus heat input.

**Figure 11.** Measurement of dendrite arms taken by optical and scanning electron microscopies at the boundary of fusion zone using a laser with an average power of 60 W, welding velocity of 2 mm.s<sup>−</sup>1, and absorption coefficient of 0.36.
