Comparison of Dynamical and Empirical Simulation Methods of Secondary Dendrite Arm Coarsening

The physical and mechanical properties of an entirely (wrought alloys) or partly (cast alloys) dendritically solidified alloy strongly depend on the secondary dendrite arm spacing (SDAS). The casting practice and the simulation of solidification need a usable but simple method to calculate the SDAS during and at the end of solidification as a function of the cooling rate. Based on many solidification experiments, a simple equation to calculate the SDAS (empirical method) is known to use the local solidification time, which can be obtained from the measured cooling curves (equiaxed solidification), or can be calculated from the temperature gradient and front velocity (directional solidification). This equation is not usable for calculating the SDAS during solidification. Kirkwood developed a semi-empirical method based on the liquid phase’s diffusion, which contains only one geometric factor that seems constant for different alloys. This equation contains some physical parameters that depend on the temperature, so the equation cannot be integral in closed form. In the present work, first, we show the effect of the curvature of the solid/liquid interface on the equilibrium concentrations and then the different processes of SDA coarsening. In our earlier paper, we demonstrated that using the empirical method, the final SDAS can be calculated with acceptable correctness in the case of four unidirectional solidification experiments of Al-7wt%Si alloy. The present work shows that numerically integrated Kirkwood’s equations used the known cooling curve; the SDAS can be calculated at the end and during solidification in good agreement with these experimental results. Compared to the two calculation methods, we stated that the correctness of the methods is similar. Still, the results of the solidification simulation (the microsegregation) will be more correct using the dynamical method. It is also shown that with the dynamical method, the SDAS can be calculated from any type of cooling curve, and using the dynamical method, it is proved that some different SDASs could belong to the same local solidification time.