## 2 Materials and Methods
The first method proposed is based on the determination of the zero-intersection point of the second and third temperature derivatives with respect to the time curve after the maximum temperature of liquidus (Method 3). It is based on previous work by [PERSON] developed for iron alloys, in which he justified that taking into account that at this point the dendrites have finished growing together and that the eutectic begins to grow and release energy, the rigidity point is also the starting point for the arrest of eutectic solidification [18]. We can observe the first proposed method for the determination of the RPT in Figure 3 with the RP close to the minimum of the first derivative. In Figure 2 we can observe how the determination of the exact minimum of the first derivative is not intuitive, and in the case of the proposed method we can obtain the RPT value directly. Figure 1: Rigidity point temperature (RPT) determination with the second minimum of the d2T/dt2 curve. Figure 2: Method 2: RPT determination with the first minimum of the dT/dt curve: Hypoeutic AlSi\({}_{10}\)Mg alloy. The second proposed method (Method 4 in the following Figure 4) for obtaining RPT using a single thermocouple is based on the determination of the elbow point at which the dT/dt curve suddenly deviates from the horizontal tangent in the first derivative of the temperature curve with respect to time (dT/dt) but represented vs. temperature. This method assumes the fact that the use of derivatives with respect to temperature is not influenced by the size of the thermal cup and also that an acceleration of dT/dt versus temperature corresponds to an increase in speed of the heat extraction from the sample due to the higher conduction of the heat in a solidified net. The process is based on the same procedure that [PERSON] proposed to determine the dendrite coherence temperature point [19]. He justified for hypoeutectic ductile iron alloys that the moment where the second and third derivative curve cross the zero line after the liquidus temperature corresponds to the DCP, with the first minimum of the first derivative after the eutectic minimum temperature for eutectic iron alloys and the first minimum of the second derivative after the minimum liquidus temperature arrest for hypereutectic iron alloys. We can observe in Figure 4 the determination of RPT in the elbow of the dT/dt curve versus time and the corresponding RPT in the sudden deviation from the horizontal tangent. However, it is often not easy to find the exact point at which the horizontal tangent deviates, since there is not always a clear loop in the obtained curve and, consequently, it is not possible to determine the exact position of the elbow point on the dT/dt versus temperature curve. Therefore, another method (Method 5) is proposed based on the previous one in which the determination of the RPT coincides with the point of intersection with the zero of the second and third derivative after the maximum temperature of liquidus, to obtain a more accurate rigidity temperature point. We can observe in Figure 5 the determination of the RPT for the third proposed method. The last step of the study was to analyze the microstructure to correlate the RPT values with the microstructure of the alloy. Several ingots of approximately 50 mm (length) \(\times\) 50 mm (width) \(\times\) 50 mm (thickness) were obtained for each alloy. The samples for optical microscopy (OM) were cut from these ingots and prepared according to standard metallographic procedures, by hot mounting in conductive resin, then grinding, and polishing. The microstructure and the determination of the possible metallographic phases of each sample were investigated by an optic microscope model DMI5000 M (LEICA Microsystems, Wetzlar, Germany). A scheme of the complete method applied is shown in the Figure 6.  
Figure 3: Method 3: RPT determination in the crossing point of the second and third derivative dT/dt vs. time for a hypoeutectic AISi\({}_{10}\)Mg alloy.