# Solid Fraction Determination at the Rigidity Point by Advanced Thermal Analysis
## 1 Introduction
In the in situ neutron diffraction determination, neutron diffraction is measured continuously, but it is an expensive method with less accurate results [19]. The two-thermocouple method records the melt temperature at the core and at the inner wall of the test vessel, determining the RP from the differences between the core and wall temperatures versus the time curve. The single thermocouple method uses only the signal from a thermocouple in the center of the test vessel and is a less expensive method, due to the reduction in commercial test cups price and by avoiding doubling the number of needed calculations. It is sometimes difficult to determine the exact point because there is not always a clear inflection point in the first derivative curve, as the signal corresponding to this event is very weak. Recent developments employ higher order derivatives to determine the characteristic points of solidification, calculating the total amounts of the different precipitated phases and other low energy reactions [20; 21], such as SF. The authors of this article have developed new methods based on the application of higher-order derivatives applied to the DCP and RPT [15; 22]. The FRP can be determined using different experimental and/or arithmetic methods [16; 23]. The Newtonian and Fourier methods [24] are the most commonly used. In the Newtonian method, the solid fraction at each point or temperature is calculated by determining the integration or cumulative area between the cooling rate (first derivative (dT/dt) of the cooling curve) and the baseline dTBL/dt (BL). BL corresponds to the cooling rate curve without phase transformations. In the Newtonian method, it is necessary to exactly define the solidus and liquidus temperatures to define the integration interval [25]. [PERSON] et al. obtained values of around 72% FRP at 564 \({}^{\circ}\)C for AlSi\({}_{8}\)Cu\({}_{3}\) alloys [18] and values ranging from 58 to 68% for FRP different AlSiCu alloys [26]. Another study obtained a value close to 89% for FRP at a temperature of 561 \({}^{\circ}\)C for AlSi\({}_{7}\)Cu\({}_{0.5}\)Mg [27]. Finally, a recent work determines the RPT values and SF of various AlSi and AlSiCu alloys [28]. A modification of the alloy composition can also lead to a variation of the solidification parameters, with solidification parameters being affected by the major and minor alloying elements [7]. The effect of major and minor alloying elements on FRP, especially in the AlSi\({}_{10}\)Mg alloy, is not well known. The aim of this work is to go one step further by using TA to predict the SF at the stiffness point in a more accurate, economical, and direct way. Considering the lack of data in the literature, the determination of the solidification characteristics at the RP in function of a specific alloy composition will allow a more accurate control of the process, quality, and an improved simulation calculation, promoting a better prediction of casting defects. Statistical analysis of the results obtained for a wide range of alloy chemical compositions shows a good correlation of the predicted and obtained values for FRP.