A Novel Method to Predict Phase Fraction Based on the Solidification Time on the Cooling Curve

Abstract

The phase fraction plays a critical role in determining the solidification characteristics of metallic alloys. In this study, we propose a novel method (f_s = (t − t_l)/(t_s − t_l)) for estimating the phase fraction based on the solidification time in cooling curves. This method was validated through an experimental analysis of Al-18 wt%Cu and Fe₄₂Ni₄₂B₁₆ alloys, where the phase fractions derived from cooling curves were compared with quantitative microstructure evaluations using computer-aided image analysis and the box-counting method. Then, a comparison between the analysis using the present novel method and Newtonian thermal analysis demonstrates good agreement between the results. The present method is easier to operate, since it does not need derivative and integral operations as in Newtonian thermal analysis. In addition, based on the characteristics of the cooling curve, we also found two other relationships—V/R_c = D/ΔT_c and RΔt = constant, where V is the solidification rate, R_c is the recalescence rate, D is the diameter of the focal area of the pyrometer, ΔT_c is the recalescence height, R is the cooling rate, and Δt is the solidification plateau time. These findings establish an operational framework for quantifying phase fractions and solidification rates in rapid solidification.

1. Introduction

Rapid analysis techniques for phase fractions are useful for adjusting the process for the control of the material microstructure and properties [1,2,3]. Neutron transmission spectroscopic imaging has enabled the quantitative analysis of solid–liquid phase evolution during solidification through Bragg-edge profile analysis [4,5]. This wavelength-resolved technique has been further extended to characterize martensite phase fractions in bulk ferritic steels [6]. While X-ray computed tomography provides alternative phase quantification capabilities [7], and solidification phase fraction correlations have demonstrated predictive value for mechanical properties in additively manufactured Ti-6Al-4V alloys [8], these advanced characterization methods remain constrained by high operational costs and technical complexity. However, the temperature curve of the alloy solidification process is helpful for obtaining the phase fraction quickly at a low cost, which has been used to investigate the effect of alloying elements on the properties of hypereutectic Pb–Sn alloys [9]. There are several methods allowing for the determination of the phase fraction from temperature curves, such as differential scanning calorimetry (DSC) [10], differential thermal analysis (DTA) microstructure [11,12], cooling curve thermal analysis [13,14], and the baseline method [15]. Techniques like DTA and DSC are used for laboratory testing and cannot be used in cases that are far from ideal or in non-equilibrium systems [16]. Cooling curve thermal analysis is regarded as one of the most effective methods for the online monitoring of molten metal solidification processes, with the following two methods: Newtonian thermal analysis [17] and Fourier thermal analysis [18]. Both of these methods involve deriving the cooling curve and then integrating it [17,18]. Newtonian thermal analysis uses only one thermocouple that is centered in a solidifying sample, where the sample is assumed to be spatially isothermal. The most basic zero curve is an exponential that is fitted to the derivative of the cooling curve in the solid region [19]. However, due to differing heat capacities between the liquid and the solid, the zero curve rarely fits the derivative of the cooling curve in the liquid region. Fourier analysis also uses a zero curve and the relative area between the zero curve and the derivative of the cooling curve to calculate solid fraction [18]. The difference between Fourier and Newtonian analysis is in how the data are gathered and how the zero curve is determined. Both methods require derivative and integral operations, which are not easy to operate. Furthermore, after the derivation of some temperature curves with particularly dense collected data, the obtained curve fluctuates greatly, and the baseline is sometimes difficult to find. Xu et al. report a baseline method that uses the extension of the liquid phase and solid phase region to calculate the baseline [15]. It can avoid the problem of derivation cooling curves, but the extension line equation is not always easy to determine.

For the cooling curve of undercooled alloy solidification, there is a solidification plateau time, which was studied by W.C. Roberts-Austen in 1897 [20]. After that, there have been many studies that focus on the recalescence process of undercooled solidification [21,22,23,24,25,26]. Patel et al. discuss a method for handling the rapid solidification recalescence in laser spot welding by solving the coupled transient conservation equations of mass, momentum, and energy [21]. He et al. studied the recalescence curves of invar alloy and found that with increasing solidification undercooling, the microstructure changes from large dendrites to small columnar grains and then to fine equiaxed grains [22]. Galenko et al. studied the anomalous kinetics and pattern formation in recalescence, finding that changes in the shape of the recalescence front occur and multiple nucleation events form the growth front [23]. An et al. studied the effect of Co on the solidification characteristics and microstructural transformation of non-equilibrium solidified Cu-Ni alloys and found that the addition of a small amount of Co weakens the recalescence behavior of the Cu₅₅Ni₄₅ alloy and significantly reduces the thermal strain in the rapid solidification phase [24]. Yang et al. studied the maximal recalescence temperature T_R upon the rapid solidification of bulk undercooled Cu₇₀Ni₃₀ alloys and found that T_R is affected by the difference in Gibbs free energy between the residual liquid and solid after recalescence [25]. Mullis et al. studied why re-melting is not a plausible explanation in Ag-Cu alloys and found that at low undercooling, the volume fraction of anomalous eutectic near the nucleation site is around an order of magnitude greater than the calculated recalescence solid fraction [26]. Despite so many reports about recalescence, there is still a lack of systematic reports on the solidification fraction, solidification rate, and solidification plateau time.

As concerns the transition heat, the released rate in the solidification temperature for liquid and solid does not vary significantly, and all the transition heat dissipates from the sample surface; the heat release time for liquid and solid phases is considered equivalent for the same sample size. Based on that, we propose a novel method to quickly estimate the solid fraction during solidification upon the heat release time in this study. This approach eliminates computational dependencies while maintaining operational simplicity. Then, the validation is conducted through synchronized cooling curve analysis and the microstructural characterization of Al-18 wt%Cu and Fe₄₂Ni₄₂B₁₆ alloys. After that the solidification fraction, the solidification rate and solidification plateau time will be discussed further.

2. Method Description

2.1. Phase Fraction Prediction Method

The method is demonstrated in Figure 1, which shown an example for a typical cooling curve with two transitions in solidification. If one sample was cooled from a high temperature to room temperature in the liquid state (never solidified), the cooling process should be from A to B. If a solid sample that was not melted at high temperatures is cooled to room temperature, the temperature curve should change from point C to point D, as shown in Figure 1. The actual process involves the sample’s temperature changing from high to low, the liquid phase transforming to a solid phase, and the temperature curve gradually changing along AB to CD. Thus, AB and CD can be seen as the baseline for the liquid and solid phases, respectively. To estimate the solid phase fraction during solidification, the method should be easy to operate and practical, so the baselines AB and CD can be fitted using a parabolic equation. The detailed process for solid fraction calculation can be described as follows.

Fitting the pure solid phase part of the cooling curve CD using a parabolic equation, the following is obtained:

$$\mathrm{CD}:T=a{t}^2+b_{s}t+c_s$$

(1)

where t is time, and α, b_s, and c_s are the fitting coefficients.

Because the specific heat of the liquid and solid phases of the same sample is slightly different, different equations should be used for the cooling rates of the solid and liquid phases at the same temperature. The overall shapes of AB and CD are similar; in order to minimize the number of fitting parameters as much as possible, we can use the parabolic equation with the same quadratic coefficient α from Equation (1) to fit the liquid phase region of the cooling curve AB, as follows:

$$\mathrm{AB}:T=a{t}^2+b_{l}t+c_l$$

(2)

where b_l and c_l are the fitting coefficients.

Then, if a suitable equation can be used to describe the experiment data, it can be expressed as follows:

T = T(t)

(3)

Note that all the equations above correspond to a single value of temperature and time.

The solid phase fraction of each point on the cooling curve can be estimated as follows:

$$f_s={\displaystyle \frac{t-t_l}{t_s-t_l}}$$

(4)

The process is as shown in Figure 1. For any time, t, the corresponding temperature T on the experimental data curve can be found using Equation (3) (①→② in Figure 1). Then, with the same T, we can calculate t_s and t_l from Equation (1) and Equation (2), respectively, (②→③ in Figure 1). In this way, the solid phase fraction for each temperature and time can be determined through Equation (4).

2.2. Microstructure Measurement Method

To identify the predicted result, there are two methods to measure the phase fraction from microstructure images. One method is computer-aided image analysis, which has been described by Gandin et al. [27]. It is based on a binary representation derived from microstructure images. For this method, the sample must be corroded well with a suitable etchant; then, the phase fraction can be accurately obtained by having a high contrast between the two phases in the photograph. When the microstructure images are of insufficient quality, this method cannot be used.

The second method is the box-counting method improved by Xu et al. [28], which can solve the measurement difficulties caused by low image quality and uneven color distribution. It can be described as follows:

(1)

The image is covered by the boxes of L × L;

(2)

The number of boxes containing primary phase (including that full of primary phase and that partially filled with primary phase) is counted as α;

(3)

The box number is counted, only containing the primary phase as α_f;

(4)

The fraction of the primary phase is calculated as follows:

$$f={\displaystyle \frac{a_f+a}{2(L\times L)}}$$

(5)

Equation (5) will give a more accurate result if the box size is smaller than the primary phase size, and the error-analyzed details can be found in [28].

3. Experimental

In order to verify the present method, the Al-Cu alloy sample from the electromagnetic levitation technique, as well as the Fe-Ni-B alloy sample from the melting glass-flux technique, was investigated here. The schematic diagram of the device’s structure has been described in our previous report [29]. For Al-Cu alloys, the samples were produced from pure elements (99.99% Al, 99.99% Cu) using levitation melting in an argon atmosphere. In total, 500 mg of alloy sample was cut for electromagnetic levitation experiments. For the electromagnetic levitation experiments, the vacuum chamber is first evacuated to 6 × 10⁻⁶ mbar and then refilled with high-purity He (6 N) to a pressure of 350 mbar in order to limit the evaporation of the sample. When the sample is completely melted, the temperature is continuously increased to 200  K above the liquidus temperature. Then, the sample is cooled using cooling gas (He). The temperature curve is measured at the sample surface using a one-color infrared-pyrometer (model IGA 140 MB 30 L, Fa. LumaSense Technologies, Santa Clara, CA, USA) with a sampling rate of 50 Hz. The solidification process was monitored using a high-speed camera (PHANTOM v7.3, Vision Research, Inc., Wayne, NJ, USA). The camera’s highest frame rate is 125,000 fps and its two-dimensional resolution is 100 × 100 µm² (128 × 32 pixels). After solidification, the samples were cut into halves and were prepared metallographically for the observation using SEM (Zeiss Evo 40, Carl Zeiss AG, Jena, Germany).

For the Fe-Ni-B alloy experiment, the high-frequency-induced melt-flux technique was used to obtain the undercooled solidification temperature curves. A sample weighing 5 g was put into a quartz crucible covered by small amounts of B₂O₃ glass; then, the crucible was placed in the induction coils of a high-frequency induction. Then, the sample was subjected to cyclic heating and cooling by air until the required undercooling was achieved. A one-color pyrometer (PYROSPOT DG 54 N, DIAS Infrared GmbH, Dresden, Germany) with a 10 ms delay time was then used to record the cooling process. A calibration was required to calculate the actual temperature, T, using the following equation: 1/T−1/T_pyr = 1/T_L−1/T_Lpyr, where T_pyr is the temperature recorded by the pyrometer, T_Lpyr is the liquid temperature measured by the pyrometer, and T_L is the liquid temperature read on the phase diagram or measured using differential scanning calorimetry.

4. Results

Figure 2 shows the solidification process record of the Al-18 wt%Cu alloy sample undergoing the primary phase transition at an undercooling rate of ΔT = 44K, as captured by the high-speed camera. It can be found that for the primary transition (L→α-Al), solidification seems to start from several places. The primary transition results in an α-Al dendrite primary phase, the branches of which extend from the inside of the sample to the surface, as all the liquid is in the undercooled state. The several yellow regions in Figure 2 are the branches of one dendrite. The solid phase fraction cannot be predicted from high-speed video images directly, but can be predicted from temperature curves.

Figure 3a is the temperature curve measured from the solidification of the Al-18 wt%Cu alloy at an undercooling rate of ΔT = 44 K. Then, with the method described in Section 2, the pure solid phase part of the cooling curve can be fitted as $T=a{t}^2+b_{s}t+c_s$ = 1.48t² − 531.48t + 47,545. The pure liquid phase part can be fitted as $T=a{t}^2+b_{l}t+c_l$ = 1.48t² − 493.80t + 41,625. Then, from Equation (4), we can obtain the solid phase fraction curves of f-t (Figure 3a) and T-t (Figure 3b). From the inflection point of the curve, the fraction of primary phase dendrites (α-Al) is determined as 67.3%.

Figure 3c shows the SEM microstructure of the Al-18 wt%Cu alloy sample. By adopting the computer image segmentation method, we can obtain the binary image as shown in Figure 3d, and the dendrite phase fraction is determined by the area of the red region as 65.2%. At present, an accuracy comparison (67.3% or 65.2%) remains challenging because the phase fraction measured in two-dimensional cross-sections inherently differs from the three-dimensional volumetric distribution. The calculation error from the cooling curve comes from the difference in the specific heat of the solid and liquid phases. Nonetheless, the results derived from Equation (4) show close consistency with the microstructure measurements.

To further validate the phase fraction method derived from cooling curves, Figure 4a presents the cooling curves of the Fe₄₂Ni₄₂B₁₆ alloy. By using Equations (1)–(4), we can calculate the solid fraction curve f-t. From the eutectic transition point (the inflection point of the solid fraction curve), we can determine the primary phase α-Fe(Ni) as being about 33.5%. The corresponding microstructure is shown in Figure 4b. For this microstructure, because the color distribution is not uniform and the score cannot be obtained by computer image analysis, we adopt the box-counting method studied by Xu et al. [28]. The microstructure image in Figure 4b is covered by boxes of 50×50. The first step is counting the number of the full box as α_f = 473 (Figure 4c). The second step is counting the number of full and half boxes as α = 1170 (Figure 4d). The total box number is 2500. Then, by using Equation (5), the value of the primary phase α-Fe(Ni) fraction is obtained as f = (α_f + α)/2L² = (473 + 1170)/2 × 50 × 50 = 0.328, which is very near the predicted result (0.335) from the cooling curve. This method is not quick but is more reliable, as has been discussed in [28]. Therefore, the present method can be used to estimate the solid fraction during solidification.

In order to show the advantage of the present method, we need to compare the results of the present analysis of cooling curves with those of other methods. There are currently two commonly used methods for analyzing temperature curves—the Fourier thermal analysis method and the Newtonian thermal analysis method [30]. For the Fourier thermal analysis method, the heat conduction behavior inside the sample is considered, and at least two measuring points in the one-dimensional temperature field are required to solve the temperature gradient. Although this method has a good accuracy, the high requirement for the number and position of thermocouples on the sample makes it difficult to obtain the necessary temperature data in the practical experiment [1,30]. So, the present experimental data in Figure 3 and Figure 4 cannot be directly analyzed using the Fourier thermal analysis method. For the Newtonian thermal analysis method, only one single cooling curve measured in the experiment is required. The idea of the Newtonian thermal analysis method is to obtain the first derivative of the cooling curve with respect to time and the baseline, which is the first derivative of the reference curve without phase transformation [1,30]. The main calculation equation for Newtonian thermal analysis method is as follows [1,30]:

$$L={\displaystyle \underset{t_0}{\overset{t_e}{\int }}{C}_v\left(\frac{dT}{dt}-baseline\right)dt}$$

(6)

where C_v is the volumetric heat capacity of the melts system, dT/dt is the temperature rate, t₀ is the onset point, and t_e is the end point. The solid fraction can be given as follows:

$$f_s={\displaystyle \frac{1}{L}}{\displaystyle \underset{t_0}{\overset{t}{\int }}{C}_v\left(\frac{dT}{dt}-baseline\right)dt}$$

(7)

where the baseline is determined by the temperature-time at the beginning and at the end of the phase transformation on the cooling curve.

This is determined by the thermophysics values and the cooling rate at the onset (dT(t₀)/dt) and end point (dT(t_e)/dt) of the solidification process [1].

Based on Equations (6) and (7), the Al-18 wt%Cu alloy cooling curve in Figure 3a was analyzed again. Figure 5a shows the derivative result (dT/dt) of the experimental temperature curve without smoothing treatment. It can be found that the derivative curve has significant fluctuations, which will pose certain difficulties for further integration operations. Figure 5b displays the phase fraction versus time derived from integrating the derivative curve in Figure 5a according to Equations (6) and (7). From the result, the primary phase fraction is determined as 67.5%, which is consistent with the present novel method result (67.3%). For the convenience of the integration calculation, we can first smooth the temperature curve and then take its derivative, as shown in Figure 5c. It can be seen that the derivative curve has become smoother. Then, from the smoothing derivative curve, the phase fraction versus time can be derived, as shown in Figure 5d. Then, the primary phase fraction is determined as 68.7%. Although this result is also near that of the novel method’s analysis, the discrepancy between the two values (67.5% vs. 68.7%) suggests that the smoothing process affects the solid faction calculated results. Overall, the result of the cooling curve analysis method (using Equation (4)) proposed in this study has little difference from the calculation of the Newtonian thermal method, but it does not require the process of differentiating and integrating the temperature curve, which is more convenient to use.

5. Discussions

5.1. Effect of the Fitting Coefficient

For the present method (using Equations (2) and (3)), there are five fitting coefficients, i.e., α, b_s, c_s, b_l, and c_l. The values of these parameters have a certain effect on the calculated result. After the fitting is completed, the solid phase and liquid phase regions of the temperature curve will be extended through Equations (2) and (3) and these parameters. Generally speaking, the longer the liquid region of the cooling curve obtained from the experiment, the more accurate the fitted values of coefficients b_l and c_l. The longer the solid phase of the cooling curve, the more accurate the fitted value of coefficients α, b_s, and c_s. Coefficient α determines the curvature (cooling rate change) of the fitted curve. c_s and c_l determine the lowest temperature of the fitted curve. The present method uses the same fitting coefficient α for both the liquid phase and solid phase regions, assuming that the temperature curves before and after solidification have relatively small changes in cooling rate. That is to say that extending the part in the solid phase region towards the high-temperature region will make it almost parallel to the part in the liquid phase region. The closer to equilibrium, the higher the calculation accuracy of this method. If the cooling rate of the cooling curve changes irregularly before and after solidification, the accuracy of this model will decrease.

5.2. Solidification Fraction

Our previous study shows that the cooling curve can be calculated from the high-speed photographs of the solidification process [31]. Combining the present study results, the solid fraction can be directly estimated from the high-speed photographs. Firstly, the cooling curve is calculated from the high-speed video. Then, the phase fraction from the cooling curve is analyzed based on Equation (4). In this way, high-speed photography enables not only the quantitative tracking of solid–liquid interface velocity and morphology dynamics but also the calculation of solid phase fractions during solidification. When all the parameters are acquired from a single device, the time correspondence should be better than that from multiple device measurements. Therefore, a more complete solidification theoretical model can be developed based on these techniques.

5.3. Solidification Rate

In addition to the solidification fraction related to the cooling curve, the solidification rate and the cooling rate can also be estimated from the cooling curves. Figure 6 shows the solidification interface move a distance in the recalescence process. The second layer of Figure 6 is the schematic of the sample in the tube and the area for pyrometer focus. The third layer of Figure 6 is the temperature curve in the sample recalescence process, which is measured using the pyrometer. Given Δτ as the time between the recalescence start point and finish point (the unit is s), from Figure 6, the following can be found: VΔτ = D and R_cΔτ = ΔT_R. Thus, Δτ can be expressed as follows:

Δτ = D/V = ΔT_R/R_c

(8)

where D is the diameter of the temperature measurement range, which is determined by the parameter of the thermometer itself (the unit is m). V is the velocity of the solidification interface (the unit is m/s). R_c is the recalescence rate (the unit is K/s), R_c = ΔT_R/Δτ, where ΔT_R is the recalescence height in the temperature curve (Figure 6; the unit is K). Then, from Equation (8), the relation between the solidification rate and the recalescence rate in the cooling curve can expressed as follows:

V/R_c = D/ΔT_R

(9)

Thus, if the cooling curve is accurate (i.e., there are enough temperature data points per second in the recalescence process), Equation (9) can be used to estimate the solidificaition rate (V; m/s) from recalesence rate (R_c; K/s) directly from the temperature curve. ΔT_R can be obtained from the temperature curve, and D is provided in the thermometer manual. If the value of D is unknown, it can be determined using the known values of V, R_c, and ΔT_R.

5.4. Solidification Plateau Time

There is a further relation between the cooling rate and the solidification plateau time (Δt, s) in the cooling curve. Figure 7a shows the schematic diagram of the thermal history of a metallic sample in the solidification process. For convenience of study, it is assumed that the specific heat (C_p) is a constant so that the cooling curve can be seen as straight before or after solidification. As BC//ED, if we move BC to ED, where point E can be seen as the nucleation point; it is easy to obtain the relationship relating to BD (the unit is K) from geometric knowledge, as follows:

BD = RΔt

(10)

where R is the cooling rate (=dT/dt, K/s), and Δt is the solidification plateau time (the unit is s) from point E to B. Equation (10) indicates that for an alloy, the cooling rate and the solidification plateau time is as follows:

RΔt = constant (K)

(11)

If we assumed that there was no latent heat release in solidification, the thermal history may go along with the AED from the liquid phase to the solid phase. Because the latent heat of the alloy in the solidification process is released, the temperature of the sample will increase or decrease at the same rate as the cooling rate, and the thermal history of the sample follows the path of AEBC (Figure 7a). At a time, t, that does not consider the latent heat, the system temperature is observed at point D; due to the latent heat release, the system temperature is seen at point B. Therefore, at the same time, the temperature difference between point D and point B is attributed to the latent heat work, as follows:

BD = ΔH/C_p = ΔT_hpy

(12)

Here, ΔT_Hpy is the temperature rise due to the latent heat release (K). From Equations (10) and (12), we can obtain the following:

RΔt = BD = ΔT_hpy

(13)

From Equation (13), the solidification plateau time is decided by the cooling rate. The larger the cooling rate, the longer the solidification plateau time (Δt), as is shown in Figure 7b. From Equation (13), we can also predict the solidification plateau time according to the known cooling rate.

It is not difficulty to find the experiment reports about cooling curve follow this relationship [32,33]. Farahany et al. obtained the temperature curves during solidification of Al–Si–Cu–Fe alloy under varying cooling rates. It can be found that when the cooling rates for the curves are 2 °C/s, 1 °C/s, 0.6 °C/s, the corresponding plateau times are about 144 s, 288 s, 480 s, respectively [32], which confirms the validity of Equation (11). Figure 8 presents the cooling curves obtained during rapid solidification of Fe-Ni-Cr alloys with different undercoolings [33], where the cooling rates are 100 K/s and 148 K/s, and the corresponding plateau times are about 2.76 s and 1.9 s. It further confirms the relationship of RΔt = constant in solidification processes.

The heating process also follows a similar relationship as reported in Equation (11), which is not discussed here. Equations (4), (9) and (11) are three simple relationships for the cooling curves of alloys in solidification processes.

6. Conclusions

From the cooling curve of the rapid solidification sample, the phase fraction change can be estimated using the proposed method based on the heat release time ($f_s={\displaystyle \frac{t-t_l}{t_s-t_l}}$). Its accuracy and practical applicability were validated through analyses of cooling curves and microstructures in Al-Cu and Fe-Ni-B alloys. The results were further verified by computer-aided image analysis and the box-counting method for microstructural characterization. Then, a comparison between the cooling curve analysis using the present novel method and the Newtonian thermal analysis demonstrates good agreement between the results. Unlike Newtonian thermal analysis, which requires derivative and integration operations, the present method offers a greater operational simplicity. Additionally, two other equations related to solidification were identified from the cooling curves—the growth velocity–recalescence (V/R_c = D/ΔT) and the cooling rate–solidification plateau time (RΔt = constant). These relationships, combined with the phase fraction estimation, enable the prediction of solidification microstructure and solidification rate in rapid solidification processes.