Crystallization kinetics of GdYScAlCo high-entropy bulk metallic glass published by Ural Federal University eISSN 2411-1414 chimicatechnoacta.ru ARTICLE 2023, vol. 10(2), No. 202310207 DOI: 10.15826/chimtech.2023.10.2.07 1 of 9 Crystallization kinetics of GdYScAlCo high-entropy bulk metallic glass V.A. Bykov a , D.A. Kovalenko b * , E.V. Sterkhov a, T.V. Kulikova a a: Institute of Metallurgy, Ural Division of Russian Academy of Sciences, Ekaterinburg 620002, Russia b: Ural Federal University, Ekaterinburg 620009, Russia * Corresponding author: darya.k.2000@list.ru This paper belongs to a Regular Issue. Abstract The thermal stability and non-isothermal crystallization of a new bulk- amorphous high-entropy (HE-BMG) equiatomic GdYScAlCo alloy were studied by differential scanning calorimetry (DSC). The alloy shows a four-stage crystallization process. The kinetic parameters (activation en- ergy (Eα)), the pre-exponential factor (logA) and glass-forming ability indicators (kinetic fragility index, characteristic temperatures) for the GdYScAlCo alloy were obtained. The Eα values obtained by isoconversional methods indicate a nonlinear Arrhenian behaviour and a complex process. The Avrami equation modification proposed by Jeziorny and the multivariate nonlinear regression method were applied on the nonisothermal crystallization. In the case of primary crystallization of the amorphous GdYScAlCo alloy under nonisothermal conditions, the kinetics of the nucleation process is best described by an autocatalytic reaction. Keywords high-entropy bulk metallic glass activation energy glass-forming ability primary crystallization Received: 13.03.23 Revised: 04.04.23 Accepted: 11.04.23 Available online: 18.04.23 Key findings ● New bulk-amorphous high-entropy equiatomic GdYScAlCo alloy was produced. ● Non-isothermal crystallization kinetics of GdYScAlCo metallic glass was investigated. ● The multivariate nonlinear regression method suggested a combined auto-catalysis reaction model. © 2023, the Authors. This article is published in open access under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction Since the first discovery in 1960, interest in metal glasses and other metastable materials has been growing [1]. For more than 60 years of research, several hundred bulk me- tallic glasses (BMG) were synthesized with dimensions up to tens of centimeters. The majority of the synthesized BMGs are alloys based on one or two basic elements with small additions of elements that increase glass-forming ability and characteristics of materials [2–4]. In addition to high glass-forming ability, such materials have a number of unique properties: high thermal stability, high plasticity and soft magnetic properties. Also, amorphous alloys based on Sc–Al–Co have unique strength and corrosion characteristics [5]. Additionally, ad- ditives of rare-earth metals (REM) affect the glass-forming ability (GFA) of these alloys. For example, REM additives (Gd, Y) increase the glass-forming ability and improve the mechanical properties of Sc–Al–Co alloys [5]. However, the GFA and thermal stability of GdYScAlCo alloys, as well as the mechanisms of their crystallization, is not studied. We chose an equiatomic GdYScAlCo alloy for research because it easily amorphizes under arc melting conditions. To estimate the GFA, we used various experi- mental indicators of glasses, such as glass transition tem- perature (Tg), on-set crystalline temperature (Tx) and liqui- dus temperature (Tl), etc. To obtain given properties of the GdYScAlCo alloy, as well as to predict the optimal compositions and heat treat- ment modes, the calculation of kinetic parameters (activa- tion energy Ea, pre-exponential factor) using iso-conversion methods of thermal analysis was carried out. http://chimicatechnoacta.ru/ https://doi.org/10.15826/chimtech.2023.10.2.07 mailto:darya.k.2000@list.ru http://creativecommons.org/licenses/by/4.0/ https://orcid.org/0000-0003-4851-8223 https://orcid.org/0000-0001-5076-6715 https://orcid.org/0000-0002-0674-1139 https://crossmark.crossref.org/dialog/?doi=https://doi.org/10.15826/chimtech.2023.10.2.07&domain=pdf&date_stamp=2023-04-18 Chimica Techno Acta 2023, vol. 10(2), No. 202310207 ARTICLE 2 of 9 DOI: 10.15826/chimtech.2023.10.2.07 2. Experimental procedure 2.1. Materials and synthesis The master equiatomic GdYScAlCo alloy was prepared by arc melting Al (99.99% purity), Gd (99.9%), Y (99.9%), Sc (99.9%) and Co (99.9%) in a helium atmosphere. Fur- ther, it is re-melted five times to achieve complete melting and compositional homogeneity. The composition of the samples was controlled by weighing them before and after synthesis, and the mass loss did not exceed 0.1 wt.%. Then, amorphous alloy was produced by performing suction cast- ing of the arc-melted metal liquid into a copper mold to pro- duce a 3 mm diameter rod. 2.2. Differential scanning calorimetry (DSC) and X-ray diffraction measurements The as-cast rod structure was examined by X-ray diffraction at room temperature in a 2θ angular range of 25°–100° at a step of 0.004° using a Shimadzu XRD7000 diffractometer and Cu Kα radiation; the exposition time was 3 s. Thermal reactions in the sample were investigated by differential scanning calorimetry using a Netzsch STA 449C device, cal- ibrated with indium, tin zinc, aluminum, silver and gold standards. The selected sample mass was 20.1±0.1·10−6 kg. DSC scans were performed at different heating rates (5, 10, 20, 40 K/min) in the temperature range from 273 to 1100 K at an argon flow of 60 ml/min. To describe the initial crys- tallization process and to determine the kinetic parameters, NETZSCH Kinetics Neo software (NETZSCH, Selb, Ger- many) was applied using model-free and the multivariate nonlinear regression methods. 3. Results and Discussion 3.1. Thermal analysis and GFA indicators Figure 1 illustrates an X-ray pattern of the GdYScAlCo rod that shows an amorphous halo without distinct crystalline peaks, i.e., the resulting sample is X-ray amorphous. The non-isothermal DSC of of GdYScAlCo HE-BMG are shown in Figure 2. All the DSC curves exhibit the four exo- thermic reactions corresponding to the crystallization pro- cesses. Table 1 shows the thermal characteristics of the GdYScAlCo alloy: glass transition temperature Tg, on-set crystalline temperature Tx, melting temperature Tm, liqui- dus temperature Tl, as well as indicators of glass-forming ability: supercooled liquid interval ΔTx = Tg – Tx, Trg = Tl/Tg and melting interval ΔTl = Tl – Tm. GdYScAlCo alloy shows high values of glass transition temperature Tg. GFA is related to physical nature of alloy and reflects how well the alloy forms metallic glass during casting. There are various criteria for evaluating GFA, including γ and δ, which are based on the classical theory of nucleation [6]. These criteria can be easily obtained by thermal analy- sis and show a strong correlation with GFA in metallic glasses. Calculation of γ and δ criteria was carried out ac- cording to equations: 𝛿 = 𝑇𝑥 (𝑇𝑙 − 𝑇𝑔 ) , (1) 𝛾 = 𝑇𝑥 (𝑇𝑙 + 𝑇𝑔 ) . (2) In paper [7] for the four-component Sc36Al24Co20Y20 al- loy the value of the criterion γ = 0.444 was presented, which agrees well with our data for the five-component GdYScAlCo alloy (γ = 0.407). Table 1 shows the criteria γ and δ, which indicate a good glass forming ability of the GdYScAlCo alloy. The term “fragility” introduced by Angel [8], shows the degree of deviation of the temperature dependence of the viscosity from the Arrhenius curve and is a GFA indicator. From the physical perspective, it characterizes how well the material transitions to the glass state when cooled. Sub- stances with high values of this parameter have a narrow range of glass transition temperatures, while substances with low values have a relatively wide range [8]. Figure 1 XRD pattern of the GdYScAlCo rod. Figure 2 Experimental DSC curves of GdYScAlCo amorphous alloy at different heating rates (inset: temperatures Tm, Tl for heating rate of 10 K/min). https://doi.org/10.15826/chimtech.2023.10.2.07 https://doi.org/10.15826/chimtech.2023.10.2.07 Chimica Techno Acta 2023, vol. 10(2), No. 202310207 ARTICLE 3 of 9 DOI: 10.15826/chimtech.2023.10.2.07 Table 1 Thermal characteristics for GdYScAlCo alloy according to DSC measurements obtained at a rate of 10 K/min. Alloy Tg, К Tx, К Tm, К Tl, К ΔTx Trg ΔTl γ δ GdYScAlCo 559 631.5 960 992 32 0.563 32 0.407 1.458 Fragility is determined by the slope of the viscosity curve according to the dependence of log (viscosity) on Tg/T when approaching Tg, which gives the kinetic index of a Fragility (m). The kinetic a Fragility index can be obtained as follows [9]: 𝑚 = 𝐷𝑇0𝑇𝑔 (𝑇𝑔 − 𝑇𝑜 ) 2ln10 , (3) where D is the strength parameter, T0 is the asymptotic value of Tg at an infinitely slow cooling and heating rate, and Tg is the glass transition temperature [8]. D and T0 can be determined through the relationship between Tg and heating rate (β) by an equation with the Vogel-Fulcher-Tammann form [10]: lnβ = ln(𝐵) − 𝐷𝑇0 𝑇𝑔 − 𝑇0 . (4) Fragility index for the sample was calculated at heating rates 5, 10, 20, 40 K/min. The data obtained from Equations 3 and 4 are summarized in Figure 3. The results show that m is equal to 33. The liquids with the fragility index m close to 17 are usually referred to “strong” glass-formers. The liq- uids with m much higher than 17 are usually referred to “fragile”. In the case of GdYScAlCo HE-BMG, it can be clas- sified into “fragile” glasses. The parameter F1, introduced by Senkov [11], is an indicator of glass-forming ability, which establishes a correlation between the fragility parameter and the critical cooling rate at which the amorphous state is formed. The value F1 = 0 corresponds to an extremely fragile liquid, and F1 ~ 0.8 – extremely strong liquid. The F1 parameter can be defined as: 𝐹1 = (𝑇𝑔 − 𝑇0) 0.5(𝑇𝑙 + 𝑇𝑔 ) − 𝑇0 , (5) where Tg is the glass transition temperature, Tl is the liquidus temperature, and T0 is the on-set temperature. The calculation results are presented in Table 2. Compared to other metallic glasses: La55Al25Ni20 (m = 42, F1 = 0.455), La55Al25Ni10Cu10 (m = 35, F1 = 0.540) [11], the alloy GdYScAlCo shows rather high values in frazilite and the F1 parameter, which indicates its good glass forming ability. 3.2. Primary crystallization kinetics of GdYScAlCo 3.2.1. Activation energy One of the most important kinetic parameters of the crystal- lization process is the activation energy. It represents the en- ergy barrier that must be overcome by the system to start the nucleation and growth process leading to crystallization. According to Figure 2, there are four crystallization peaks on the DSC curves. Since the first peak overlaps with the second and third ones, we used a mathematical proce- dure of peak shape separation to correctly estimate the ki- netic parameters of the nucleation and primary crystalliza- tion process (Figure 4) [12]. In the calculations we used the DSC peaks at different heating rates obtained by deconvolu- tion, so they should be compared. The separated peaks have the same shape at different heating speeds, i.e., they are identical to each other. As can be seen in Figure 4, the posi- tions of onset and peak maximum temperatures of the origi- nal DSC signal and the separated ones are almost the same. The baseline was a straight line. Because the crystallization of amorphous alloy is a multistage process, it requires addi- tional structural studies. Here, we limit ourselves to calcu- lating the kinetic parameters only for the first peak and de- scribing the mechanism of nucleation and primary crystalli- zation. Thus, the DSC curves (Figure 2) show peaks corre- sponding to the nucleation and crystal growth processes. We calculate the activation energies corresponding to these tem- peratures to understand the nucleation and crystallization in general. In the methods of non-isothermal kinetics based on the free model, it is assumed that the single-step processes oc- curring can be described following rate equation [13]: 𝛽 𝑑𝑎 𝑑𝑡 = 𝐴𝑓(𝑎)𝑒 −𝐸𝑎 𝑅𝑇 , (6) where 𝐴 is the pre-exponential factor, 𝛽 – the constant heating rate, Eα is the activation energy, and the concentration de- pendence of the reaction rate (reaction model) is 𝑓(𝑎), where a is the degree of transformation of the substance in the range from 0 to 1. Figure 3 Fragility index for GdYScAlCo alloy. Table 2 Tg, Tl, m and F1 parameter for GdYScAlCo alloy. Tg (K) Tl (K) m F1 559 992 33 0.463 https://doi.org/10.15826/chimtech.2023.10.2.07 https://doi.org/10.15826/chimtech.2023.10.2.07 Chimica Techno Acta 2023, vol. 10(2), No. 202310207 ARTICLE 4 of 9 DOI: 10.15826/chimtech.2023.10.2.07 Additionally, it was assumed that the reaction rate at a constant conversion value depends only on temperature (iso-conversion principle). It should be noted the im- portance of determining the preexponential factor A in the framework of kinetic analysis without using a model. Ac- cording to Vyazovkin [14] model-free method of estimating the preexponential factor A is suitable for both single- and multi-step kinetics. Model-free methods of analysis allow us to determine the activation energy Eα of the reaction pro- cess without making hypotheses about the kinetic model of the process and without knowing the type of reaction [13]. Various model-free methods are used to calculate the acti- vation energy Eα of non-isothermal reactions at different heating rates [15–17]. First, we used the most common of them, the Kissinger method. The basic equation of the Kis- singer method is written as ln(β 𝑇2⁄ ) = − 𝐸 𝑅𝑇⁄ + const, (7) where β is the heating rate, R is the gas constant, T is the temperature. Figure 5 shows the dependences described by equation (2), which correspond to the following processes: the glass transition at Tg(Eg); the start of nucleation at Tx(Ex); the crystal growth at Tp1(Ep1). The calculation results of the ac- tivation energy are shown in Figure 5 and Table 3. Table 4 shows that the highest activation energy is at the beginning of the nucleation process, Ex = 314 kJ/mol. This suggests that the most energy-consuming process is the beginning of crystallization (the appearance of the first nuclei of the crystalline phase in the amorphous matrix). Thus, we determined the activation energy of the primary crystallization process (Ep) using the Kissinger equation (Equation 7). However, from a physical viewpoint, this method cannot be applied directly to amorphous alloys, since the crystallization process in them proceeds through the nucleation and crystal growth processes rather than through the n-order reaction [18, 19]. Also, as noted by Vyazovkin [13], Kissinger's method is not very accurate. The method gives a single activation energy in accordance with the assumption of one-step kinetics, which creates a problem for most applications. Figure 4 Separated DSC curves for a heating rate of 10 K/min. We performed calculations of Eα using other model-free methods (Vyazovkin [15], Friedman [16] and Ozawa [17]) in the NETZCH Kinetics Neo software. The calculation results by Kissinger, Vyazovkin, Friedman and Ozawa methods are presented in Table 4 and Figure 5. The analysis of the de- pendence of Eα versus α (conversion) of the crystallization process of the HE-BMG GdYScAlCo alloy obtained from iso- conversional methods (Ozawa, Friedman and Vyazovkin) allows us to check the applicability of the one-step kinetics according to Equation 7. Figure 5 Kissinger plots for calculating activation energies in GdY- ScAlCo BMG. Table 3 Activation energy Eg, Ex, Ep1 for temperatures Тg, Тх, Tp1. Alloy Activation energy, kJ/mol Eg Ex Ep1 GdYScAlCo 160 314 227 https://doi.org/10.15826/chimtech.2023.10.2.07 https://doi.org/10.15826/chimtech.2023.10.2.07 Chimica Techno Acta 2023, vol. 10(2), No. 202310207 ARTICLE 5 of 9 DOI: 10.15826/chimtech.2023.10.2.07 Figure 4 presents the dependence of the Eα and logA on the degree of conversion calculated using Ozawa, Friedman and Vyazovkin methods. We can see from Figure 6 that the values of E(α) change nonlinearly, where as the dependences of E(α) are similar. The Eα values indicate a nonlinear Arrhenian behaviour and a complex process. The close average E(α) values obtained by the conversional methods differ from those calculated with Kissinger's method. Therefore, the Kissinger method can only be used for preliminary estimation of the activation energy of a one-step process. Due to the fact that the values of E(α) change nonlinearly and indicate a complex process, naturally, the Kissinger activation energy values cannot be applied when simulating the complex process of crystallization of amorphous materials, where multistage processes are observed. Table 4 Crystal growth activation energy Ep1 and pre-exponential factor log A for GdYScAlCo alloy. Kinetic parame- ter Vyazov- kin Kissin- ger method Fried- man Ozawa- Flynn- Wall Aver- age value Ep1, kJ/mol 250 227 250 260 247 logA 17.6 13.1 17.6 17.8 16.5 Figure 6 Аctivation energy and logA values of versus conversion during the crystallization process using the Friedman (a), Vyazov- kin (b), and Ozawa (c) methods. 3.2.2. Avrami model using Jeziorny method One of the most commonly used models for describing the crystallization kinetics of polymers, metals, and glasses is the Avrami model (also known as the Johnson–Mehl–Av- rami–Erofeev–Kolmogorov (JMAEK) model [20–25]). To isothermal conditions, the Avrami equation is typically used in the following form: 𝑎 (𝑇) = 1 − 𝑒 −𝑘(𝑇)∙𝑡 𝑛 , (8) where 𝑎 is the degree of conversion, k(T) represents the rate constant, t is time and n is the local Avrami index. For nonisothermal experimental conditions, Equation (8) is frequently expressed as the following linear equation: 𝑑𝑎 𝑑𝑡 = 𝑘(𝑇) 𝛽 ∙ 𝑛 ∙ (1 − 𝑎) ∙ [− ln(1 − 𝑎)] ∙ (𝑛 − 1) 𝑛 . (9) The volume fraction of crystals (conversion 𝑎) can be determined by the crystallization heat using the following equation [26]: 𝑎 = ∫ (𝑑𝐻/𝑑𝑇)𝑑𝑇 𝑇 𝑇0 ∫ (𝑑𝐻/𝑑𝑇)𝑑𝑇 𝑇𝑖𝑛𝑓 𝑇0 = 𝐴0 𝐴𝑖𝑛𝑓 , (10) where T0 and Tinf are the temperatures of the beginning and end of crystallization, respectively. dH corresponds to the enthalpy of crystallization released during an infinitesimal temperature interval dT. A0 and Ainf correspond to the region between the initial and specific temperature and the end of crystallization, respectively. Figure 7 shows the relationship between the volume fraction of crystallization and the temperature for primary crystallization. From Figure 7 we see that with an increase in the heating rate, the S-curves shift to the region of higher temperatures. Many attempts have been made to derive Equation 8 under non-isothermal temperature with constant heating or cooling rates [27]. The main difficulty in modifying the JMAEK model under non-isothermal conditions is that these experiments are much faster than isothermal experiments. Figure 7 Сrystallized volume fraction versus temperature plots. https://doi.org/10.15826/chimtech.2023.10.2.07 https://doi.org/10.15826/chimtech.2023.10.2.07 Chimica Techno Acta 2023, vol. 10(2), No. 202310207 ARTICLE 6 of 9 DOI: 10.15826/chimtech.2023.10.2.07 In isothermal experiments, the material under study is heated to a certain temperature in a time much shorter than the transformation time. Crystallization of metallic glasses under linear heating conditions is mainly studied using the JMAEK method modified by Jeziorny [28], who converted the linear Avrami equation to linear heating conditions us- ing the following assumption: ln 𝑘𝐴 = ln𝑘(𝑇) 𝛽 , (11) where 𝑘(𝑇) = 𝐴𝑒 −𝐸𝑎 𝑅𝑇 – rate constant, β – heating rate However, as Vyazovkin states [29], this transformation contradicts the basic principle of equating physical quanti- ties and also leads to incorrect Avrami indices, usually larger than the actual value. It should be noted that the Jeziorny’s method is often used to calculate the Avrami in- dex of non-isothermal crystallization kinetics of amorphous metallic glasses due to the developed algorithm for the in- terpretation of the polymer crystallization mechanism. [30]. However, the crystallization processes in amorphous metallic materials and polymers can be radically different. Further, the mechanism of primary crystallization for the amorphous GdYScAlCo alloy under non-isothermal condi- tions by the Jeziorny method will be tested in the multivar- iate nonlinear regression method. To determine the mech- anism of nucleation and its growth, we used the approach of determining the local Avrami index n(α) with Jeziorny's assumption (Equation 11), which was proposed in [31]. According to [31], the local Avrami index n(α) can be found from a modification of the JMAEK equation, given that t–t0 = (T–T0)/β, where T0 is the temperature at the crystallization onset. Therefore, it can be written as: 𝑑ln[−ln (1 − 𝛼)] 𝑑{ln [(𝑇 − 𝑇0)/𝛽]} = 𝑛 {1 + 𝐸 𝑅𝑇 (1 − 𝑇0 𝑇 )}, (12) where T0 is the initial crystallization temperature and 𝐸 is the activation energy of the crystallization process. The val- ues of E in Equation 10 were used as E(α) calculated by the Vyazovkin method. For non-isothermal DSC curves, the local Avrami index (n(α)) can be obtained by plotting the dependence of ln[–ln(1–α)] on ln[(T–T0)/β]. The plots of these dependences are shown in Figure 8, where the value of n(α) can be obtained from the slope of these curves. In addition, Figure 9 shows the local Avrami index indices as a function of the volume fraction of crystallized matter α at different heating rates. The resulting curves are not linear, indicating that n(α) changes during the crystallization process. The n(α) is a basic parameter that is necessary to understand the mechanisms of nucleation and grain growth with increasing α during the transformation phase. The local Avrami index is expressed as: 𝑛 = 𝑏 + 𝑝𝑚, (11) where b is the nucleation index, m is the dimensionality of grain growth and p is the type of growth. The germination index contains four conditions: (1) b = 0 suggests a zero germination rate; (2) 01 indicates an increase in the germination rate with time. The grain growth magnitude m is 1, 2, or 3; p = 1 represents surface-controlled growth, and p = 0.5 indicates diffusion-controlled growth. Before describing the primary crystallisation mechanism using the n(α) dependence, we present an important remark. Since the peak separation procedure performed allows determining the approximate baseline, it must be considered that extreme values of n(α) (both α< and α~1) are unreliable, since they are affected by the baseline. Figure 8 shows that at the start of the crystallization process, the n(α) values are 2.2–3.5, indicating that the growth mechanism is surface-controlled. The appearance of these curves clearly shows the course of nucleation and grain growth at different heating rates. All n(α) at different heating rates are greater than unity at x = 0, which corresponds to the nucleation process with increasing nucleation rate; n(α) of the three curves (10, 20, 40 K/min) firstly increases and then decreases at 0.05≲x≲0.91, which indicates that the nucleation rate and growth dimension always change at this stage. The continuous decrease of n(α) values after a certain percentage of crystallization volume fraction (≳ 5% for speeds 10–40 K/min) shows that nucleation rates decrease to about zero and the grain growth process dominates; all n(α) curves increase rapidly at 0.91≲x≲1. In the initial stage of crystallization, nucleation dominates, with a constant or increasing rate and no grain growth process when 1≤n(α)<2; there is one (n(α) = 2), two (2B logAA–>B, log(s –1) logAB–>C, log(s –1) Ea A–>B, kJ/mol Ea B–>C, kJ/mol CA–>B CB–>C R 2 2.1 13.7 25.8 200.1 354.3 0.48 0.52 0.9678 https://doi.org/10.15826/chimtech.2023.10.2.07 https://doi.org/10.15826/chimtech.2023.10.2.07 Chimica Techno Acta 2023, vol. 10(2), No. 202310207 ARTICLE 8 of 9 DOI: 10.15826/chimtech.2023.10.2.07 Table 6 Kinetic parameters and the regression coefficient R2 for the fitting of multivariate nonlinear regression for combined auto- catalysis reaction (heterogeneous reaction with n-th order and m- power autocatalysis). n m logA, log(s –1) Ea, kJ/mol R 2 1.8 0.8 12.7 216.6 0.9950 4. Conclusions The thermal stability and primary crystallization kinetics of the GdYScAlCo HE-BMG alloy were studied using DSC. The main conclusions are presented below. 1. GdYScAlCo HE-BMG exhibited four different crystallization events. The characteristic temperatures (such as Tx1, Tp1, Tg...) increased with increasing heating rate. In terms of m = 33 and Angel’s GdYScAlCo classification, HE-BMG belongs to the “fragile” glasses. 2. The activation energy (Eα) of the process was calculated using the methods of Vyazovkin, Ozawa, Friedman and Kissinger. The Kissinger method gives inaccurate activation energy values in comparison to the other methods. The Eα values obtained by isoconversional methods indicate a nonlinear Arrhenian behaviour and a complex process. 3. Analysis of the crystallization kinetics by Jeziorny’s modification of the Avrami model showed that values of the local Avrami index are 1.23.0.CO;2-P 16. Friedman HL. Kinetics of thermal degradation of char-form- ing plastics from thermogravimetry. J Polym Sci Part C Polym Symposia. 1964;6(1):183–185. 17. Ozawa Takeo. A new method of analysis thermogravimetric data. bulletin of the chemical society of Japan. Bull Chem Soc Japan. 1965;38:1881–1886. doi:10.1246/bcsj.38.1881 18. Cheng YT, Hung TH, Huang JC, Hsieh PJ, Jang JSC. Thermal stability and crystallization kinetics of Mg–Cu–Y–B quater- nary alloys. Mater Sci Engin. 2007;449:501–505. doi:10.2320/matertrans1989.32.609 19. Matusita K, Komatsu T, Yokota R. Kinetics of non-isothermal crystallization process and activation energy for crystal growth in amorphous materials. J Mater Sci. 1984;19(1):291– 296. doi:10.1007/bf02403137 20. Avrami M. Kinetics of phase change. I: General theory. J Chem Phys. 1939;7:1103–1112. doi:10.1063/1.1750380 21. Avrami M. Kinetics of phase change. II Transformation-time relations for random distribution of nuclei. J Chem Phys. 1940;8:212–224. doi:10.1063/1.1750631 22. Avrami M. Granulation, phase change, and microstructure ki- netics of phase change. J Chem Phys. 1941:177–184. doi:10.1063/1.1750872 23. Shiryayev AN. On the statistical theory of metal crystalliza- tion (Ed.), in: A. N. Shiryayev (Ed.), Sel. Work. A. N. Kolmo- gorov Vol. II Probab. Theory Math Stat. 1992:188–192. doi:10.1007/978-94-011-2260-3_22 24. Barmak K. A commentary on reaction kinetics in processes of nucleation and growth. Mater Sci. 2010;41:2711–2775. doi:10.1007/s11661-010-0421-1 25. Erofe’ev BV. Generalized equation of chemical kinetics and its application in reactions involving solids. Dokl Akad Nauk SSSR. 1946:511–514. 26. Shan Zhang, Chao Wei, Jingwang Lv, Haoran Zhang, Zhilin Sh, Xinyu Zhang, Mingzhen Ma. Non-isothermal crystalliza- tion kinetics of the Zr50Cu34Al8Ag8 amorphous alloy. Mater Lett. 2022;307. doi:10.1016/j.matlet.2021.130996 27. Ruitenberg G, Woldt E, Petford-Long AK. Comparing the Johnson-Mehl-Avrami-Kolmogorov equations for isothermal and linear heating conditions. Thermochim Acta. 2001;378:97–105. doi:10.1016/S0040-6031(01)00584-6 28. Jeziorny A. Parameters characterizing the kinetics of the non- isothermal crystallization of poly(ethylene terephthalate) de- termined by d.s.c. Polymer (Guildf). 1978;19:1142–1144. doi:10.1016/0032-3861(78)90060-5 29. Vyazovkin S. Jeziorny method should be avoided in avrami analysis of nonisothermal crystallization. Polym. 2023;15:197. doi:10.3390/polym15010197 30. Ozawa T. Kinetics of non-isothermal crystallization. Polymer. 1971;12;3:150–158. doi:10.1016/0032-3861(71)90041-3 31. Blázquez JS, Conde CF, Conde A. Non-isothermal approach to isokinetic crystallization processes: Application to the nano- crystallization of HITPERM alloys. Acta Mater. 2005;53(8):2305–2311. doi:10.1016/j.actamat.2005.01.037 32. Opfermann J. Kinetic analysis using multivariate non-linear regression. I. Basic concepts. J Therm Anal Calorim. 2000;60:641–658. doi:10.1023/A:1010167626551 33. Sourour S, Kamal MR. Differential scanning calorimetry of epoxy cure: isothermal cure kinetics. Thermochim Acta. 1976;14(1–2):41–59. doi:10.1016/0040-6031(76)80056-1 https://doi.org/10.15826/chimtech.2023.10.2.07 https://doi.org/10.15826/chimtech.2023.10.2.07 https://doi.org/10.1063/1.1736628 https://doi.org/10.1103/PhysRevB.46.11318 https://doi.org/10.1007/s11661-007-9255-x https://analyzing-testing.netzsch.com/ru/blog/2021/separating-overlapping-effects-in-analytical-measurement-curves https://analyzing-testing.netzsch.com/ru/blog/2021/separating-overlapping-effects-in-analytical-measurement-curves https://analyzing-testing.netzsch.com/ru/blog/2021/separating-overlapping-effects-in-analytical-measurement-curves https://doi.org/10.3390/molecules25122813 https://doi.org/10.3390/molecules26113077 https://doi.org/10.1002/(SICI)1096-987X(199702)18:3%3c393::AID-JCC9%3e3.0.CO;2-P https://doi.org/10.1002/(SICI)1096-987X(199702)18:3%3c393::AID-JCC9%3e3.0.CO;2-P https://doi.org/10.1246/bcsj.38.1881 https://doi.org/10.2320/matertrans1989.32.609 https://doi.org/10.1007/bf02403137 https://doi.org/10.1063/1.1750380 https://doi.org/10.1063/1.1750631 https://doi.org/10.1063/1.1750872 https://doi.org/10.1007/978-94-011-2260-3_22 https://doi.org/10.1007/s11661-010-0421-1 https://doi.org/10.1016/j.matlet.2021.130996 https://doi.org/10.1016/S0040-6031(01)00584-6 https://doi.org/10.1016/0032-3861(78)90060-5 https://doi.org/10.3390/polym15010197 https://doi.org/10.1016/0032-3861(71)90041-3 https://doi.org/10.1016/j.actamat.2005.01.037 https://doi.org/10.1023/A:1010167626551 https://doi.org/10.1016/0040-6031(76)80056-1