Theoretical and Experimental Parameters of the Structure and Crystallization Kinetics of Melt-Quenched As30Te64Ga6 Glassy Alloy

The present framework reports the structural, fundamental parameters, and crystallization kinetics of the melt-quenched As30Te64Ga6 chalcogenide glass. The energy dispersive X-ray analysis of the As30Te64Ga6 glassy system reveals that the constituent element ratio of the investigated bulk sample agrees with the nominal composition. Also, X-ray diffraction (XRD) and Differential Scanning Calorimetry (DSC) were used to characterize structural properties, and crystallization kinetics; respectively. Four characteristic temperatures related to various phenomena are observed in the investigated DSC traces. The first one is Tg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{g}$$\end{document} that corresponds to the glass transition temperature. The second one is (Tc1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{c1}$$\end{document}, and Tc2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{c2}$$\end{document}) that corresponds to the onset of the double crystallization temperatures. The third one (TP1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{P1}$$\end{document}, and TP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{P2}$$\end{document}) identifies the double peak crystallization temperatures. The last characteristic temperature (Tm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{m}$$\end{document}) is the melting point. The XRD analysis indicates the amorphous structure of the as-prepared glassy alloy, while the annealed samples are polycrystalline. The crystallization kinetics of the As30Te64Ga6 bulk are studied under non-isothermal conditions. In addition, the values of various kinetic parameters such as the glass transition activation energy, weight stability standard, and Avrami exponent were determined. The activation energy of the crystallization process for As30Te64Ga6 glass alloy was calculated using classical methods. The results indicated that the rate of crystallization is related to thermal stability and the ability to form glass. Kinetic parameters have been estimated with some conventional methods and found to be dependent on heating rates (β).


Introduction
Chalcogenide-based materials are the subject of intensive studies because of their utilizing in potential technological applications specially in enhancing energy storage and solar cell efficiencies [1][2][3], as well as optoelectronics and electrical switching devices [4]. Chalcogenide glasses are usually synthesized via mixing chalcogen elements in group VI (such as S, Se, and Te) with other elements. Chalcogenide materials are utilized for many electronic applications such as IR optical elements, switching devices, reversible phase change, optical fibers, transmission media, and records [5,6]. One of these materials, As-Te alloy that characterized by a high crystallization ability as well as a high glass-forming ability. Therefore, the formation of the As-Te alloys should be performed at relatively low temperatures [7]. The As-Te glassy alloys are commonly synthesized by the melt quenching technique even the concentration of As is constantly increased at the expense of the Te ratio [8].
In order to improve the physical properties of the binary As-Te system, doping with other elements or/and heating treatment are required. For example, the electrical response of the As-Te glassy system reinforces by doping Cu, Ge, Si, or Ga elements, for this reason, the ternary alloys are beneficial in the electrical field not to mention the great benefit of these alloys, whether in the optical or thermal field [9][10][11]. The addition of such elements for forming ternary alloys leads to a change in the characteristic's temperatures such as T g [12]. Furthermore, the transformation from semiconductor to metal trend in the ternary glassy system As-Te-Al encourages after addition of the Al element to the As-Te system [13]. Studies of the physical properties, including the optical, electrical, and structural properties of the As-Te-Ga system is limited [11,[14][15][16][17]. For example, some efforts have been done to study the effect of heat treatment (e.g. annealing) on the optical properties for few compositions of As-Te-Ga thin films [11,16,18]. To the best of our knowledge, the thermal analysis and crystallization kinetics of the As-Te-Ga system has not been extensively investigated [19]. Thus, thermal kinetics and analysis is an important tool that helps for predicting the structure of the bulk alloys and hence modifying the performance of the material under investigation. Accordingly, the study of thermal analysis and crystallization kinetics of As-Te-Ga is very important because it could help in further progress in the development of many physical properties and technological applications based on the As-Te-Ga system.
Moreover, after listing the physical properties and mentioned scientifically interesting contours of the studied glass system, it is worth noting here for several thermal parameters and the main concepts for glass transformation and crystallization process, i.e. thermal stability of chalcogenide glassy alloys. Thermal stability has a magnificent benefit in several technological applications. This is due to determine the useful ranging of working temperatures via the structure changes and thus definitive crystallization for the operating temperatures. Consequently, amorphous materials seek to complete the transformation to the crystalline state in order to stabilize. In all the solid-state systems, thermally activated transformations can be studied via isothermal and non-isothermal examinations [20][21][22]. In the isothermal methods, the investigated glassy alloys are brought quickly to a temperature above the temperature of the glass transition, and then the resulted heat as a function of time is recorded to observe the crystallization processes. The isothermal techniques have a clear disadvantage presentation in the impossibility of reaching an exam temperature immediately and during the periods in which the glassy system needs to stabilize. In contrast, the constant heating rate experiments do not have the mentioned drawback [23]. Generally, the differential thermal analysis (DTA) and DSC are the common techniques used for investigating the phase transitions and the crystallization kinetics of the chalcogenide glasses and polymers among other techniques [22,[24][25][26][27].
The main aims of the present work: (1) investigate the theoretical and physical aspects of the structure of the As 30 Te 64 Ga 6 system, (2) study the crystallization kinetics, and thus evaluate the crystallization parameters of As 30 Te 64 Ga 6 glassy alloy via DSC under a non-isothermal method, (3) determine the crystal structure for the asprepared and annealed bulk samples in terms of the XRD measurements via Cu-K α radiation, and (4) evaluate the thermal stability, glass-forming ability, and other thermal criteria for the As 30 Te 64 Ga 6 system.

Samples Preparations
The bulk of the As 30 Te 64 Ga 6 glassy alloy is formed by the conventional melt quenching method from a high-purity (~ 99.999%) of As, Te, and Ga (purchased from Sigma-Aldrich Company). The powders of As, Te, and Ga are weighted accordingly to their atomic concentrations (at.%) and then put into a clean silica-glass ampoule. After that, the ampoule is closed when the inside vacuum reached 10 −5 Torr. Then the mixture is melted by introducing into a controlled furnace at 1100 K for 24 h. Through the melting process, the ampoule is frequently shaken to intermix the constituents and to increase the homogenization of the As 30 Te 64 Ga 6 alloy. Finally, after this treatment, the melt is followed by quick quenching in an ice-water mixture to avoid the crystallization process through the melt solidification. The thermal annealing of the bulk samples of As 30 Te 64 Ga 6 glass is carried out at selected annealing temperatures of 450, 475, and 575 K for 1 h under a vacuum of 10 −5 Torr.

Samples Characterizations
The concentration of elements in the as-prepared bulk sample is examined by the energy dispersive X-ray (EDX) spectroscopy. Whereas, the crystal structures of the asprepared, as well as the annealed As 30 Te 64 Ga 6 samples, are identified using a Phillips X-ray diffractometer. The X-ray diffractometer is composed of a Cu-K α radiation source with a graphite monochromator and the radiation wavelength (λ) equals 1.54178 Å. The XRD measurements are performed within the range of 2θ = 10°-70° with a scanning step and speed of 0.02°, and 0.06° s −1 , respectively.
The characteristic temperatures, phase change, and crystallization kinetics are investigated based on the monitored DSC (Model: TA -Q20). For this objective, around 10 mg of the As 30 Te 64 Ga 6 glass sample is used for each β (ranging from 5 to 25 K min −1 ), and the heat scanning was recorded from 310 to 675 K. The value of T g , crystallization onset temperatures ( T c1 , and T c2 ), the crystallization peak temperatures ( T P1 , and T P2 ) for the first and second crystallization peaks, respectively, and the melting temperature ( T m ) are deduced with an accuracy of ± 1 K by the microprocessor of the thermal analyzer. Structure  of the As 30 Te 64 Ga 6 3.

The Coordination Number and the Number of Constraints
The coordination number ⟨r⟩ is defined as the average number of the adjacent atoms of the components in a substance and describes the properties of the covalent chains that make up the glass [28]. Accordingly, the value of ⟨r⟩ for the As 30 Te 64 Ga 6 alloy have been computed using the subsequence equation [29]: Here α, β and γ are the atomic ratios of As, Te, and Ga, and in the present study are proposed to be 30, 64, and 6 at.%, while N As , N Te , and N Ga denote their coordination number and equal to 3, 2, and 5, respectively. The deduced value of ⟨r⟩ for the studied system is 2.48 and this is greater than the critical value (2.4), which is known as the rigidity percolation threshold. Based on the ⟨r⟩ value, the atoms in the proposed As 30 Te 64 Ga 6 glass over-constrained or rigidly connected. In an optimal glass configuration, the mechanical constraints, i.e., the bond expansion constraints (N α ) and the bond flexion constraints (N β ) of each atom are enumerated by N α = ⟨r⟩ /2 and N β = [2 ⟨r⟩ − 3] [30], and equals to 1.24 and 1.96 for the As 30 Te 64 Ga 6 glass, respectively. The average number of constraints is N con = N α + N β , and also can be expressed by the following equation [31]: Both the two equations give the same value (3.2) of N con for the studied As 30 Te 64 Ga 6 system.

Lone Pair Electrons
In chalcogenide material, the lone pair electrons ( L P ) are responsible for the glass formation. It was proposed [29], that the L P is a number must be > 1 for the ternary system. The value of L P in the present work is computed by the subsequence equation [32]: where V is the valence electrons and given from [32]: Here [V] As , [V] Te , and [V] Ga are the valence of As, Te, and Ga and equals 5, 6, and 3, respectively, while α, , and are their atomic concentrations. Accordingly, the values of V and L P for the studied glass are equal to 5.52 and 3.04, respectively. It is observed that the computed value of L P for As 30 Te 64 Ga 6 alloy is much greater than 1 (value of L P for ideal glass in the ternary system), this emphasizes that the proposed As 30 Te 64 Ga 6 composition strongly tends to be formed in the amorphous nature.

The Deviation of Stoichiometry
The deviation of stoichiometry (S) is given by the ratio of covalent bonding possibilities of chalcogen atom, such as Te in the present system, to that of a non-chalcogen atom, such as As and Ga. The value of S is computed using the subsequence equation [33]: For S > 1, the system is considered as chalcogen-rich and for S < 1, the system is considered chalcogen-poor, while at S = 1, the system is stoichiometry. The obtained value of S equals 1.07 and hence the As 30 Te 64 Ga 6 glass belongs to the chalcogen-rich type.

The Average Heat of Atomization
The average heat of atomization ( H S ) for the As 30 Te 64 Ga 6 glassy system can be evaluated from Eq. (6) [34]: The values of heat of atomization of individual elements As, Te, and Ga equal to 72.37, 47.03, and 65.06 kcal mol −1 , respectively [34]. The estimated values of H S , and the average single heat of atomization

The Bond Energy
The bond energies for heteronuclear bonds ( D (A−B) ) have been determined using Pauling's relation [35]: bond energies for heteronuclear bonds are D (As−Te) and D (Te−Ga) and equal to 75.61 and 42.13 kcal mol −1 , respectively, while D (As−As) , D (Te−Te) and D (Ga−Ga) equal to 92.30, 61.62, and 25.46 kcal mol −1 and As , Te and Ga equal to 2.18, 2.10, and 1.81, respectively.

The Cohesive Energy
The cohesive energy (CE) is referring to the energy needed to separate constituent atoms away from each other and to bring them to an assembly of neutral free atoms. The value of CE gives information about the chemical stability of the material which is deduced by summing the bond energies of all the bonds expected in the material under consideration as shown in Eq. (8) [36]: Here C i is the number of expected chemicals and bonds mentioned above in Sect. 3.1.5. In As 30 Te 64 Ga 6 , D i is the energy of each bond and equal to 75.61, 42.13, and 61.62 kcal mol −1 for (As-Te), (Te-Ga), and (Te-Te) bonds, respectively. The estimated value of CE for As 30 Te 64 Ga 6 glassy equals 83.15 kcal mol −1 .

The Fraction of Floppy Modes and Cross-Linking Density
Thorpe [37] suggested that there is a finite fraction of the modes of normal zero-frequency vibration called floppy modes (f). These floppy modes are found in the absence of weaker long-range forces and the f fractions are given by: Using ⟨r⟩ = 2.48 for the As 30 Te 64 Ga 6 glassy, hence the value of the fraction of floppy modes equals − 0.07. This means that the system has high rigidity. Moreover, the cross-linking density (X) can be obtained from the following expression: The deduced value of X for the As 30 Te 64 Ga 6 system is 0.48.

The Electronegativity
According to principle Sanderson [38], the electronegativity ( CM ) of the studied glassy system has been obtained from the following relation: If the molecule contains P atoms (same and/or different), the value of k will equal 1, 2,…, P, while k denote their isolated atom electronegativities. The obtained CM for As 30 Te 64 Ga 6 equals 2.04 where As , Te and Ga equal to 2.18, 2.10, and 1.81, respectively.

Elemental Analysis
The EDX analysis is used for the quantitative elemental composition of the as-prepared As 30 Te 64 Ga 6 alloy as shown in Fig. 1. The EDX chart reveals the spectral distribution of the constituent element and the atomic percentage ratio for As, Te, and Ga found to equals 30.13, 63.93, and 5.94 at.%, respectively. The percentage errors of the proposed concentrations are 0.43, 0.11, and 1% for As, Te, and Ga, respectively.

The Crystal Structure of the As 30 Te 64 Ga 6 Bulk Alloy
The crystal structure (either amorphous or crystalline) of the as-prepared and annealed As 30 Te 64 Ga 6 bulk system is confirmed by the XRD measurements as shown in Fig. 2. The absence of any sharp peak in the XRD plot of the asprepared sample confirms the non-crystalline structure (amorphous) of the as-prepared melt-quenched As 30 Te 64 Ga 6 alloy. Whereas, the XRD charts of annealed samples at 450, 475, and 575 K reveal few sharp as well as small peaks as shown in Fig. 2. The annealing temperatures were selected Fig. 1 Energy dispersive X-ray (EDX) spectrum of the As 30 Te 64 Ga 6 bulk alloy to cover both the two crystallization peaks according to the DSC traces (see Fig. 3). The presence of these diffraction peaks attributed to the partial transition from amorphous to the polycrystalline state during the annealing process of the As 30 Te 64 Ga 6 bulk sample. The diffraction of dominant peaks for an annealed sample at 450 K was indexed with the monoclinic As 2 Te 3 phase associated with the diffraction plane (112  Fig. 2. The observed phases are denoted by popular symbols in the XRD charts in Fig. 2, while the crystal structure parameters such as the average crystallite size, interplanar distance, strain, dislocation density, etc., were determined in our previous work [18].

DSC Analysis
To study the crystallization kinetics of melt-quenched As 30 Te 64 Ga 6 chalcogenide glassy and confirming its glassy nature, DSC of As 30 Te 64 Ga 6 at different β started from 5 to 25 K min −1 and in the temperature range, 310-670 K is carried out as presented in Fig. 3. It is observed for all values of β, four characteristic phenomena are detected in the investigated temperature range for the As 30 Te 64 Ga 6 system. The first one ( T g ) corresponds to the glass transition which is associated with absorbing energy to overcome the rigidity of the lattice. The second one ( T c1 and T c2 ) corresponds to the onset of the crystallization temperatures for the first and second crystallization peaks, respectively. The third characteristic temperature ( T P1 and T P2 ) identifies the peak crystallization temperatures for the two observed crystallization stages. The last characteristic temperature ( T m ) is the melting point. As mentioned, the DSC thermo-grams reveal a single glass transition, while two exothermic crystallization peaks. The values of the characteristic temperatures ( T g , T c1 , T c2 , T P1 , T P2 and T m ) of the studied glassy system at different β, are estimated from the DSC traces and listed in Table 1. From Table 1, it is observed those values are shifted to a higher temperature with the increase of β. This observation confirmed that all these transitions and transformations are thermally activated. The observation of two endothermic peaks for the melting phenomena could be attributed to the formation of more than one phase through the heating which is confirmed by the XRD analysis. The existence of a single T g reveals that the formed As 30 Te 64 Ga 6 glass is more stable than those glasses revealing multi-T g . The concentration of Ga in the As-Te-Ga alloys affect the number of crystallization peaks, as there is one, or two crystallization peaks observed for the As 30 Te 70 , As 30 Te 67 Ga 3 , and As 30 Te 60 Ga 10 glasses at different β [16,[39][40][41]. It was observed the value of characteristics temperatures are shifted to higher values as the Ga concentration increase or Te decrease at the same β.

Thermal Stability Parameters
The common criterion employed to obtain the glass thermal stability against crystallization is Deiztal's criterion, which is defined by the subsequent equation [42]: The glassy becomes more stable as ∆T increases. The obtained values of the thermal stability ΔT ΔT 1 , and ΔT 2 based on both crystallization peaks for the studied system are computed and summarized in Table 2. The values of (12) ΔT = T c − T g ΔT 1 , and ΔT 2 increase as presented in Table 2 with the increase in β. In other words, the glass thermal stability against crystallization is improved with increasing β. The investigated glassy alloy is more stable whenever the β increase but the tendency for stability is more for the difference ΔT 2 compared to the difference ΔT 1 .
There are two other criteria that give information about thermal stability, which was obtained by Saad and Poulain [43]. The first one is called the S-parameter (S P ) while the second is called the weighted thermal stability criterion ( H ′ ), and they are expressed by: The thermal stability S P reveals the resistance to devitrification after the formation of the glass. The difference between T p , and T c in Eq. (13), is related to the rate of devitrification transformation of the glassy phases. The values of S P ( S P1 , and S P2 ) and H ′ H ′ 1 , and H ′ 2 for the As 30 Te 64 Ga 6 glassy system at different β are computed for both crystallization stages and presented in Table 2. It has been observed that the values of S P1 , S P2 , H ′ 1 , and H ′ 2 are increased with the increase in β. These combined reasons and all these criteria in both crystallization stages are utilized to determine the stability in terms of the β in the studied glassy alloy. In the sense that the studied system has two stages of stability but the first stability does not last long and the studied system is more stable at the second crystallization stage. Compared to other studied compositions of As-Te-Ga glasses, the increase of Ga or decrease in Te enhances the thermal stability of the formed glass [16,[39][40][41].

Hrubý Criterion and the Glass-Formation Factor
Hrubý's criterion ( H r ) which utilized to determine the thermal stability of the glass via the characteristic temperatures according to the following relation [44]: On the other side, the glass-formation factor or ability parameter (k H ) of the investigated system is given by: The k H a parameter used to measure the glass-forming tendency, for an instant, the higher values of criterion parameters reveal an enhancement in the thermal stability of the formed glass. The values of all these parameters are estimated for the As 30 Te 64 Ga 6 and summarized in Table 2. Both of these parameters (H r and k H ) are enhanced by increasing the value of β. Figure 4 plot shows the exothermic heat flow ( ΔQ exo ) against temperature (T) at β = 5 K min −1 in the first crystallization region, as an example. In Fig. 4, T is the crystallized fraction at a given T expressed by the relation T = A T ∕A, where A is the total area of the peak and A T is the area of the crystallized part (the area between T i and a given temperature T). Both T i and T f are the initial and final crystallization temperature, respectively.

The Exothermic Heat Flow Analysis
The same way, the area under the crystallization peak is directly proportional to the total amount of the alloy being crystallized. The ratio between the selected coordinates and the total area of the exothermic peak shows the corresponding crystallization rates, making it possible to plot the curves between the volume crystallization fraction and the temperature.
The experimental graph of the crystalline volume fraction (χ) shows the typical sigmoid curve as a function of T at different β for the studied system as shown in Fig. 5a  where s is several points of ΔQ exo , (namely, s = 0 to 99). Now, the relationship between crystallization rate d dt and the temperature produced by differentiating of the volume crystallized fraction as a function of time (t), which given by the following equation for the maximum crystallization rate: Figure 6a and b present the crystallization rate d dt as a function of temperature for the first and second crystallization peaks, respectively. The values of the maximum crystallization rate evaluated using Eq. (17) and listed in Table 3. From Figs. 5a and b and 6a and b, one can conclude that the saturation of crystallization shifts towards the higher temperature as the β for the As 30 Te 64 Ga 6 glassy alloy increases.

The Glass Transition
The dependence of T g on β in the glass transition region could be estimated via two basic formulas; the first equation is the empirical equation originally proposed by Lasocka [45] as follows: where, A g and B g are constants depending on the studied glass alloy. The noticed value of A g articulates or indicates the T g at β = 1 K min −1 , while the observed value of B g indicates the temporal response to the configurational changes within the glass transition region. The validity of this relation to the studied system is shown in Fig. 7. The values of A g and B g constants are computed from the intercept and slope of the fitted line and equal to 5.95 ± 0.08 K, and 383.10 ± 0.21 min, respectively.  Continuously, the apparent activation energy of the glass transition (E t ) of the investigated glassy alloy has been estimated using Kissinger's formula [46] and the approximation of Moynihan et al. [47]: (20) ln ( ) = − E t RT g + const. Figure 8 plots the relationship between ln T 2 g against 1000/T g and also the plot of ln ( ) versus 1000/T g for As 30 Te 64 Ga 6 chalcogenide glass. From the slopes of these fitted lines, the activation energy for the glass transition E t was computed and equal to 213.93 kJ mol −1 (51.18 kcal mol −1 ) and 220.54 kJ mol −1 (52.78 kcal mol −1 ) using Kissinger`s formula and approximation of Moynihan et al., respectively. The observed values of E t lie in the range generally observed for other chalcogenide glasses [42,48,49]. The slight difference between the value of E t estimated from the two methods could be related to the individual and proposed approximations.

Fragility Index
The liquid fragility is an important parameter that has beneficial as a basis for organizing data on glass-forming liquids and then facilitating the investigation of relaxation processes and glass transition in the super-cooled liquid state. The fragility index (F) can be computed via the subsequent relation [50,51]: where E t is the activation energy for the glass transition is estimated based on Kissinger's equation. Figure 9a shows the plot of the liquid fragility as a function of ( 1000∕T g ) according to Eq. (21). From this figure, we note that the value of F increases with the decrease of T g .  Fig. 7 The test of the validity of relation's Lasocka for bulk As 30 Te 64 Ga 6 glass alloy at different heating rates (β) Fig. 8 The plot of ln T 2 g versus 1000/T g and the plot of ln ( ) versus 1000/T g for bulk As 30 Te 64 Ga 6 glass alloy at different heating rates (β)

The Cooling Rate
The dependence of the cooling rate (Q c ) on the T g measured during cooling obeys the following equations [52][53][54]: The plot of the cooling rate as a function of the T g is shown in Fig. 9b. We note from this figure that the cooling rate increases with increasing the T g .

Activation Energy Assessment by the JMA Model
The DSC thermograms have been analyzed by several methods to determine the activation energy of amorphouscrystalline transformation ( E c ), the Avrami exponent (n), the frequency factor ( K 0 ), and other constants for the investigated glassy alloy.

Kissinger and Mahadevan Methods
The activation energy of the amorphous-crystalline transition is originally computed utilizing Kissinger's formula [46] as follows: values of E c , which are computed from the slope of a straight line, equals 50.24 and 85.87 kcal mol −1 for the two exothermic peaks (peak1 and peak 2), respectively.

Augis and Bennett Method
The activation energy was estimated using a method intended by Augis et al. [55] according to the following equation:

T g [K]
Fig . 9 The plot of the liquid fragility (F) versus 1000/ T g and plot of the cooling rate (Q c ) as a function of the glass transition temperature ( T g ) for bulk As 30 Te 64 Ga 6 glass alloy at different heating rates (β) Figure 12 shows the relation between ln

Gao and Wang Method
The activation energy of the As 30 Te 64 Ga 6 alloy has also been evaluated through the method used by Gao et al. [56]. In this method, the maxima of the DSC curves are used and the relationships deduced from the following equations [56]: Here is the volume fraction of crystallization, and ( d dt ) p is the maximum crystallization rate which is proportional to Fig. 12 The plot of ln 1 3 the peak height. Figure 13 shows the plot of d dt p versus 1000 T P . According to this method, the value of E c for the studied glassy is computed and it equals 51.17 and 86.99 kcal mol −1 for the two exothermic peaks (peak1 and peak 2), respectively. The crystallization reaction rate constant, K p corresponding to T P was computed from the thermo-grams by Eq. (26). The values of K p1 , ⟨ K p1 ⟩ for the first exothermic peak and K p2 , ⟨ K p2 ⟩ for the second exothermic peak are computed and listed in Table 3. Here, ⟨ K p1 ⟩ and ⟨ K p2 ⟩ represent the average of crystallization reaction rate constants for the exothermic peaks at different β.
On the other hand, one can compute the kinetic exponent from the experimental values of the d dt p by the subsequence equation: Here 1 and 2 refer to the first and second crystallization peaks, respectively. The values of n 1 and n 2 for studied glassy alloy are computed and listed in Table 3.
The numerical values of the experimental data, T p , and  27), the kinematic exponent, n, for each of the experimental β for the peaks for the As 30 Te 64 Ga 6 , whose values are also presented in Table 3. The kinetic exponent n was calculated based on the crystallization mechanism of (27) Mahadevan et al. [42] where it was shown that n may be 4, 3, 2, or which correlated to different glass-crystal transformation mechanisms: n = 4, volume nucleation, three-dimensional (3D) growth; n = 3, volume nucleation, two-dimensional (2D) growth, n = 2, volume nucleation, one-dimensional (1D) growth; n = 1, surface nucleation, 1D growth from the surface inward. Therefore, taking into account the average values obtained above, ⟨n 1 ⟩ = 4.58 and ⟨n 2 ⟩ = 4.81 (as shown in Table 3) for the two crystallization peaks, mean size nucleation, 3D growth. The computed n values are not integers. This means that the crystallization occurs through more than one mechanism together [57,58].

Matusita Method
Furthermore, the activation energy of the investigated glass was computed via the Matusita relationship [59]: where (m = n − 1) is the dimension order parameter and (n) is constant related to the crystallization mechanism.

Kinetics Parameters of the Crystallization
The reaction rate constant or the kinetic parameter has a temperature dependency based on the Arrhenius equation: where E c is the crystal growth effective activation energy, K 0 is the pre-exponential factor or so-called the frequency factor, and R is the universal gas constant (8.314 J K −1 mol −1 ). Besides, a kinetic parameter, K(T), with an Arrhenian temperature dependence, describes the stability parameters. According to Surinach et al. [60] and Hu and Jiang [61] the thermal stability of glassy materials estimates via two criteria as follows: The values of these two parameters indicate the tendency of the glass to dissolve upon heating, whereas the glass formation is a kinetic process. The higher their values, the greater the tendency to depreciate.
According to the obtained values of E c and K 0 , that resulted from the Kissinger method, the kinetic parameters K(T) of the studied alloy were computed by Eq. (30). These calculations were made to compare the stability sequence of the studied substance from the parameters quoted with the corresponding sequence computed from the stability parameters based on the characteristic temperatures. The K T g and K T p as a function of T g and T p , respectively, for the (29) studied sample are shown in Fig. 15a and b. According to the previous work [62,63], the smaller the values of K T g and K T p parameters, the better the ability of the material to form glass. So the data for both K T g and K T p indicate that the thermal stability decrease with increasing β.
The parameter H r is a stability factor depends on characteristic temperatures. Here, a stability criterion is defined as follows [64]: where T is an arbitrary temperature between T g and T p . The theoretical definition of the parameter K r (T) has relied on the analysis of the relation between the parameters K(T) and K r (T). By differentiating Eqs. (29) and (31) concerning temperature and rewriting each parameter results, one can get the following relations: Then, using Eq. (31), the data for K r (T g ) and K r (T p ) were also calculated as a function of as shown in Fig. 16 confirming that the stability orders increased with increasing β.
It can be concluded that the change in the parameter K r (T) is equal to H r times the parameter K(T) . Therefore, similar to K(T) criteria, the smaller the K r (T) values, the greater the thermal stability of the glass. The obvious advantage of this method is that, it can evaluate the stability of the glass over a wide range of temperatures other than only one temperature, such as T g or T p . Then, using Eq. (31), the data of K r T g and  Fig. 15 The plot of a K T g , and b K T p as a function of T g and T p , respectively, for the As 30 Te 64 Ga 6 sample K r T p were also computed as a function of β as shown in Fig. 16 confirming that that the stability orders increased with increasing β.

Conclusions
In summary, the chalcogenide of the As 30 Te 64 Ga 6 bulk alloy is prepared using the melt-quenching technique. Fitting of the kinetic model to isothermal transformation-rate data, is exhibited by DSC scans. The characteristic temperatures (T g , T c , T p , and T m ) were determined, and hence the glass transition activation energy, amorphous crystallization, and crystallization transformations. The kinetics of the crystallization was studied by non-isothermal DSC, XRD, and the values of the activation energy and Avrami index values were calculated by the Matausita method also the activation energy was deduced by other methods. The results appear strong dependence on the heating rate of the activation energy for the crystalline peaks. Avrami exponent is close to 4.58 and 4.81 thus corresponding to complex processes that include different mechanisms. On the other hand, the thermal stability for As 30 Te 64 Ga 6 glassy alloy at different heating rate has been computed via various criteria. The K r (T) criterion has been considered in the present framework for the determination of glass stability based on DSC data. The obtained results of the K(T g ), K(T p ), K r (T g ), and K r (T p ) agree satisfactorily with the ΔT, H r , H ′ and S criteria for the studied glassy alloy. The observed data of the thermal stability criteria indicate that the studied glass is the most stable, and the stability orders at various heating rate increases as the criterion parameters have high values.  Fig. 16 The plot of a K r T g , and b K r T p as a function of T g and T p , respectively, for the As 30 Te 64 Ga 6 sample