Pre-Crystallization Criteria and Triple Crystallization Kinetic Parameters of Amorphous–Crystalline Phase Transition of As40S45Se15 Alloy

This framework focuses mainly on a detailed study of the pre-crystallization criteria that characterize the As40S45Se15 glassy alloy in various heating rates ranging from 5 to 40 (K/min) by Differential Scanning Calorimetry (DSC). These criteria aim to clarify the relationship of the tendency of glass-forming by the heating rate for the investigated glassy alloy. The crystallization parameters were calculated using different methods.The activation energy of crystallization Ec(χ) as a function of conversion (χ) was obtained using the iso-conversional models of Flynn–Wall–Ozawa (FWO), Starink and Kissinger–Akahira–Sunose (KAS). The results show a slight increase of Ec(χ) with conversion (χ) which accounts for a single-step mechanism controlling the crystallization process. Moreever, the conversion dependence of the Avrami exponent n(χ) show an increase with conversion (χ), average values of n(χ) can be accounted for two and three-dimensional crystal growth with heterogeneous nucleation. On the other hand, the fitting of the experimental DSC data to the calculated DSC curves indicated that the crystallization process of the studied glasses cannot be satisfactorily described by the Johnson–Mehl–Avrami (JMA) model. On the contrary (SB) model is more suitable to describe the crystallization process for the studied of As40S45Se15 Alloy. Finally, the crystalline structure of the study sample was recognized by X-ray diffraction (XRD) and electron scanning microscope (SEM).

Chalcogenide may be prepared either in a crystal form or in an amorphous form. Commonly, the crystallographic state of the solid is restricted to a few frame structures, while the amorphous Chalcogenide is meaningful in amorphous systems regarding the efficacy as excellent materials for optical ultrafast nonlinear devices such as demultiplexer (or demux), wavelength adapters and Kerr optical shutters [7,8].
The structure of chalcogenide components has been extensively studied in binary compositions, whether bulk or thin-film. Several binary substances can be formulated by attaching one of the chalcogens with another element like As, Sn, Ag, In, Pb, Al…etc. [2,9,10].
The molecular frameworks of As-S, As-Se binary glasses premised on the phase diagrams for these systems [11] were investigated by Abrikosov and his co-workers. Thus, As-S glassy system can be formed when As element concentration 1 3 is high in the chemical compound to be obtained comparing to the concentration of sulfur element which is combined with the arsenic (As) element in the alloy which prepares it. In contrast, the As-Se system formed when the concentration of As element increased to almost 60%. Low the content of As of glassy system can transfer easily to the crystallization materials, for an example, if the As concentration is almost 6%, the glass will crystallize at room temperatureat 298 °C-in a short time, namely, almost 24 h in weight equals to 5-16%, in a few days at 60 °C while it appears to take one month for As-S at 280 °C [11] As-Se glass composites can crystallize throughout the entire compound range, but this was to be completed under pressure and at high temperatures.
Ternary glass alloys could be formulated by inserting an appropriate additive element into the binary matrix. A ternary (As-S-Se) system has a very broad glass-forming region [12][13][14][15][16][17]. Solid solutions could be developed across the line As 2 S 3 -As 2 Se 3 , which has been proven by Velinov and his coworkers through IR spectra and x-ray evaluation [10].
As-rich glassy alloys can be formed in several bands with another As atom or with Se and S atoms. The same thing happens with selenium and sulfur, so we find that the atoms are either bound to similar ones or with non-similar atoms. The primary weighted sum of each of the above-mentioned bonds is anticipated to be directly proportional to the overall look of the glass itself [18][19][20][21].
Great efforts have been done to study (As-S-Se) systems in all research aspects such as the optical, electrical, and thermal properties [22][23][24].
The focus of the current research is to broaden the past findings and to study in detail these alloys to clarify the structural and thermal properties of them where the knowledge of crystallization kinetic parameters, the glass-forming, and also thermal stability, is very important for the usage of these glassy alloys in modern technology, to assess their applicability. Furthermore, the experimental investigation of the kinetics is analyzed with different theoretical methods, including the iso-conversional approaches, which allow getting more information on the reaction mechanism. Among the widely used iso-conversional approaches, those developed by Flynn-Wall-Ozawa [25,26], Starink [27], Kissinger-Akahira-Sunose [28,29] The analysis by such method could help not only to get understanding of the structure of chalcogenide materials but also the development and improving of these materials to be suitable for different technological applications. For our knowledge, the thermal analysis, in particular the thermal stability as well as the crystallization kinetics of As-S-Se glasses have not been reported yet.
The target of the present framework the crystallization kinetic parameters corresponding to the crystallization process of the studied glassy alloy and the thermal stability, ability of glass-forming, and other thermal criteria on different heating rates of As 40 S 45 Se 15 glassy alloy are investigated. The activation energy of glass transition and crystallization are calculated using different methods. The crystallization activation energy E c (χ) as well as the Avrami exponent n(χ) were calculated and discussed according to some iso-conversional methods. In addition, both Sestak-Berggren (SB) and Johnson-Mehl-Avrami (JMA) models were applied to analyze the crystallization kinetics of the As 40 S 45 Se 15 composition. The nature of the structure is determined in terms of X-ray diffraction (XRD) and (SEM) measurements.

Experimental Methods
The bulk of As 40 S 45 Se 15 was formed via the conventional melt-quenching method. High purity components, As, S, and Se (purchased from Sigma-Aldrich, 99.999%) were measured depending on their atomic ratios utilizing the type of electrical balance. The three elements were placed according to their proportions in highly sealed silica tubes in pressure equals to . After that, the studied samples were heated in a vibrating furnace to ensure homogeneity of the mixture formed, and the heating temperature continued for 32 h at a temperature of 1000 °C. On the other hand, the powder sample was produced from the grinding of the yielded bulk samples in a mortar. The compositional contents of the prepared sample were checked by the EDAX technique. The proportion ratios of the constituent parts in bulk samples are close to the measured ratios for the thin films. The nature of the structural the As-obtained alloys was identified by a Phillips X-ray diffractometer. The XRD data were carried out within the range of 2θ = 5°-75°. The X-ray diffractometer is composed of Cu K α radiation source with a graphite monochromator where λ = 1.54178 Å. The crystallization kinetics was monitored by DSC (Model: TA-Q20). For this purpose, ~ 10 mg of the bulk glass sample was utilized. The measurement was carried out for different compositions ranging and temperature scan in the range of (295-575 K). The temperature of glass transition, T g , the beginning temperature of crystallization, T c , the peak temperature of crystallization, T P and the melting temperature T m were deduced by the thermal analyzer. The thermal annealing process for the investigated alloy was performed under the vacuum of 10 -5 torr at numerous temperatures (370, 390, 400, and 410 K) for 30 min. Such temperatures are selected to start lying between the transition temperature and the amorphous-crystal transition areas that occur in the differential calorimeter scanning. The amorphous-crystallization transformation is clearly shown in SEM analysis.

Compositional Analysis
The EDXA spectrum diagram for the researched alloy demonstrated a constitutive element percentage is close to the nominal composition. The percentage ratios of the constituent elements are As 40 ± 0.3 S 45 ± 0.1 Se 15 ± 0.6 depending on the measurement performed utilizing electron microprobe analysis. Figure 1a offers the EDX analysis for the investigated sample.

DSC Analysis
To investigate the pre-crystallization criteria and the crystallization kinetic parameters of melt-quenched As 40 S 45 Se 15 chalcogenide alloy at various heating rates from 5 to 40 (K/ min) and for confirming its glassy nature, DSC of the powdered of As 40 S 45 Se 15 glassy alloy is carried out as illustrated in Fig. 1b. DSC curve represents the scheme of thermal transitions in a semi-crystalline solid material that provides the thermal information for the heat flow in terms of temperature. The thermo-gram DSC-in our framework-illustrates four characteristic phenomena arrange from lowest to highest temperature. The first characteristic corresponds to the glass transition temperature, (T g ) which is associated with absorbing energy to overcome the rigidity of the lattice. The second characteristic corresponds to the beginning temperature for the crystallization, T c . The third one describes the top crystallization temperature, (T p ). The last characteristic represents the melting point, (T m ). As mentioned, the DSC thermo-grams reveal a single glass transition and a single exothermic crystallization peak. The obtained values for such quantities are reported in Table 1. From Table 1, it is observed those values were shifted to a higher temperature as heating rates increased. This observation confirmed that all these transitions and transformations are thermally effectual. The observation of one endothermic peak for the melting phenomena could be attributed to the formation of one phase through the heating which is confirmed by the XRD analysis. The existence of a single glass transition temperature reveals that the formed samples are more stable than those glasses revealing multi-glass transition temperatures.

Pre-crystallization Criteria
There are sundry criteria used to know the glass formation ability (GFA). Based on the nucleation theory, Kauzmann and later Turnbull [30,31]  utilized as a good criterion to deduce the ability of glass formation. such quantity is defined as the ratio between the temperature of glass transformation, T g to the temperature of melting point, T m . Such ratio is called a lowered glass transition temperature, T rg , (where T rg ≥ 2/3) and generalized by the subsequence relation [30]: The ability to form glass increases with increasing values T rg more than 2/3 until the value of the criterion approaches one, at which time we describe the ability to form a glass as super [30]. The obtained values of T rg are summarized in Table 1. It has been observed from the obtained values of T rg that they decreased as the heating rate increased, and this, in turn, indicates that the ability to form a glass decreases as the heating rates increase. Thus, the lower the T rg value is to 2/3, the lower the material's glass-formation ability, GFA [30,31].
The common criterion utilized also to determine the glass thermal stability in term of crystallization is Deiztal's criterion, which is defined by the subsequent relation [32]: The kinetic resistance of crystallization is determined utilizing the difference ΔT = T c − T g [32,33]. Such difference is also called the thermal stability of the investigated glass alloy. The glass system becomes more stable as the ΔT increases. On the other hand, the temperature difference ΔT is a perfect signification of the thermal stability because the higher value of this difference gives a higher resistance of crystallization and thus more the retard in various the nucleation processes [33].
The glass alloy with lower ΔT values is predicted the greater electrical conduction [34][35][36]. The obtained values of ΔT are reported in Table 1. Such values increase with the increase of the heating rates (as presented in Table 1). In other words, the thermal stability against crystallization is getting good with increasing the heating rates in the (2) ΔT = T c − T g studied system. Thus, the investigated glassy alloy is more stable whenever the heating rate increases.
As well, there are other two criteria utilize to evaluate thermal stability, which are suggested by Saad and Poulain [37]. The first one is called S-parameter (S P ) and the second is called the weighted thermal stability criterion (H � ) , they are given by: Here S P refers to the resistance to devitrification after the glass has been formed. The difference between, T p , and T c in Eq. (3), is linked to the rate of devitrification changes of the glassy states. The obtained values of S P and H′ are presented in Table 2. It has been noticed that such quantities increase with an increase in the heating rates. These criteria in the crystallization stage are utilized to determine the stability in terms of the heating rates for the investigated glassy alloy.

Hruby Criterion and the Glass-Formation Factor
Hruby's criterion (H r ) which utilized to deduce the probability thermal stability of the glass alloy via the characteristic temperatures depending on the following equation [38]: the probability of the thermal stability of the glass which rises as the difference T m − T p reduces and ΔT rises. The obtained values of H r indicate that the thermal stability of the glass alloy is almost improved since the values of H r are moving away from 0.1 whenever the heating the rate increases, where if the criterion H r is less or equals 0.1, the thermal stability will be difficult, while good the thermal stability can be obtained if the criterion H r is more or equal to 0.4 [38]. In our case, the thermal stability of the glass is improved because such values are close to 0.4 and also it exceeds 0.4 (see Table 2).
On the other side, the thermal stability factor or ability criterion (k H ) of the investigated system is expressed by: The k H criterion is used as a measure of the thermal stability tendency, for an instant, the greater values of k H reveal the greater thermal stability of the formed glass alloy. The values of all these parameters are estimated for the studied glassy alloy and then summarized in Table 2. Both of these criteria (H r and k H ) are perfective with increasing the heating rate. Thus the ability to the thermal stability of the glass is progressed. Generally, the volume fraction crystallized (χ) presented the typical sigmoid curve in terms of temperature at various heating rates as demonstrated in Fig. 3, which expressed as follows [24]:

The Crystallized Fraction at a Given Temperature
where s is the total points of exothermic heat flow, ΔQ exo . Based on Eq. (6), the value of χ varies between 0 and 1 corresponding to a minimum and maximum crystallization fraction, respectively.
Mathematically, we can express the surface resistance of the bulk alloy in the crystallization region, namely, in ΔQ exo the region between onset and end-set of crystallization as follows: The offset values of sheet resistance in terms of temperature, T are yielded according to Eq. (7) and plotted in Fig. 4. One can be observed that the bulk sheet resistance rises with increasing the heating rate for the studied glassy alloy. Now, the relation between the crystallization rate (d ∕dt) and the temperature given by the subsequence relationship for the crystallization rate [24]: And for the maximum crystallization rate: Figure 5 represents the crystallization rate (d ∕dt) in terms of temperature, for the exothermic crystallization peak. The values of the maximum crystallization rate were evaluated utilizing Eq. (9) and reported in Table 2. From Figs. 3, 4, and 5, one can conclude that the crystallization concentration shifts toward the higher temperatures as the heating rate for the glass alloy under investigation increases.

Validity of Lasocka's Relation
The viability of the relationship's Lasocka can be tested in the glass transition region according to the following [39]: where A g and B g are constants for a given glass composition and β is the heating rate utilized in DSC thermo-grams. The computed value of A g indicates the T g for 1 K/min whereas the obtained value B g indicates the time response of changes in the glass transition area. Figure 6 offers the validity of such a relation. The A g and B g constants equal to (333.35 ± 0.046 K) and (6.18 ± 0.016 min), respectively. One can obtain that values of the glass transition temperature, T g for the investigated glassy system increase with increasing heating rate, as a result of the system doesn't get abundant time for full nucleation and growth [39,40]. On the other hand, the heating rate linked to the relaxation time ( g ) for glass transition of atoms via the following equation [40]: Here ΔT g is the difference between T g (endset) and T g (onset) which obtained from DSC thermo-gram curves. The ΔT g is the temperature bandwidth describing the liquid-glass transition region. The observed values of such quantities are reported in Table 1. Figure 7 plots the relationship between the relaxation time, g and for the investigated system. It is observed that the value g reduced when raised. Thus, one can obtain that the increasing of the ΔT g with increasing denotes a slower process for glass transition.

The Activation Energy of the Glass Transition
Depending on Kissinger's formula [41] and its approximation obtained by Moynihan et al. [42], the activation energy E t of the glass transition has been extracted from the slope of the following relationships: Figure 8 depicts the extrapolation of ln T 2 g versus 1000/T g and also the extrapolation of ln (β) against 1000/T g

Fragility Index in the Glass Transition Region
The fragility index in the glass transition zone, (F) is obtained via the subsequent relation [45,46]: where E t is mentioned above and this quantity yielded from Kissinger's relation. Figure 9a offers of the liquid fragility, F in terms of (1000/T g ) depending on Eq. (14). From such a figure, one can obtain that the value of F increases as T g decreases. In addation, the dependence of the index F i of As 40 S 45 Se 15 Alloy on the heating rate is listed in tabile. 1. It is found that the avarege value F i ≈ 22 for the studied composition. This means that the prepared  Fig. 9 The plot of the liquid fragility F as a function of. (1000/T g ) (a) and also the cooling rate (Q c ) as a function of the glass transition temperature T g (b) for the studied glassy alloy As 40 S 45 Se 15 Alloy sample correspond to the strong glassforming liquids. Moreover, this composition is considered as the most stable glass [47].

The Cooling Rate in the Glass Transition Region
The relationship between the cooling rate (Q c ) and T g measured during cooling obeys the subsequence equation [48][49][50]: The extrapolation of the cooling rate ain terms of T g is presented in Fig. 9b. From this figure, the cooling rate has the same behavior as the glass transition temperature.

The Crystallization Kinetic Parameters
Mainly, DSC thermo-grams have been analyzed by several methods to compute the triple kinetic parameters for the crystallization, namely, the activation energy, E c , Avrami index, n, and the frequency factor, K 0 . The triple kinetic parameters for crystallization have been computed in the following sections:

Kissinger Method
Utilizing the equation derived by Kissinger [41], the activation energy E c of the transition from an amorphous to the crystalline state for the studied system is extracted: Figure 10 offers the extrapolation of ln T 2 p against (1000 T p ) for the investigated system. The activation energy E c and frequency factor K 0 are then extracted by the famous least-squares fitting method from the slopes and intercepts of the straight lines, respectively. The values E c and K 0 equal to 85.98 kJ/mole and 4.5 × 10 −6 (min −1 ). On the other hand, the activation energy E c has been computed also utilizing the approximation of Mahadevan et al. [33] as:

Augis and Bennett Method
To confirm the results, the E c was computed utilizing the method intended by Augis et al. [51] as: Figure 11 offers the extrapolation between ln T p versus (1000 T p ).The E c equals to 89.24 kJ/mole in this (18) ln T p = −E c RT p + const.

Gao and Wang Method
The E c of the analyzed alloy has also been computed through the method utilized by Gao et al. [52]. In this model, the following is achieved [53]: where is the volume fraction of crystallization as we stated, and d ∕ dt p is the maximum crystallization rate.  Here, β represents the heating rate as we know and is the full width at half maximum (FWHM) of the exothermic peak. The values for the investigated glassy alloys are reported in Table 2. The numerical values of the experimental data, and d ∕ dt p , are present in Tables 1 and 2, respectively, and the value of the activation energy of the crystallization process for the crystallization peak which gets from the Kissinger method (85.98 kJ/mole), make it possible to be deduced, through Eq. (16), the kinetic exponent, n, for the studied glassy alloy, whose values are also summarized in Table 2 have been computed based on the mechanism of crystallization Mahadevan et al. [33]. In our case, the obtained average values, < n >= 3.38 based on Eq. (20) and < n >= 2.53 according to Eq. (21) (as shown in Table 2) for the crystallization peak, means (volume nucleation, 2 D growth). The determined Avrami index has not integer values, which implies that the crystallization takes place by more of one technique [54,55].

Ozawa Method
In addition to the two Eqs. (20,21), Avrami exponent, n at selected temperature computed also by the method suggested by Ozawa [56]. In this method, the values of n are computed from the slopes of yielded straight lines [56] (see Fig. 12): The average value of n for the studied composition at T = 416 and T = 425 K equal to 2.69 and 2.82, respectively.

Matusita Method
Efficient activation energy needed to achieve crystallization, thus, we resort to using the Matustia equation to compute this energy as given [57]: where (m = n-1) is the dimension order parameter and (n) is constantly linked to the crystallization process. Figure 13 offers against at various heating rates. We take into consideration the linear region of this plot. The best linear region that we must take into account is the region in which the crystallization energy takes the lowest value and which products in most cases when values of the vertical axis are confined between − 2 and 2. From the average n values and mE c , where (n = m + 1), the E c for the studied system are listed in Table 2, where (n = 2.53 and m = 1.53). Moreover, one can see that the plots in Fig. 13 are linear over a wide (22)

Fig. 12
The plot of ln(− ln(1 − )) against ln( ) at two constant temperatures for the studied glassy alloy temperature range. At higher temperatures a break in the linearity for all heating rates is observed, which could be considered due to the saturation of the nucleation sites at the final stage of crystallization [58,59]. It is always noted that the resulting activation energy from the application of the Matusita's equation is much greater than that produced by the Kissinger's equation and others. The reason for this due to the resulting activation energy from the Matusita's equation is a produced effective energy without a selection of one temperature only while the one temperature is determined in the Kissinger's equation and other methods (we use T p ).

The Yield Parameters from Arrhenius's Equation
Depending on Arrhenius's extrapolation, Surinach et al. [60] inserted K(T g ) parameter and Hu and Jiang [61], inserted K(T p ) parameter as follows (see Fig. 14a): where is accurate activation energy for the crystallization process in case of the growth, is the frequency factor, and R is the general gas constant, (8.314 J/mole K). The calculated values for such two criteria allude to the predisposition of glass to devitrify on heat treatment. Based on the literature [62,63], the smaller the values of such two criteria, The glass-forming abilities of the material should be better. The parameter of H r is linked to Arrhenius's extrapolation as given [61]: where: T is any temperature between T g and T p . Thus, by utilizing Eq. (25), the information of K r (T g ) and K r (T p ) were determined in terms of heating rates (see Fig. 14b). The results confirm that the order of stability rise with rising the heating rate. Fig. 14 The plot of K(T g ) and K(T p ) as a function of T g and T p (a) and the plot of K r (T g ) and K r (T p ) as a function of heating rates (b) for the studied sample

Classification of the Iso-conversional Methods
There are several iso-conversion methods or models [64,65] to compute the local activation energy, E χ , which vary in the integral temperature approximations. as given: Such approximations are given in the following general form [28]: where i is the heating rate, E c ( ) is the local activation energy, and χ is the volume of the crystallized fraction. On the other hand, there are three types of approximations depending on the values of B and C parameters in Eq. (27) determined the type of the temperature integral approximation as follows:

The
First Approximation for B = 0 and C = 1.052 Such approximation is called Flynn-Wall-Ozawa {FWO} model [25,26] and it's relationship given by: The extrapolation of ln( i ) versus 1000/T χ for the fractional conversion in the range of [0.1, 0.9] is illustrated in Fig. 15. The local crystallization activation energy which computed from the slops of the yielded straight lines and summarized in Table. 3.  Table 3.

The Third
Approximation for B = 2 and C = 1 for B = 2 and C = 1, the Kissinger-Akahira-Sunose {KAS} are yielded [29]: Figure 17 presents the plot of function fitted and from the slops of the straight lines; the values of E c ( ) are reported in Table. 3. In general, the average E c ( ) values for the three mentioned approximations are 88.92, 86.79, and 86.58 kJ/mol, respectively. Such results are agreement with other results that produced form the isothermal models. Also, the dependence on the conversion (χ c ) helps to recognize the kinetic pattern of the thermal processes of decomposition [66,67]. In general, the values of being independent on the conversion (χ c ) within the range of 0.3 ≤ χ c ≤ 0.8. however, some variations occurred for lower and higher (χ c ) values, especially for faster processes. Fig. 15 The plot of ln( i ) versus 1000/T χ where 0.1 ≤ χ ≤ 0.9 for bulk As 40 S 45 Se 15 glass alloy These variations are possible due to the errors in the baseline approximation at the peak tails [59]. Generally, for all the three methods its noticed from with (χ c ) can be accounted for the one-step mechanism controlling the crystallization process [68,69]. Moreover, the local Avrami exponent n( ) is computed depending on the following relationship [70,71] (see Fig. 18): Using the computed E c ( ) , the local Avrami exponent n( ) at 10 K/min was computed via Eq. (31). The reduction in values of n indicates the reduction in the nucleation rate attributed to saturation [72,73]. In addition, It is clear that the value of n in the range 2.63 ≤ n ≤ 2.76. The variation of n with transformation fraction could be attributed to the volume nucleation followed by two-dimensional growth. Additionally, this process might be controlled by diffusion as well as an increase of the nucleation rate of bulk crystallization. This agrees with the results obtained using the previuse mentioned methods in the present work [73,74].

The Validity of JMA or SB Model
To describe the crystallization process, there are two suggested functions y(χ) and z(χ) as given [75]: Depending on JMA and SB models, the theoretical heat flow, is given by the subsequence equation [76]: Here, ΔH c is the enthalpy of crystallization (see Table 2), A is frequency factor in the exponential region, E c is total activation energy, and χ represents the degree of conversion, and f ( ) is the analytical expression of the kinetic model function. The analytical functions that characterize the JMA and SB model can be expressed by the subsequence relations, respectively [77][78][79]: Here n is connected to formation and crystal growth processes, while M and N are variables that indicate the unique influence of the acceleratory and decomposing areas to the process of transformation, respectively.
If the values of χ p ∞ for z(χ) curves fall into the range from 0.61 to 0.65, the JMA methodology is appropriate for describing the crystallization process [80]. While if 0 < χ M < , where is the higher value in the y(χ) curve, SB model in this case is utilized to describe the thermal process [77]. The y(χ) and z(χ) curves are normalized in the range of (0, 1), so that, one can be expressed the two functions as follows (see Fig. 19): To determined the curve of y(χ), the obtained average values of E(χ) computed by the KAS model were utilized. The values of χ M and χ p ∞ correspond to the maxima of such functions are reported in Table 4. The noticed values decrease with increasing heating rates in the studied framework as presented in Table 4. Besides, the value of χ p ∞ is substantially lower than the so-called fingerprint value of the JMA method for the alloy under investigation at all heating rates. Thus, it can be concluded that the JMA model is not appropriate for the study of crystallization kinetics for the alloy under investigation. Also, the general results χ p ∞ values are lower than 0.632 indicating a complicated and accelerated crystallization techniques [81]. Such crystallization behavior could be particularly characteristic in the evidential SB (M, N) method. M and N are two main parameters that identify the unique influence of the accelerator and decomposition components of the crystallization operating [82]. The mentioned two parameters are computed in terms of χ M [83] as given: According to the average values of E c , computed by the KAS model from Eq. (35) and values computed from Eq. (38), the N values are extracted via the following equation [82].
The validity of this equation is for falls in the range of (0.   Table 4 summarizes all the mentioned parameters.

The Theatrical DSC Extrapolation
The theatrical DSC curves depending on the SB and JMA models are extracted (see Fig. 21a, b). The theoretical DSC curves for JMA model do not agree with the experimental curve in all heating rates for the investigated alloy. On the contract, the DSC curves computed via the SB model as shown in Fig. 21b are fully applicable to the experimental curve at all heating rates. This indicates that the nucleation process appear at the early stages of crystallization transformation and are neglected at the final stages [84]. According to these results, one can say that the rate of crystallization is dependent on the thermal history of the of As 40 S 45 Se 15 alloy [85]. On the other hand, the nonvalidity of the JMA could (41) ln exp E c ( ) RT = ln (ΔHA) + ln n(1 − )[− ln(1 − ) 1−1∕n ] be due to the approximations applied for the derivation of Eq. (35) are not suitable for analyzing the crystallization of of As 40 S 45 Se 15 Alloy. From the last discussion, one can say that the SB model is the most appropriate model for quantitatively describing the process of crystallization in the studied alloys. It can be obtained that some values of M and N correspond to a given value kinetic exponent (when n ≥ 1) for the JMA model are accepted to describe the process of crystallization [80].

Identification of the Crystalline Phases by X-ray and SEM Analysis
The X-ray diffraction trends of the glass alloy are used to identify potential phases that crystallize throughout heat processing at 370, 390, 400 and 410 K for 30 min (see Fig. 22a). For this purpose, Fig. 22b offers the most pertinent parts of the diffractometer traces for glass (after T g ). The curve in Fig. 22a has a wide hump personality trait of the amorphous state in the angle range from 5° to 70°. The existence of one amorphous hump (at 370 K) can be Max. "(y( ))" Max. "(z( ))" SB JMA  (n−1) n (b) for the studied system 1 3 described as a finding of one amorphous state of glass. The micrographs of the changed material after the crystallisation process identifies the existence of micro-crystals of As 2 (Se, S) 3 has the monoclinic crystal system as the dominant phase depending to JSTM (No. Card: 040-0469) [86] at 410 K, see Fig. 22a. Figure 22a obtains that the intensity of the peaks rise with rising temperature of heat treatment, while there remains also a residual amorphous phase. The results of the scanning electron microscope analysis were completely consistent with the results of the X-ray analysis (see Fig. 23). From such a figure, it can be seen that the size of the crystallisation increases with an increase in the annealed temperature, and that the created crystals also take a similar shape to the circular particles.

Conclusions
In this framework, the studied chalcogenide is prepared by the melt-quenching technique. The activation energy of the glass transition was determined. The crystallization kinetics, namely, the activation energy, the frequency factor, K 0 and Avrami index, n, were computed. one the other hand, applying the iso-conversional model show a slight increase of Ec(χ) with (χ) which accounts for one-step mechanism controlling the crystallization process of the studied glasse, farther, the results shows the strong heating rate dependence of the activation energy. Avrami exponent, n, is non-integer value, which accounts for two and three-dimensional crystal growth with heterogeneous nucleation. Applying JMA and SB(M, N) as a model fitting approaches indicates that the SB(M, N) model can be used to describe crystallization process of As 40 S 45 Se 15 glassy, while as JMA falls.The present study showed the role of doping and heating rate in controlling the crystallization parameters. Finally, the interesting variation of the thermal stability and crystallization parameters could help in the development of the investigated compositions to be promising for the practical applications.