Thermal stability and crystallization process in a Fe-based bulk amorphous alloy: The kinetic analysis

Journal of Non-Crystalline Solids xxx (xxxx) xxx–xxx

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Thermal stability and crystallization process in a Fe-based bulk amorphous alloy: The kinetic analysis P. Rezaei-Shahreza, A. Seifoddini, S. Hasani⁎ Department of Mining and Metallurgical Engineering, Yazd University, 89195-741 Yazd, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Fe-based amorphous alloy Bulk metallic glasses (BMGs) Kinetic analysis JMAK method Isoconversional methods Avrami's exponent

In this work, the thermal stability and non-isothermal kinetic analysis of crystallization process in (Fe41Co7Cr15Mo14Y2C15)94B6 amorphous alloy were investigated. The research findings revealed that crystallization process was done in four stages whereby the crystalline phases including Fe23 (B, C)6, Fe3Mo3C, and Mo3Co3C were formed. In order to determine the activation energy of every crystallization stages, the isoconversional methods were used. The activation energies were calculated about 550, 620, 550, and 820 kJ/mol for I, II, III, and IV crystallization stages, respectively. Also, values of the rate constant at the peak temperature of crystallization, Kp; Avrami exponent, n; and growth dimensional, m; were determined by means of Augis-Bennett and Gao-Wang methods. Furthermore, for a more accurate determination of kinetic parameters including n, and m, these kinetic parameters were obtained by the Johnson-Mehl-Avrami-Kolmogorov (JMAK) and Matusita. The values of m were calculated equal to 2, 1, 2, and 3 for I, II, III, IV crystallization stages, respectively. Also, the obtained nucleation index, showed that the rate of nucleation in I, II, and III crystallization stages were zero. Therefore, it can be acceptable that a large number of nuclei related to these stages already exist in the specimen.

1. Introduction Bulk metallic glasses (BMGs) are a relatively new class of materials with a specific combination of properties and thereby very interesting for researchers. High price of the elemental additions (e.g. Ln, Ga, Zr, Hf, Nb, Pd, Pr, and Y) which lead to easiest glass formation in nonferrous-based amorphous alloy systems, limit the wide commercial applications of them. While, low cost Fe-based BMGs using abundant and inexpensive elements such as B, C, Si, and P provides a great advantage for the transition to commercial applications when material cost is a critical issue [1]. In addition, the Fe-based of these materials have a unique combination of structural and functional properties including very high strength, large elasticity, and glass forming ability (GFA) as well as excellent soft magnetic properties and corrosion resistance [2–6]. In these amorphous alloys, there are various elements including not only B and C which are effective on the GFA, thermal stability, crystallization and mechanical properties of the BMGs, but also rare earth elements which increase the negative mixing heat (Δ Hmix) [7–10]. Therefore, the excellent thermal stability of these BMGs provides an experimentally accessible time and temperature window to investigate nucleation and growth behavior during the crystallization process [11]. On the other hand, the thermal stability of BMGs is affected by this transformation. Hence, considerable interest in

⁎

crystallization kinetics of these BMGs has been aroused again in recent years [2,4–6,12–14]. It is well known that the kinetics of crystallization process in BMGs is sensitive to the kinetic parameters which mainly include the activation energy; mechanisms of nucleation and growth; nature of crystalline phases; and crystallization time and temperature. The crystallization kinetic analysis of BMGs is investigated in two ways: (i) isothermal crystallization in which samples are quickly heated up and held at a temperature in the super cooled liquid region and (ii) isochronal crystallization in which samples are heated up at a constant heating rate. In non-isothermal condition, the kinetic and mechanism of the transformations are studied by thermo-analytical techniques, namely differential scanning calorimetry (DSC) [15,16], dilatometry (DIL) [17], thermogravimetry (TG) [18], and differential thermal analysis (DTA) [19,20]. Kinetic parameters of crystallization process in BMGs can be determined using various kinetic models including isoconversional and isokinetic methods. The most commonly used isoconversional methods are Flynn-Wall-Ozawa (FWO) [21,22]; Kissinger-Akahira-Sunose (KAS) [23,24]; Augis-Bennet [25]; and Gao-Wang [26] methods. Also, most of the isokinetic methods are based on Johnson-Mehl-Avrami-Kolmogorov (JMAK) [27–33] method. According to the published results [11,12,14,34], the complementary use of both the methods (isokinetic

Corresponding author. E-mail address: [email protected] (S. Hasani).

http://dx.doi.org/10.1016/j.jnoncrysol.2017.05.044 Received 20 February 2017; Received in revised form 3 April 2017; Accepted 31 May 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.

Please cite this article as: Rezaei-Shahreza, P., Journal of Non-Crystalline Solids (2017), http://dx.doi.org/10.1016/j.jnoncrysol.2017.05.044

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dα = A exp(−Eα RT ) f (α ) dT

and isoconversional) is more useful for the understanding of the kinetics of the crystallization process. Hence, thermal stability and nonisothermal crystallization kinetic of TieZreBe [35], Ti41.5Cu42.5Ni7.5Zr2.5Hf5Si1 [36], Cu47.5Zr47.5Al5 [37], TiZrHfCuNiBe [38], Zr60Al15Ni25 [39], Fe83B17 [2], Fe80P13C7 [40], FeCoBSiNb, FeCSiBPCu [41], and FeCoBSiNbCu [42] bulk metallic glasses were investigated by both isoconversional and isokinetic methods. Therefore, in this study, the non-isothermal kinetic analysis of crystallization process in Fe41Co7Cr15Mo14Y2C15B6 as a Fe-based bulk amorphous alloy was performed using complementary isoconversional methods (including FWO [21,22], KAS [23,24], Augis-Bennet [25], and Gao-Wang [26]) and isokinetic methods (including Matusita [43,44] and JMAK [29–33]). Furthermore, phase analysis and microstructural evaluations of specimens were investigated by X-ray diffraction (XRD) and field emission scanning microscopy (FE-SEM), respectively.

To solve the temperature integral, several approximations have been introduced and all of these approximations lead to a direct isoconversional method. Therefore, there is a wide range of theoretical models and mathematical treatments to estimate the activation energy of a reaction. The most popular isoconversional methods used for the calculation of activation energy of crystallization process are: 2.1.1. Flynn-Wall-Ozawa (FWO) [21,22] FWO is derived from integral isoconversional method. Using Doyle's approximation [50] for the integral which gives ln p(Ea / RT) ≈ − 5.331 − 1.052 Ea / RT

ln β = constant‐1.0516

2. Theoretical background

(4)

2.1.2. Kissinger-Akahira-Sunose (KAS) [23,24] This method is derived from integral isoconversional method like FWO. But, compared to the FWO method (based on the Doyle's approximation [50]), the KAS method (based on the Coats-Redfern approximation [51]) offers a significant improvement in the accuracy of the Eα values.

β E ln ⎛ 2 ⎞ = constant‐ α RT ⎝T ⎠

(5)

Thus, for α = const., the plot ln(β / T ) vs. 1 / T should be a straight line whose slope can be used to evaluate the apparent activation energy. 2

2.1.3. Augis-Bennet [25] Augis-Bennett method [25] is an extension of Kissinger method for its applicability to heterogeneous reaction described by Avrami expression. In this kinetic model, apart from the peak crystallization temperature, it also incorporates the onset temperature of crystallization and is supposed to be a very accurate method of determining E through the Eq. (6):

β ⎞ E ln ⎛⎜ + ln A ⎟ = ‐ − T T RT p 0 p ⎝ ⎠

(6)

where Tp and To are the peak and the onset temperatures of crystallization, respectively. The values of E are obtained from the plot (ln (β / (Tp − To)) vs. 1000 / Tp. Furthermore, Avrami exponent (n) is determined by the Eq. (7) in this method.

2.1. Isoconversional methods Model-free isoconversional methods are the most reliable methods for the calculation of the activation energy of thermally activated reactions [19,45,48,49]. These methods can be classified as differential and integral methods [45] and are based on the basic kinetic equation:

n = 2.5

Tp2 ΔT

(1)

() E R

(7)

where ΔT is the full width at half maximum of the DSC curve. Boswell [52] founded a limitation in the Augis-Bennett method so that if

where k (T), f (α), and α are the reaction rate constant, the reaction model, and the conversion fraction that represents the volume of the crystallized fraction, respectively. The rate constant, k(T), usually has an Arrhenian temperature dependence:

k (T ) = A exp(−Eα RT )

Eα RT

Thus, for α = const., the plot ln β vs. 1/T, obtained from thermograms recorded at several heating rates, should be a straight line whose slope can be used to evaluate the apparent activation energy.

In this section, a brief description is presented on the various kinetic methods based on the experimental data obtained by thermal analyses such as DTA, TG, DSC, and dilatometry. The methods presented in this section focus on the non-isothermal transformations such as crystallization process in BMGs in non-isothermal condition. These methods are generally based on either the isoconversional or the isokinetic methods. The isoconversional methods are generally used for non-isothermal conditions, assuming that the reaction rate at a constant extent of conversion is only a function of temperature. The use of isoconversional methods is widespread in physical chemistry to determine the activation energy of thermally activated solid-state reactions [34,45]. The activation energy obtained by isoconversional methods can be varying with the degree of conversion dependent on the reaction mechanism. This is contrary to the isokinetic hypothesis that the activation energy obtained by this method is constant and independent of the degree of conversion. Most of the thermal phase transformations (like crystallization) are controlled by the nucleation and growth processes [34]. Therefore, the mechanism of these transformations can be complex, e.g. interfacecontrolled and diffusion-controlled growth. However, the kinetic analysis of these transformations is done by isokinetic hypothesis alongside with isoconversional methods. Although, only in a few studies [46,47], it is reported that the kinetic parameters are varied with the degree of conversion in a crystallization process, in this study like most publications, the complementary use of both the isokinetic and isoconversional methods is used for better understanding of the kinetics of the crystallization process.

dα = k (T ) f (α ) dT

(3)

Tp − T0 Tp

=1

(8)

then Augis-Bennett gives crude results. Therefore, Boswell method determines the activation energy at peak temperature using the following equation:

(2)

where A (min− 1) is the pre-exponential factor; Eα (kJ/mol) is the local activation energy; and R is the universal gas constant. Thus, the kinetic equation combined with the Arrhenius approach can be described as Eq. 3:

β E ln ⎛⎜ ⎟⎞ = ‐ + ln A RTp ⎝ Tp ⎠ 2

(9)

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2.1.4. Gao-Wang [26] This method is derived from differential isoconversional method. The value of the rate constant at the peak temperature for all crystallization stages, Kp, and Avrami exponent (n) can be determined using the Gao-Wang method from the following equations:

for increasing nucleation rate); and p is the growth index with values of 1 and 0.5 for interfaced and diffusion controlled growth, respectively.

E dα + costanat ln ⎛ ⎞ = ‐ RTp ⎝ dt ⎠ p

In this study, the well-established melt-quench technique was used to prepare the bulk material. The alloy ingots with the nominal composition of Fe41Co7Cr15Mo14Y2C15B6 (at.%) were prepared by vacuum arc melting of high pure elements (99.99 wt%) of Fe, Co, Cr, Mo, Y, C, and B with Ti gettered under argon atmosphere. In order to ensure good composition homogeneity, not only the ingot was remelted four times but the tube was also frequently shaken during the melt process. Cylinder samples with a diameter of 2 mm and a length of 70 mm were then prepared by water-cooled copper mold casting under argon atmosphere. The thermal behavior of the glassy samples was evaluated with a Perkin Elmer, DSC-7, differential scanning calorimeter (DSC) from ambient to 1200 °C. The DSC analysis was carried out on approximately 25 mg quantities of powder samples with sensitivity of ± 10 μW. The heating rates were varied from 5 to 20 °C/min under argon flow supplied at the rate 30 mL/min. Also, to confirm the crystallization process, the as-cast specimen was heated in non-isothermal condition by DSC at a heating rate of 20 °C/min up to 696, 737, and 928 °C (determined by DSC curves) with argon flow. The structure of the as-cast and annealed samples was examined by using the X'Pert MPD Philips diffractometer fitted with diffracted-beam monochromator set for Cu kα radiation (λ = 0.1540 nm). Also, microstructural investigations of annealed specimen were carried out using a FE-SEM (MIRA3 TESCAN). For this purpose, the surfaces of specimens were mechanically polished with silicon carbide papers (up to 2000#) and then electrochemically etched in 1 mol/L HCl and 0.5 mol/L H2SO4 solution operated at a potential of 2 V.

Kp =

3. Materials and methods

(10)

βE RTp2

(11)

⎛ dα ⎞ = 0.37n Kp ⎝ dt ⎠ p

(12)

2.2. Isokinetic method Crystallization of amorphous materials usually involves nucleation and growth [53]. The Avrami's exponent commonly is used to determine the details of nucleation and growth mechanism during the crystallization process which can be determined by JMAK method. A solution of JMAK model under isothermal conditions can be obtained assuming that nucleation and growth rates are time independent [33].

α = 1‐exp[‐(k (T ). t )n]

(13)

where α is the transformed phase fraction; k(T) is the rate constant that generally depends on temperature and incorporates the rates of nucleation and growth followed by Eq. (2); n is usually known as Avrami's exponent; and t is time. Accordingly, JMAK equation can be rewritten as [6,54,55]:

ln(‐ln(1‐α )) = n ln k (T ) + n ln t

(14)

Therefore, the value of n can be obtained by plotting ln (−ln (1 − α)) vs. ln(t). Furthermore, according to Eq. (14), the local Avrami's exponent, n (α), can be determined by ln(t) derivation, as shown in the following equation [56,57]:

n (α ) =

4. Results and discussion 4.1. Thermal stability and recrystallization Fig. 1 illustrates the XRD pattern of the as-cast alloy which exhibits a diffuse halo peak at 2θ ≈ 50° and no diffraction peaks suggesting its amorphous state without any crystalline phases. Fig. 2 shows the DSC curves of the bulk glassy alloy in continuous heating condition at heating rates of 5, 10, and 20 °C/min from ambient temperature to 1200 °C. It can be observed that all of the DSC curves exhibit four exothermic events corresponding to the crystallization process, so the Fe41Co7Cr15Mo14Y2C15B6 glassy alloy crystallized in multiple stages. Also, glass transition temperature (Tg) followed by a super-cooled liquid region (ΔTx = Tx − Tg); the onset exothermic peak of crystallization temperature (Tx); solidus (Tm); and liquidus (Tl) temperatures were distinguished in these DSC curves. As the heating rate increases from 5

R∂ln(‐ln(1‐α )) Eα ∂

() 1 T

(15)

where Eα is variation of local activation energy calculated by FWO or KAS method. Hence, by plotting ln (− ln(1 − α)) vs. 1000 / T, the value of n(α) at each heating rate can be obtained from the slope of these straight lines. Moreover, the growth dimensional, m, and Avrami's exponent, n, can be calculated by Matusita equation [43,44]:

ln(‐ln(1‐α )) = n ln(β )‐1.052

mE + constant RT

(16)

In order to obtain the ratio m/n, Eq. (16) is rewritten as [44]

ln(β ) = ‐1.052

m E 1 ‐ ln(‐ln(1‐α )) + constant n RT n

(17)

Thus, the plot of ln(β) vs. 1 / T, where T is the temperature at which the crystal volume fraction reaches a specific value, gives a straight line and the slope gives the value of 1.052 (m / n)E. The ratio m/n can be obtained when the activation energy is known. Since it is known that the volume fraction of crystals at the peak temperature, Tp, in DSC or DTA curves is almost the same irrespective of β; this equation should be applied for the peak temperature [44]. Also, the mechanism of nucleation and growth at each heating rate can be determined by Ranganathan–Heimendahl equation [37–40]:

n = pm + b

(18)

Where n is Avrami's exponent; m is growth dimension; b is a nucleation index (b = 0 for zero nucleation rate, 0 < b < 1 for decreasing nucleation rate, b = 1 for constant nucleation rate, and b > 1

Fig. 1. XRD pattern and FE-SEM micrograph of the as-cast Fe41Co7Cr15Mo14Y2C15B6 bulk metallic glass.

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Fig. 3. (a) XRD patterns and FE-SEM micrographs of the investigated alloy annealed from ambient temperature up to (b) 696, (c) 737, and (d) 928 °C, at a heating rate of 20 °C/ min.

rate, the atoms rapidly face the temperature level changes and don't have sufficient time to locate defined crystalline sites; therefore, crystallization takes place at higher temperatures (Fig. 2). Furthermore, according to characteristic transformation temperatures obtained by DSC curves, some criteria such as Trg = Tg / Tl [58], γ = Tx / (Tg + Tl) [59], δ = Tx / (Tl − Tg) [60], and β = 1 + (Tx / Tl) [61] were calculated to explain GFA of Fe41Co7Cr15Mo14Y2C15B6 BMG. These calculated parameters, listed in Table 1, indicate good GFA for this Fe-based BMG. Fig. 3 illustrates the XRD patterns and FE-SEM micrographs related to specimens annealed at a heating rate of 20 °C/min up to 696, 737, and 928 °C which shows either the formation of crystalline phases (including Fe23 (B, C)6, Fe3Mo3C, and Mo3Co3C) or the crystallized nanostructure.

Fig. 2. DSC curves for the investigated alloy at heating rates of (a) 5, (b) 10, and (c) 20 °C/min. Table 1 Thermal properties of investigated BMG at various heating rates. Heating rate (°C/ min)

Tg (°C)

Tx (°C)

Tm (°C)

Tl (°C)

Trg

γ

δ

β

5 10 20

530 558 565

595 601 610

1103 1103 1103

1188 1191 1195

0.45 0.47 0.47

0.35 0.35 0.35

0.90 0.95 0.97

1.50 1.50 1.51

4.2. Kinetic analysis As mentioned, the non-isothermal DSC data can be used for the nonisothermal kinetic analysis of the crystallization process. Therefore, the curves of α vs. T were obtained by data extracted from DCS curves. In order to evaluate the dependence of E on α for these transformations, FWO and KAS methods were used. For the selected α = const., the plots ln β vs. 1 / T and ln(β / T2) vs. 1 / T are obtained and the values of E are calculated from their slopes by means of FWO and KAS methods, respectively. The E values obtained by means of FWO and KAS methods

to 20 °C/min, all the characteristic temperatures are shifted to higher temperatures (Table 1), indicating that the crystallization process depends on the heating rate caused by the fact that crystallization is a thermally activated process. Therefore, there is enough time for the formation of crystalline nucleus at lower heating rates so that crystallization occur at lower temperatures; while, with an increase in heating 4

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Fig. 4. The dependence of E on α evaluated for the non-isothermal crystallization processes by means of KAS method.

Fig. 5. Plots of the ln(β / Tp) vs. 1000 / Tp for different crystallization by means of Augis & Bennet method.

are in good agreement with each other. Hence, the dependence of E on α evaluated is shown in Fig. 4. As seen, the activation energies of crystallization stages are found to be practically independent on α for a very wide conversion range. This means that the investigated processes are simple (one-step process) and can be described by unique kinetic triplet. Therefore, the average values of E within the conversion range are presented in Table 2 in which E values are practically constant for all of the crystallization stages. The activation energy for crystallization process of the investigated alloy is more than other amorphous alloys [35,41,62]. Also, these results are in good agreement with activation energies calculated for the recrystallization process of this alloy by using the Starink isoconversional methods [63]. The model-free isoconversional methods are definitely superior to the isokinetic methods for the accurate determination of kinetic parameters like E and A. However, the knowledge of accurate E and A is not sufficient for the detailed investigations of the dimensionality of the growth and the grain size using thermal analysis; it should be noted that the determination of other kinetic parameters including the values of the rate constant at the peak temperature of crystallization, Kp; Avrami exponent, n; and growth dimensional, m (apart from E) is essential to obtain the mechanism of this transformation. Therefore, in this study, the Augis-Bennett and Gao-Wang methods are used to determine these kinetic parameters. According to Eq. (9), plotting ln(β / Tp) vs. 1000 / Tp provides the E values from the slope of the curve by means of AugisBennett method (see Fig. 5) whose results are presented in Table 2. Also, n is calculated by means Augis-Bennett method (Eq. (7)) at

different heating rates for all of the crystallization stages listed in Table 2. Furthermore, to obtain the activation energy by means of GaoWang method, plotting the volume fraction crystallized, α, and the crystallization rate, dα/dt, vs. the time for four crystallization stages of Fe41Co7Cr15Mo14Y2C15B6 BMG is depicted in Fig. 6. The plots of ln (dα / dt)p vs. 1000 / Tp for four crystallization stages are shown in Fig. 7. Effective activation energies for four crystallization stages are determined from the slope of straight lines, presented in Table 2. Also, the values of calculated parameters, Kp and n at different heating rates for all of the crystallization stages are listed in Table 2. As seen, there is a good agreement between kinetic parameters calculated by means of various kinetic methods. On the other hand, JMAK, as an isokinetic model, is the most suitable method for describing the nucleation and growth process during the non-isothermal crystallization of metallic glasses [64]. According to Eq. (14), the value of Avrami exponent, n, can be obtained by plotting ln (− ln(1 − α)) vs. ln(t). Also, by determining of the local Avrami exponent (n (α) in Eq. (15)), it can be shown that the nucleation and growth behavior during crystallization transformation is constant or not. Hence, by plotting ln(− ln(1 − α)) vs. 1000 / Tα the value of n(α) at each heating rate can be obtained from the slope of these straight lines. Fig. 8 shows the ln(− ln(1 − α)) vs. 1000/Tα curves at different heating rates for all the crystallization stages. Also, the plots of local Avrami exponent, n (α), vs. the crystallized volume fraction (α) at different heating rates for all the crystallization peaks are shown in Fig. 9. As seen, the values of n(α) at different heating rates show the same

Table 2 Kinetic parameters of the crystallization process for the investigated alloy as determined by the different methods. Peak no.

E (kJ/mol)

Heating rate (°C/min)

FWO

KAS

Augis & Bennet

Gao-Wang

I

546.0 ± 11.0

559.3 ± 11.5

581.8 ± 2.0

583.2 ± 2.4

II

616.2 ± 4.5

632.3 ± 4.7

659.0 ± 4.3

636.6 ± 3.3

III

513.5 ± 2.3

591.6 ± 2.4

617.1 ± 3.9

592.7 ± 5.1

IV

826.5 ± 9.5

808.0 ± 9.5

861.3 ± 6.3

935.2 ± 6.3

5 10 20 5 10 20 5 10 20 5 10 20

5

Kp

n

Gao-Wang

Gao-Wang

JMAK

Augis & Bennet

0.449 0.882 1.735 0.445 0.875 1.724 0.378 0.745 1.460 0.412 0.810 1.600

1.66 1.67 1.72 1.43 1.45 1.49 1.98 2.10 2.30 3.31 3.85 4.2

1.88 1.92 1.95 1.48 1.49 1.53 1.88 1.92 1.90 3.71 3.85 3.91

1.71 1.82 1.90 1.45 1.47 1.49 1.78 1.82 1.89 3.61 3.74 3.82

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Fig. 6. Plots of α and dα / dt vs. t at different heating rates for (a) I, (b) II, (c) III, and (d) IV peaks.

methods is to be practically independent of α, it can be accepted that the activation energy is constant for either a very wide conversion range or for all the used heating rates. Therefore, based on the results presented in Fig. 4, it can be concluded that the activation energies for all the crystallization stages is independent of α and heating rates of 5, 10, and 20 °C/min. Therefore, according to Ep calculated by means of

tendency and don't vary with an increase in crystallized volume fraction during the whole crystallization process. Therefore, the average of n(α) values at different heating rates are presented in Table 2. On the other hand, the ratio of m/n can be obtained by Eq. (18) when the activation energy is known. Since it is known that if the value of local activation energy obtained by means of isoconversional 6

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Fig. 7. Plots of ln(dα / dt) vs. 1000 / Tp for different crystallization stages by means of Gao-Wang method.

Fig. 9. Local Avrami exponent vs. α at different heating rates for I, II, III, and IV crystallization peaks.

Gao-Wang method, the ratio of m/n can be determined by Eq. (18). Also, as shown in Fig. 9, the local Avrami exponent is constant and independent of α for all the crystallization stages. Hence, the values of growth dimensional, m, can be obtained by means of Matusita method in which they are equal to 2, 1, 2, and 3 for I, II, III, IV crystallization stages, respectively. As mentioned, the mechanism of nucleation and growth can be determined by Eq. (18). The values of calculated parameters, n, m, p, and b for all the crystallization stages are listed in Table 3.

The maximum value of n is 4 and its minimum value is 1, while the corresponding m-values must be 3 and 1, respectively. When n = 2, the corresponding m-value is 2 or 1; and when n = 3, the corresponding mvalue is 3 or 2, where m = 3 for the three-dimensional growth of crystal particles, m = 2 for two-dimensional growth, and m = 1 for one-dimensional growth. One way for determining the m-value is to use Matusita method when the activation energy is known, as performed in this study. Also, p is growth index with values of 1 and 0.5 for

Fig. 8. JMAK plots of ln(− ln(1 − α)) vs. 1000 / Tα at different heating rates for (a) I, (b) II, (c) III, and (d) IV peaks.

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Table 3 The values of calculated kinetic parameters related to the mechanism of crystallization stages. Peak No.

n

m

p

b

I II III IV

2 1.5 2 4

2 1 2 3

1 1 1 1

0 0 0 1

interfaced and diffusion controlled growth, respectively, and b is a nucleation index (b = 0 for zero nucleation rate; 0 < b < 1 for decreasing nucleation rate; b = 1 for constant nucleation rate; and b > 1 for increasing nucleation rate). As presented in Table 3, the b value in I, II, and III crystallization stages are zero. Therefore, it can be acceptable that a large number of nuclei related to these stages already exist in the specimen which is in good agreement with TEM investigations by researchers [65,66]. But, the rate of nucleation for IV crystallization stage was equal to 1 (constant nucleation rate). 5. Conclusions In this study, the non-isothermal kinetic of Fe41Co7Cr15Mo14Y2C15B6 as a Fe-based bulk amorphous alloy was analyzed by the isoconversional and isokinetic methods. According to the research findings, the main conclusions are drawn as follows:

• Fe • • • •

41Co7Cr15Mo14Y2C15B6 BMG manifested crystallization process in four steps. During this transformation, the crystalline phases including Mo3Fe3C, Fe23(B, C)6, and Mo3Co3C were formed. Furthermore, formation of the nano-crystalline phases was confirmed by FE-SEM micrograph. Local activation energies were estimated by isoconversional FWO and KAS methods which have a good agreement with each other. The average of local activation energies for four exothermic crystallization peaks was reported about 550, 620, 550, and 820 kJ/mol, respectively. On the other hand, the activation energies were obtained by AugisBennett and Gao-Wang methods in which their results were in good agreement with those obtained by isoconversional methods. In addition, the Avrami exponent, n, as another kinetic parameter was calculated by means of Augis-Bennett and Gao-Wang methods. The rate constants at the peak temperature of crystallization stages, Kp, were determined by means of Gao-Wang method at various heating rates. Furthermore, the mechanism of crystallization stages as processes controlled by nucleation and growth were determined by means of JMAK and Matusita methods.

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