Introduction

The Fe-based bulk metallic glasses (BMGs) have attracted too much attention by exhibiting impressive mechanical properties [1], excellent corrosion resistance [2,3,4,5,6], and good magnetic properties [7,8,9,10] to use in widespread applications such as sensors [11], transformers [12, 13], and magnetic tapes [14]. However, these BMGs exhibit a limited plasticity at room temperature and are a common example of brittle BMGs [15,16,17,18]. Therefore, despite their advantages, these amorphous alloys were unattractive for commercial applications due to lack of their ductility [19,20,21,22,23]. To improve ductility, two solutions have been proposed: (a) the addition of alloy elements and (b) partial crystallization leading to the formation of amorphous matrix–nanocrystalline composite. Hence, in the last decade, numerous researchers tried to enhance the mechanical properties of Fe-based BMGs by the addition of alloy elements such as Cu [24], Ni [23], Zr [25], and Mo [26] or by the formation of nanocrystalline phases/precipitates in amorphous matrix [27,28,29]. In the case of partial annealing, it has been accepted that volume fraction, morphology, and type of crystalline precipitates have a strong effect on mechanical properties [30, 31]. Activation energy (E), the pre-exponential factor (A), and kinetic model (g(α)) as the triple kinetic parameters are known as the essential information to control a reaction (such as crystallization); without knowing the kinetic parameters of this process, products of reactions can be uncontrollable. Therefore, kinetic analysis of partial annealing in BMGs (especially Fe-based BMGs) has attracted the attention of many researchers [32,33,34,35,36,37].

In recent years, (Fe0.9Ni0.1)77Mo5P9C7.5B1.5 BMG has been introduced as a Fe-based amorphous alloy with high ductility; furthermore, the effect of annealing treatment on the mechanical properties of this BMG was investigated [38, 39]. Nevertheless, despite the previous researches, no comprehensive investigation has been done into the kinetic analysis of crystallization process of this BMG and, therefore, there exists a knowledge gap. Therefore, non-isothermal kinetic analysis of crystallization process was investigated in [(Fe0.9Ni0.1)77Mo5P9C7.5B1.5]100−xCux (x = 0.1 at.%) BMG as a newer generation of Fe-based BMG in the present study. For this purpose, despite thermal, phase analysis, and microstructural observations, kinetic parameters of crystallization process in this BMG were calculated by different kinetic methods including the isoconversional Starink [40, 41] and Friedman (FR) [42] methods in combination with the invariant kinetic parameters (IKP) [43] and fitting methods [44].

Materials and methods

Materials and experimental procedure

The ingots of master alloy with the chemical compositions of [(Fe0.9Ni0.1)77Mo5P9C7.5B1.5]100−xCux (x = 0.1 at.%) were prepared in a vacuum arc furnace by melting the mixture of high purity (99.99 mass%) elements under a Ti gettered and argon atmosphere. To guarantee the homogeneity of the as-cast samples, all ingots were remelted for at least four times. Then, the initial metallic glassy alloys were prepared as rods with a diameter of 2 mm and a length of 100 mm by suction casting in a water-cooled Cu mold under high purity argon atmosphere. The chemical composition of the as-cast alloy was checked by inductively coupled plasma optical spectroscopy (ICP-OS) (presented in Table 1). As seen, the chemical composition of the BMG is in good agreement with the nominal composition.

Table 1 Chemical composition of the studied BMG (measured by ICP-OS)
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The thermal behavior of BMG during crystallization processes was determined by a DTA (BAHR-STA 504) at an ambient temperature up to 1473 K, using various heating rates of 10, 20, and 40 K min−1 under high purity argon flow supplied at a rate of 30 mL min−1. According to DTA curves, the various stages of crystallization process in the BMG under non-isothermal condition were determined. Then, the samples of [(Fe0.9Ni0.1)77Mo5P9C7.5B1.5]100−xCux (x = 0.1 at.%) BMGs were annealed in non-isothermal condition by DTA at a heating rate of 20 K min−1 up to the temperature of each peak under argon flow. Phase analysis of the as-cast and annealed specimens was done by XRD (X’Pert MPD Philips Diffractometer) to determine the amorphous or crystalline phases. These results were recorded on an X’Pert MPD Philips diffractometer fitted with diffracted-beam monochromator set for Cu kα radiation (λ = 0.1540 nm) by “Brag Brentano” geometry. The operation voltage, current, scan speed, and step size were 45 kV, 40 mA, 1 s, and 0.05°, respectively. Furthermore, microstructural observations of the annealed samples were investigated by an FESEM (MIRA3 TESCAN). For this purpose, the surface of all the annealed specimens was polished by silicon carbide papers (up to 3000#) and then electrochemically etched by a 1 mol L−1 HCl and 0.5 mol L−1 H2SO4 solution operated at a potential of 3 V.

Kinetic analysis

To determine the kinetic parameters, isoconversional methods are usually used in association with IKP and Fitting [43] methods. Therefore, a significant number of researches [36, 44,45,46] investigated the kinetics analysis of solid-state reactions by the model-free (isoconversional) as well as model-fitting whose basis is discussed in the following.

Isoconversional methods

Isoconversional methods are used to determine E and its dependence on α [47,48,49]. The basis of these methods has been discussed in our previous articles [36, 45, 46]. Among these methods, Starink [41] and FR [42] methods are known as more accurate integral and differential isoconversional methods, respectively [50]. Hence, these two methods are discussed below.

  1. 1.

    The integral isoconversional Starink method [41] is a new method for the derivation of Eα, as follows:

    $$\ln \frac{\beta }{{T^{ 1. 9 2} }} = {\text{const}}. - 1. 0 0 0 8\frac{E}{RT}$$
    (1)

    where α is the degree of conversion, β is the linear heating rate (K min−1), T is the absolute temperature (K), R is the general gas constant (J mol−1 K−1), and E is the activation energy (kJ mol−1).

  2. 2.

    The differential isoconversional FR method [42] is a linear differential isoconversional method which is directly based on Eq. (2):

    $$\ln \left[ {\beta \left( {\frac{{{\text{d}}\alpha }}{{{\text{d}}T}}} \right)} \right] = \ln \left[ {Af(\alpha )} \right] - \frac{E}{RT}$$
    (2)

For a constant α, the plots ln(β/T1.92) versus 1/T; and ln[β (dα/dT)] versus 1/T recorded at several heating rates (at least for three heating rates) should be straight lines and the slope of each plot lets us calculate the E parameter by Starink and FR methods, sequentially.

IKP method

IKP method needs various α–T curves recorded at different heating rates. In this method, the invariant kinetic parameters including Einv and Ainv values are achieved through the intersections of the curves of lnA versus E; this intersection is observed for the correct kinetic models. Therefore, this method is based on the existence of a linear correlation between E and lnA [Eq. (3)] achieved by theoretical kinetic models.

$$\ln A_{\text{i}} = a + bE_{\text{i}}$$
(3)

In Eq. (3), subscript “i” indicates a value of heating rate and, a and b are the compensation effect parameters.

According to Eq. (4), for each theoretical kinetic model, the values of E and lnA are obtained from the slope and interruption of ln [g(α)/T2] versus 1/T plots, respectively. These algebraic expressions for the most frequently mechanisms have been presented in previous publications [51, 52].

$$\ln \frac{g(\alpha )}{{T^{2} }} \cong \ln \frac{AR}{\beta E} - \frac{E}{RT}$$
(4)

Fitting models

To validate the results obtained by isoconversional and IKP methods, fitting models are usually used [36, 44, 53]. In one of these methods, the reaction model can be established by plotting the numerical g(a) based on theoretical and experimental data and finding the best matching between them. Therefore, the results obtained by this method can determine how and when the reaction mechanism changes during the course of transformation. The theoretical curves of g(a) versus a are plotted according to the algebraic expressions for g(a) used to describe the solid-state reactions (which have been presented elsewhere [44, 54]). While the experimental curve of g(a) versus a is plotted by Eq. (5).

$$g(\alpha ) = \frac{A}{\beta }\int_{0}^{T} {\exp \left( { - \frac{E}{RT}} \right){\text{d}}T}$$
(5)

The temperature integral in Eq (5) (\(\int_{0}^{T} {\exp \left( { - \frac{E}{RT}} \right){\text{d}}T}\)) is determined by Eq. (6) obtained by the Gorbachev approximation [55].

$$\int_{0}^{T} {\exp \left( { - \frac{E}{RT}} \right){\text{d}}T} = \frac{{RT^{2} }}{E + 2RT}\exp \left( { - \frac{E}{RT}} \right)$$
(6)

Results and discussion

Experimental observations

Thermal analysis techniques including DTA, thermogravimetry (TG), differential scanning calorimetry (DSC), dilatometry (DIL), etc. are powerful and convenient methods to study the reactions at different heating rates [56,57,58,59]. Figure 1a shows the DTA curves of [(Fe0.9Ni0.1)77Mo5P9C7.5B1.5]100−xCux (x = 0.1 at.%) BMG in four crystallization steps at various heating rates of 10, 20, and 40 K min−1. As shown, with an increase in the heating rate, critical temperatures such as peak temperature (Tp), glass transition temperature (Tg), and one-set crystallization temperature (Tx) shift to higher temperature ranges which are in good agreement with the results obtained by other researcher [49, 50]. According to temperature ranges extracted from DTA curve at a heating rate of 20 K min−1, related to the crystallization peaks, the as-cast specimens were annealed from ambient temperature up to 731, 778, 809, and 841 K at a heating rate of 20 K min−1. Figure 2 depicts the X-ray diffraction patterns of the as-cast and annealed samples. As seen, in both the as-cast and the sample annealed at 731 K, just a broad single peak (in a range of 2θ = 40–60) is observed, indicating the amorphous nature and absence of crystalline phases in these samples. While, with an increase in the crystallization temperature to 841 K, some crystalline phases such as α-Fe, γ-Fe, FeNi2P, and Fe3C are formed, so that by annealing at a higher temperature, only the intensity of these crystalline phases is increased. Furthermore, the volume fraction of crystalline phases was calculated by the rate of the peak’s areas to the total area of the XRD pattern [51] as presented in Table 2. As shown, Table 2 clearly demonstrates that with an increase in the crystallization temperature from 778 to 841 K, the volume fraction of the crystalline phases increases from 9.2 to 20.2%. Furthermore, the average sizes of crystallites related to the samples annealed at various temperatures were calculated by the Debye–Scherer method [52] as shown in Eq. (7):

$$D = \frac{K\lambda }{\beta \cos \theta }$$
(7)

where for a constant K (= 0.89), D is the average grain size, λ is the X-ray wavelength (λ ≈ 1.5456 Å), θ is the diffraction angel of the peak, and β is the full width at the half maximum of the peaks. The average of crystallite sizes of the annealed samples is presented in Table 2.

Fig. 1
figure 1

DTA curves for the investigated BMG at different heating rates

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Fig. 2
figure 2

XRD patterns of the as-cast and annealed specimens from ambient temperature up to various temperatures at a heating rate of 20 K min−1

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Table 2 Volume fraction and average size of crystalline phases formed during annealing process at various temperatures
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Also, Fig. 3a–c presents the micrographs of nano-crystalline phases related to the specimens annealed at various temperatures. To validate the Debye–Scherer results, the average size of crystalline phases was measured using the microstructural image processor (MIP) commercial software (presented in Table 2). As seen, with an increase in the crystallization temperature, the average grain size of crystalline phases is increased which is in good agreement with the Debye–Scherer results.

Fig. 3
figure 3

FE-SEM micrographs of specimens annealed from ambient temperature up to a 731, b 778, c 809, and d 841 K at a heating rate of 20 K min−1

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Kinetic calculations

Figure 4 shows the plots of α versus T at various heating rates (10, 20, and 40 K min−1) for the four steps of crystallization process. To calculate local E, Starink and FR isoconversional kinetic methods were used; the plots of E versus α for all the four crystallization steps are shown in Fig. 5. As seen, within a wide range of α (0.1 < α < 0.9), the local activation energy related to each crystallization peak is practically independent on α. It means that all crystallization steps are one-step and controlled by a unique kinetic mechanism. The average of activation energies calculated by these isoconversional methods is presented in Table 3. As shown, the activation energies calculated by both Starink and FR methods are in good agreement with each other. Furthermore, it is found that the activation energies decrease with an increase in annealing temperature. Therefore, the formation of crystalline phases in the first and second crystallization stages faces higher energy barriers compared with the crystallites formed in the third and fourth stages.

Fig. 4
figure 4

Plots of α versus T at different heating rates for four crystallization stages

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Fig. 5
figure 5

The dependence of E on α evaluated for the non-isothermal crystallization process calculated by a Starink, and b FR methods

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Table 3 Values of kinetic parameters obtained by isoconversional, IKP, and fitting methods
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The activation energies and the pre-exponential factors obtained by Coats–Redfern (CR) equation [60] were used for each crystallization stage and at each heating rate to determine other two kinetic parameters and to plot the linear relationship between lnA and E. For each stage, the linear relationship between lnA and E is shown in Fig. 6. As seen, the intersections of lnA versus E curves determine the correct activation energy, pre-exponential factor, and kinetic model. The results are presented in Table 3. These results confirm what has been achieved by isoconversional methods. In addition, it is revealed that the first, second, third, and fourth crystallization stages are controlled by A4, A4, A4, and P4 models.

Fig. 6
figure 6figure 6

The compensation relationship and enlarged region of interception for a I, b II, c III, and d IV crystallization peaks through IKP method

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Furthermore, a popular fitting method was used to ensure the results obtained using the isoconversional and IKP methods. Figure 7 shows the theoretical and experimental g(α) versus α curves for each crystallization stage and heating rate. Also, the results extracted from these curves are presented in Table 3 which confirm the results obtained by other kinetic methods. According to kinetic models obtained, it is clear that nucleation and growth of nanocrystals control the crystallization process in Fe–Ni-based BMG.

Fig. 7
figure 7

The experimental and theoretical g(α) versus α plots for a I, b II, c III, and d IV crystallization peaks

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Conclusions

In the present study, non-isothermal kinetic analysis of the crystallization process in the [(Fe0.9Ni0.1)77Mo5P9C7.5B1.5]100−xCux (x = 0.1 at.%) BMG was investigated at different heating rates up to 1473 K. The research findings revealed that:

  • DTA results showed that crystallization and melting processes took place in four exothermic peaks and one endothermic peak, respectively. The critical temperatures such as Tp, Tg, and Tx increased with an increase in the heating rate.

  • The XRD results confirmed the occurrence of crystallization. These patterns illustrated that the crystalline phases including α-Fe, γ-Fe, FeNi2P, and Fe3C were formed up to ~ 841 K. The average sizes of crystallites formed during the crystallization process at various temperatures (calculated by the Debye–Scherer method) were in the range of 40–80 nm.

  • The average size of the nanocrystalline phases increased from 38 to 81 nm by increasing the crystallization temperature from 731 to 841 K. In addition, by increasing the crystallization temperature, the volume fraction of crystallized phases increased from 9.2 to 20.2%.

  • Investigation of E diagram versus α for the crystalline phases in four crystallization steps indicated that E was approximately independent of α, within the conversion range of 0.10 ≤ α ≤ 0.90.

  • Activation energy was calculated for every stage of crystallization by isoconversional Starink and Friedman methods. For instance, the activation energies calculated by Starink method were equal to 264, 265, 166, and 190 kJ mol−1 for the first, second, third and fourth crystallization stages, respectively.

  • The results obtained by isoconversional methods were checked by IKP and fitting methods. The pre-exponential factors (A) calculated by IKP and Fitting methods were in a perfect agreement with each other.

  • Kinetic models obtained from the two methods (IKP and Fitting) in different crystallization stages indicated that the four crystallization steps were controlled by the mechanisms of A4, A4, A4, and P4, respectively.