KINETIC STUDY OF THE CRYSTALLIZATION PROCESS OF THE -Fe PHASE IN THE AMORPHOUS Fe 81 B 13 Si 4 C2 ALLOY

Summary: The kinetic study of the crystallization process of the α -Fe phase from the amorphous Fe 81 B 13 Si 4 C 2 alloy was investigated by DSC and XRD techniques. The kinetic parameters (lnA, E a ) of the investigated process were determined using the Kissinger and isoconversional (model-free) methods. It was established that the α -Fe crystallization process can be described by the JMA (Johnson-Mehl-Avrami) kinetic equation. In accordance with the XRD analysis and the calculated crystallization parameters (n = 4; m = 3), it was concluded that the crystallization stages of the considered process can be described by the bulk nucleation and the three-dimensional (3D) growth of nuclei. *


Introduction
morphous alloys are relatively new materials offering a specific combination of properties and attracting special interest of many scientists during the last two decades.The amorphous state of matter is, however, structurally and thermodynamically unstable and very susceptible to partial or complete crystallization during thermal treatment or nonisothermal compacting.The crucial limitation with respect to using amorphous alloys for high temperature applications arises from their restricted thermal stability.The onset of exothermic crystallization upon crossing the stability domain of the glassy state results in the formation of highly stable, but brittle intermetallic compounds, which renders these alloys useful only once.The latter imposes the knowledge of alloys stability in a broad range of temperature due to different crystallization processes, which appears during annealing.
Soliman, et al. (Soliman, et al., 2004, pp. 57) investigated the crystallization kinetics of melt-spun Fe 83 B 17 metallic glass ribbons.The Xray diffraction (XRD), scanning electron microscopy (SEM), and differential scanning calorimetry (DSC) were used to investigate the structural and thermal transformations of the Fe 83 B 17 metallic glass.The activation energy of the crystallization E c was evaluated using the different theoretical models, and they obtained the following values of E c : 235.3, 223.6 and 221.0 kJ mol -1 .The same authors applied the modified Johnson-Mehl-Avrami (JMA) equation and they concluded that the Fe 83 B 17 crystallization process is governed by a bulk growth in two dimensions (n = 2).
Santos and Santos (dos Santos, dos Santos, 2001, pp.47-52), (dos Santos, dos Santos, 2002, pp.56) investigated the crystallization kinetics of the amorphous alloy Metglass 2605SC (Fe 81 B 13.5 Si 3.5 C 2 ) using the ferromagnetic resonance (FMR), differential scanning calorimetry (DSC) and X-ray diffraction (XRD).Three exothermal peaks can be seen on the presented DSC curves of the investigated alloy, which shows that the crystallization process includes more than one phase.The calculated value of the Avrami exponent for the low temperature DSC peak is n = 1.24, and this value is in good agreement with the FMR results.The obtained results show that the investigated process is diffusion controlled with nucleation rate near zero.The diffraction pattern of Metglass 2605SC shows the amorphous nature of the investigated alloy.The isothermal annealing at T > 780 K leads to the appearance of the crystalline α-Fe and Fe 2 B 3 in alloy.The crystallization kinetics of Fe-B-Si metallic glasses (Fe 78 B 13 Si 9 and Fe 81 B 13.5 Si 3.5 C 2 ) was investigated by X-ray diffraction performed in-situ during Joule-heating, with simultaneous monitoring of the electrical resistence and DSC (dos Santos, dos Santos, 2001, pp.47-52), (dos Santos, dos Santos, 2002, pp.56).
The X-ray measurements show that with the increase of annealed temperature of the investigated alloy, the crystallization of the α-Fe and Fe 2 B 3 occurs, while at higher temperatures the metastable γ-Fe phase appears.The determined values of the apparent activation energy, E a (326 and 329 kJ mol -1 ) for Fe 78 B 13 Si 9 and 346, 356 and 558 kJ mol -1 for Fe 81 B 13.5 Si 3.5 C 2 are comparable with the reported values of E a for amorphous iron alloys.The values of the Avrami exponent for Fe 81 B 13.5 Si 3.5 C 2 at the first and the third peak (n I,III = 1.3) point at the diffusion controlled growth particles of a significant initial volume and the three-dimensional growth of small particles with the zero nucleation rate (dos Santos, dos Santos, 2001, pp.47-52), (dos Santos, dos Santos, 2002, pp.56).The values of the Avrami exponents (n I = 1.6 and n II = 2.0) calculated for Fe 78 B 13 Si 9 are characteristical for the diffusion controlled growth of small particles with a decreasing nucleation rate (dos Santos, dos Santos, 2001, pp.47-52), (dos Santos, dos Santos, 2002, pp.56).
Bearing in mind the above exhibited differences in the proposed mechanisms of the α-Fe crystallization from the amorphous alloy and differences in kinetic models and also in the values of kinetic parameters, in this paper, the crystallization process of α-Fe from Fe 81 B 13 Si 4 C 2 alloy was investigated from the kinetic point of view.In order to determine the kinetic triplet for the investigated process, theKissinger peak method (Kissinger, 1957(Kissinger, , pp.1702-6) -6) and the Friedman differential isoconversional method (Friedman, 1964, pp.183-195) were applied.

Materials and experimental details
The ribbon-shaped samples of the Fe 81 B 13 Si 4 C 2 amorphous alloy were obtained using the standard procedure of rapid quenching of the melt on a rotating disc (melt-spinning).The obtained ribbon was 2 cm wide and 35 μm thick.
The crystallization process was investigated by differential scanning calorimetry (DSC) in a nitrogen atmosphere using a SHIMADZU DSC-50 analyzer.The sample masses used for DSC measurements were about several miligrams and they were heated in the DSC cell from the room temperature to 650 0 C in a stream of nitrogen with N 2 flowing at a rate of 20 mL min -1 and at the heating rates of 5, 10, 20 and 30 K min -1 .
In order to investigate the structural transformations by X-ray diffraction, the samples of the amorphous alloy Fe 81 B 13 Si 4 C 2 were annealed at different temperatures (298, 473, 573, 673, 823, 873, 973 and 1103 K) in a stream of nitrogen during 30 min.The X-ray powder diffraction (XRD) patterns were recorded on a Philips PW-1710 automated diffractometer using a Cu tube operated at 40 kV and 30 mA.The instrument was equipped with a diffraction beam curved graphite monochromator and the Xe-filled proportional counter.For the routine characterization, the diffraction data were collected in the range of 2θ Bragg angles (4-100 0 counting for 0.1 seconds).For the diffraction data used for crystallite size measurements between 40-50 0 , the Bragg angles were collected using a four-second scan at 0.02 0 steps, respectively.A fixed 1 0 divergence and 0.1 mm receiving slits were used.Silicon powder was used as an external standard for the calibration of the diffractometer.All XRD measurements were done with solid samples in the form of ribbon at the ambient temperature.

Microstructural parameters
The crystallite size dimensions (D hkl ) were determined by using an interactive Windows program for profile fitting and the size analysis (Winfitt).The full-width at half-maximum (FWHM) values of the peaks at the corresponding Bragg angles were fitted assuming a Pearson VII function for a profile.
Microstrain in the investigated sample was calculated by the following equation: where ψ is the width of the diffraction line, which arises only from the structural factors and θ is the Bragg angle.

Kissinger method
Crystallization kinetics is an important subject in the research of amorphous alloys, and the crystallization apparent activation energy represents the significant part of comprenhensive crystallization kinetics.Usually, the Kissinger equation (Kissinger, 1957(Kissinger, , pp.1702-6) -6) is used for the determination of apparent activation energy by measuring crystallization temperatures as a function of heating rates.This method assumes that the reaction rate has a maximum value at the peak temperature (T p ).This assumption also implies a constant degree of conversion (α) at T p .In many cases, the extent of conversion (α p ) at T p varies with the heating rate (β).However, in the present study, the analysis of the crystallization data shows that α p is nearly constant, and we may classify the current method as a special case of the KAS (Kissinger-Akahira-Sunose) isoconversional method.The Kissinger equation (Kissinger, 1957(Kissinger, , pp.1702-6-6) can be presented in the following form as: where the plot of ln(β/T p 2 ) versus 1/T p should give a straight line with the slope, which gives the value of the apparent activation energy (E a ).The pre-exponential factor (A) can be calculated from the intercept value of the estimated straight line.

Isoconversional (model-free) method
The differential isoconversional method was firstly proposed by Friedman (Friedman, 1964, pp.183-195).The basic assumption of the differential isoconversional method is that the reaction rate at a constant conversion value is only a function of temperature.This premise supposes the fact it is accepted to different kinetic parameters, E a,α and A α , for each value of α over the entire reaction range.The Friedman isoconversional equation can be expressed in the following form: (henceforth, the subscript α designates the values related to a given conversion, and i is an ordinal number of the non-isothermal experiment conducted at the heating rate β i ).By plotting ln[β i (dα/dT) α,i ] against 1/T α,i , the value of -E a /R for a given value of α can be directly obtained.Using this equation, it is possible to obtain the values for E a over a wide range of conversions.
At the presented diffractogram of the non-annealed Fe 81 B 13 Si 4 C 2 alloy (298 K), a broad peak with the maximum at 2θ ≈ 46.8 0 , characteristic for amorphous materials, can be observed.The annealing of the sample at T = 473 K leads to the appearance of a quite sharp peak at 2θ ≈ 83.8 0 (Fig. 1).The increase of annealing temperature in the range of 473 K ≤ T ≤ 823 K does not lead to considerable changes in the form of diffractograms, except for the decrease in intensity (a.u.) of the peak at 2θ ≈ 83.8 0 .In the diffractogram of the annealed alloy at T = 873 K, in respect to the diffractograms of the same alloy annealed at lower temperatures, the appearance of a peak at 2θ = 46.8 0can be observed.Further increase of annealing temperature (873 K ≤ T ≤ 1103 K) leads to the increase in intensity for the peak at 2θ ≈ 46.8 0 and to the decrease in intensity for the peak at 2θ ≈ 83.8 0 .
The comparative semiquantitative X-ray analysiscan establish that the broad peak at 2θ ≈ 46.8 0 and the sharp peak at 2θ ≈ 83.8 0 belong to the α-Fe phase (JPC DS No. 066698).
Bearing in mind the dis-arranged ratio of the intensities of the diffraction lines for the α-Fe phase, the sharp peak at 2θ ≈ 83.8 0 can be connected with the structurally deformed α-Fe crystalline phase, while the broad peak at 2θ ≈ 46.8 0 can be connected with the amorphous phase.
If the Scherrer formula for determining the crystallite size was applied on the broad diffraction peak of α-Fe, it can be concluded that in the non-annealed alloy there are highly disordered clusters of α-Fe, with dimensions of D hkl ≈ 7.1 × 10 -10 m.
The values of the microstructural parameters of the investigated alloy annealed at different annealing temperatures are presented in Table 1.
The annealing of the alloy at the temperatures T ≤ 973 K does not lead to significant changes in the microstructure of the formed crystals; however, their relative shares in the annealed alloy decrease.The structurally deformed α-Fe (observed at T = 473 K) does not undergo structural changes with the increase of the annealing temperature in the range of 473 K ≤ T ≤ 973 K.The corresponding share of this phase in the alloy increases until a temperature of 823 K, when a newly formed crystal phase appears (2θ = 46.8 0).At T = 973 K, the dimensions of crystallites sharply decrease and microstrains in the mentioned crystallites significantly increase.
The microstructural properties of the crystalline α-Fe (2θ = 46.8 0) formed at a temperature of T = 873 K from the amorphous clusters of Fe do not change with an increase of the annealing temperature, but their relative share in the sample increases.The kinetic study of phase transformations which appear during the procedure of annealing of the Fe 81 B 13 Si 4 C 2 alloy at different temperatures is performed on the basis of the DSC curves recorded at different heating rates.Fig. 2 shows the DSC curves for the Fe 81 B 13 Si 4 C 2 ribbon, recorded at four different heating rates (5, 10, 20 and 30 K min -1 ).On all DSC curves, we can see the broad (the low temperature) peak and the sharp (the high temperature) peak.With an increase of the heating rate (β), the values of the initial (T i ), the maximum (T p ) and the final temperature (T f ) for both peaks (Table 2) were shifted to higher values.The volume fraction (the degree of conversion) (α) of the sample transformed into the crystalline phase during the crystallization event has been obtained from the DSC curve as a function of the temperature (T).The volume fraction crystallized, α, at any temperature T is given as α = S T /S, where S is the total area of the exotherm between the temperature T i , where the crystallization is just beginning and the temperature T f , where the crystallization is completed, and S T is the area between the initial temperature and a generic temperature, T, ranging between T i and T f .
The dependence of ln[-ln(1-α)] versus -lnβ at different temperatures (T) shows high values of the linear correlation coefficient (r).The values of n obtained from the slopes of the linear plots presented in Fig. 3, for the considered crystallization process, are given in Table 3.At all considered temperatures, the values of n, in the limits of the experimental errors, were approximately the same (n ≈ 4.00).From these results follows that the kinetics of the crystallization process is independent of the temperature and also, in accordance with Henderson (Henderson, 1979a, pp.325-331), (Henderson, 1979b, pp.301), the above results indicate that the nucleation process occurs with a constant rate.
The Kissinger method is used for the determination of the kinetic parameter values (E a and lnA) for both phase transformations.The Kissinger plots (ln(β/T p 2 ) versus 1/T p ) for both considered phase transformations are shown in Fig. 4   From the slope and the intercept of the linear dependence ln(β/T p 2 ) versus 1/T p , the values of the kinetic parameters (E a and lnA) are calculated, and the results are presented in Table 2 (columns 5, 6, 10 and 11  in Table 2).
In the case of continuous heating, it can be shown that a generalization of the JMA theory leads to the validation of the Kissinger analysis, through the following equation (Vázquez, et al., 1990, pp.181): ( ) ,

37
. 0 where β is the heating rate, while (dα/dt) p represents the crystallization rate at the maximum value, i.e., the rate at the peak of an exothermal reaction, detected by the DSC technique.Eq. ( 5) was derived assuming that usually E a >> RT p and that the crystallized fraction, for the maximum crystallization rate is 0.63 (Bezjak, et al., 2007, pp.1), which is independent of the heating rate and of the Avrami kinetic exponent (Vázquez, et al., 1990, pp.181).From the relation α = S T /S, it could be possible to calculate the crystallization rate function, dα/dt.Eq. ( 5) can then be used to determine the Avrami kinetic exponent, n, as an average of the set of parameters obtained for different heating rates.
The values of the Avrami kinetic exponent for the high temperature peak (Table 2) were calculated using Eq. ( 5).The following values of n were obtained as: n = 3.55, 4.23, 4.00 and 3.68 for β = 5, 10, 20 and 30 K min -1 , respectively.If we compare the average value of n evaluated from Eq. ( 5) (n av ≈ 3.90), with an average value of n calculated using Eq.(4) for three selected and fixed temperatures (n av ≈ 4.00, see Table 3), we can conclude that a good agreement exists between these two values.
By the comparison of the results obtained from the X-ray analysis and the DSC measurements, it can be concluded that the sharp X-ray peak positioned at 2θ ≈ 83.8 0 can be attributed to the processes of rearrangement of the α-Fe clusters, while the peak positioned at 2θ ≈ 46.8 0 can be attributed to the processes of the α-Fe crystallization, during alloy annealing at certain temperature.
In order to determine the influence of nucleation and crystal growth processes on the values of kinetic parameters, the differential isoconversional (model-free) method was applied, and the corresponding dependence of E a on α was established.According to the isoconversional method, the linear relationships of ln[β i (dα/dT) α,i ] versus 1/T α,i were found for the high temperature DSC peak (Fig. 5).This result describes very well the data obtained from the non-isothermal DSC measurements, in the conversion range (α) of 0.05 ≤ α ≤ 0.90.The values of the apparent activation energy (E a ) and the apparent intercepts (ln[A α f(α)]) were calculated by means of this method, and the obtained results are presented in Fig. 6.Any point in this figure (corresponding to E a for constant α) was obtained from Eq. (3), within the certain experimental error limits (specified by the error bars).
The Friedman differential method for the calculation of the apparent activation energy values removes the systematic errors due to the approximation of the temperature integral.However, it can be pointed out that the Friedman method introduces large errors for small values of (βdα/dT), such as near the onset and sometimes near the end of the reaction (Vyazovkin, Wight, 1998, pp.407), (Munteanu, et al., 2003(Munteanu, et al., , pp.1995)).
It can be observed that the apparent activation energy for the considered crystallization process is practically constant in 0.05 ≤ α ≤ 0.90 range, and its values were found to be E a = 347.3± 5.5 kJ mol -1 .
The above results indicate that there is a high probability for the presence of a single-step reaction (Vyazovkin, 2000, pp.155), (Opfermann, Flammersheim, 2003, pp.1), i.e., this statement is in agreement with the above exhibited unique process of nucleus growth with a value of E a,G = 347.3kJ mol -1 (the average value obtained by the differential isoconversional method).The value of the apparent activation energy calculated by the differential isoconversional method is lower than the value of the apparent activation energy calculated by the Kissinger method (E a,Kissinger = 351.2kJ mol -1 (attached to peak II)).Fig. 7 shows that the values of lnA follows the same trend as the values of the apparent activation energy on the degree of conversion (α) (Fig. 6).This dependence is evaluated for the JMA model function in the form of f(α) = 4(1 -α)[-ln(1 -α)] 3/4 at n = 4.0.
It can be pointed out that the crystallization exponent n is connected with the number of growth dimensions (m) and the number of nuclei forming stages (s) (Brown, Dollimore, Galwey, 1980) by the following equation: where m is the number of growth dimensions (m = 1 for one-dimensional growth; m = 2 for two-dimensional growth (for disc and cylinder) and m = 3 for three-dimensional growth (for sphere and half-sphere)), s is the number of the nuclei forming stages (s = 0 for the instantaneous nucleation; s = 1 for the constant nucleation rate, and s > 1 for the self-acceleratory nucleation rate).In order to describe in detail the considered crystallization process, the value of the parameter m should be determined.In accordance with Farjas and Roura (Farjas, Roura, 2006, pp.5573), the fractional extent of the reaction can be expressed as: where E a represents the overall apparent activation energy (E a = (E a,N + m•E a,G )/n), and C is a constant that depends on m, E a,N and E a,G .) ( The function p(E a /RT) in Eq. ( 7) represents the temperature integral, with the reduced apparent activation energy expressed as x = E a /RT.
The straight line dependence between ln[-ln(1-α)] and (1/T) at all considered heating rates was obtained.Based on the above facts, the calculation of the parameter m is possible.
The values of the parameters m and s obtained at different heating rates (β) for the investigated crystallization process of the α-Fe phase, in Fe 81 B 13 Si 4 C 2 amorphous alloy are given in Table 4. Based on the obtained values of the parameters m and s, at different heating rates (Table 4), we can assume that the nucleation process of the α-Fe phase occurs within the amorphous alloy with a constant rate.The growth of α-Fe crystallites occurs in three geometrically effective directions (the threedimensional (3D) growth) and this process also proceeds with a constant rate.

Conclusion
Bearing in mind all the above exhibit results, we can establish the following conclusions: i) In the non-annealed sample of the Fe 81 B 13 Si 4 C 2 alloy, there are highly disordered clusters of the α-Fe phase.
ii) The isothermal annealing of the alloy sample at T = 473 K leads to the formation of the structurally deformed crystalline α-Fe phase of nano-dimensions i.e., the simultaneous existing of the amorphous noncrystalline α-Fe phase in the sample.
iii) Further crystallization of α-Fe in the considered alloy at a higher annealing temperature precedently follows the phase transformation within the α-Fe cluster by a disorder-order type with E a,I =134.0 kJ mol -1 and lnA I = 18.12.
iv) The isothermal crystallization was governed on the previously formed nucleuses of the α-Fe phase, with the three-dimensional (3D) growth of α-Fe crystal germs.
v) The kinetics of the non-isothermal crystallization was described by the JMA (Johnson-Mehl-Avrami) equation with the following values of the kinetic parameters: m = 3.0, s = 1.0 (n = 4.0) and E a = 347.3kJ mol -1 .
vi) At T > 973 K, the structurally deformed crystalline phase of α-Fe was dissolved and it participated in the crystal growth of the crystalline α-Fe.
The obtained DSC (the two peaks were found -peak I and II), the XRD results, and in addition, the kinetic description of the crystallization process are contrary to the results obtained by the Santos and Santos (dos Santos, dos Santos, 2001, pp.47-52), (dos Santos, dos Santos, 2002, pp.56).These facts point to the physico-chemical and microstructural differences between the investigated samples, which are caused by the different procedures of their preparation.

Table 1 -
Values of the microstructural parameters of Fe81B13Si4C2 alloy annealed at

Table 2 -
The influence of the heating rate (β) on the characteristic temperature values of the DSC curves; The values of the kinetic parameters (Ea (I;II) and lnA (I;II)) calculated by the Kissinger method are given in the same Table.Tabela 2 -Uticaj brzine zagrevanja (β) na karakteristične temperaturske vrednosti DSC krivih; vrednosti kinetičkih parametara (Ea (I;II) and lnA (I;II)) izračunatih Kisindžerovom metodom prikayane su u istoj tabeli.

Table 3 -
The changes of the n values with temperature, for the investigated crystallization process of the amorphous Fe81B13Si4C2 alloy Tabela 3 -Promene vrednosti n sa temperaturom, za ispitivani proces kristalizacije amorfne Fe81B13Si4C2 legure

Table 4 -
The dependence of the m and s parameter values on the heating rate β, for the crystallization process of the amorphous Fe81B13Si4C2 alloy Tabela 4 -Zavisnost vrednosti m i s parametara od brzine zagrevanja β, za proces kristalizacije amorfne Fe81B13Si4C2 legure.