Fe-Based Nanocomposite Formed by Thermal Treatment of Rapid-Quenched Fe81B13Si4C2 Alloy

Amorphous alloys have been a focus of considerable scientific interest, both from fundamental and practical point of view, ever since the first of its kind (Al75Si25) was produced by Klement, Willens and Duwez in 1960 (Klement et al., 1960). It has been shown that the amorphous alloys have features that are different from those of crystalline alloys in both alloy compositions and atomic configurations. This enabled the exhibition of various characteristics which were not obtained for conventional crystalline alloys. Their soft ferromagnetic properties (saturation magnetization, high permeability, low coercivity and loss), high corrosion resistance and good mechanical properties make them suitable for use in a variety of applications, such as power devices, information handling technology, magnetic sensors, anti-theft security systems and construction materials (Minić et al., 2007). Since amorphous alloys are meta-stable, elevated temperature or prolonged performance can induce a transformation into a crystalline state, which could lead to a loss of their advantageous physical properties limiting them to single-use applications. Commercial soft magnetic nanocrystalline materials have recently been successfully obtained by crystallization of amorphous precursors. Materials like this are characterized by a microstructure of nanocrystals embedded into an amorphous matrix, exhibiting superior soft magnetic and mechanical properties to both amorphous and crystalline magnetic alloys (Blagojević et al., 2011; Minić et al., 2011a). This dependence of functional properties on microstructure can be used to produce functional materials with tailored properties (Maričić et al., 2012).


Introduction
Amorphous alloys have been a focus of considerable scientific interest, both from fundamental and practical point of view, ever since the first of its kind (Al 75 Si 25 ) was produced by Klement, Willens andDuwez in 1960 (Klement et al., 1960). It has been shown that the amorphous alloys have features that are different from those of crystalline alloys in both alloy compositions and atomic configurations. This enabled the exhibition of various characteristics which were not obtained for conventional crystalline alloys. Their soft ferromagnetic properties (saturation magnetization, high permeability, low coercivity and loss), high corrosion resistance and good mechanical properties make them suitable for use in a variety of applications, such as power devices, information handling technology, magnetic sensors, anti-theft security systems and construction materials (Minić et al., 2007). Since amorphous alloys are meta-stable, elevated temperature or prolonged performance can induce a transformation into a crystalline state, which could lead to a loss of their advantageous physical properties limiting them to single-use applications. Commercial soft magnetic nanocrystalline materials have recently been successfully obtained by crystallization of amorphous precursors. Materials like this are characterized by a microstructure of nanocrystals embedded into an amorphous matrix, exhibiting superior soft magnetic and mechanical properties to both amorphous and crystalline magnetic alloys Minić et al., 2011a). This dependence of functional properties on microstructure can be used to produce functional materials with tailored properties (Maričić et al., 2012).
The production of early amorphous alloys required very high cooling rates (as much as 10 6 K/s) to avoid crystallization. This limited the form in which they could be produced, as one dimension had to remain small enough to allow sufficiently rapid heat extraction, in order to achieve the necessary high cooling rates. The result was that the thickness of amorphous metal specimens was limited to less than 100μm. In 1976, Liebermann and Graham developed a new method of manufacturing thin ribbons of amorphous metals on a supercooled rapidly rotating disc -the melt-spinning method (Liebermann & Graham, 1976). For the next thirty years, with the production of new materials, the required cooling rate diminished, until, in 1990s, materials were developed, whose production required cooling rates as low as 1 K/s, allowing these alloys to be cast into metallic moulds to produce specimens up to 100mm thick (Ponnambalam et al., 2004). The alloys that require cooling rates below 10 3 K/s are known as bulk amorphous alloys, as the production process allows for higher specimen thickness (above 1mm), classifying them as bulk materials.

Experimental procedures
The ribbon shaped samples of 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 method). The obtained ribbon was 2 cm wide and 35 μm thick. During the preparation process of the amorphous alloy ribbon, one of the sides was in direct contact with the cooled rotating disc, while the other was in inert atmosphere. As a result, the two sides of the ribbon show an easily observable difference in reflectivity, surface morphology and structure, as can be seen in X-ray diffraction (XRD) spectra, Fig.1, as well as SEM images of the sample in the as-prepared alloy (Fig.2). The side that was in the contact with the cooled spinning disc is usually labeled as fishy or matte side and the other side, free of contact, as shiny side.
DSC was obtained using SHIMADZU DSC-50 analyzer. In this case, samples weighting several milligrams were heated in the DSC cell from the room temperature to 650 o C in a stream of nitrogen with nitrogen flowing at a rate of 20 mL min -1 at the heating rates of 5, 10, 20 and 30 o Cmin -1 .
Mössbauer spectra were taken in the standard transmission geometry using a 57 Co(Rh) source at room temperature. The calibration was done against α-Fe foil data. "CONFIT" program package was used for the spectra fitting and decomposition (T. Žák, 1999).
The X-ray diffraction (XRD) patterns were recorded on an X'Pert PROMPD diffractometer from PANalytical with CoKα radiation operated at 40kVand 30mA. For routine characterization, diffraction data was collected in the range of 2θ Bragg angles (15-135 o , step 0.0081). For a quantitative analysis and determination of crystallite size from XRD spectra, TOPAS V3 general profile and structure analysis software for powder diffraction data was used (BrukerAXS, general profile and structure analysis software for powder diffraction data, Karlsruhe, 2005). Dislocation density was obtained from the Rietveld analysis, while microstrain was calculated using Williamson-Hall method (Williamson, Hall 1953), using the XRD data. Lattice parameters obtained through XRD spectra were used to calculate the unit cell volumes, which were then compared to the standard values in JCPDS database.
Vickers microhardness tests were performed using MHT-10 (Anton Paar, Austria) microhardness tester, with loads of 0.4N and loading time of 10s (Minić, et al. 2011c). Up to seven measurements were performed on each individual sample, using the average value of microhardness for each sample. The measurements were performed on the cross-section of the ribbons, rather than on any of the sides. Error was calculated as standard deviation for each series of measurements.

Microstructure of as-prepared Fe 81 B 13 Si 4 C 2 amorphous alloy
XRD spectra of the as-prepared alloy ribbon, Fig.1, showed that the as-prepared alloy already had a degree of crystallinity caused by presence of α-Fe phase (JCPDS-PDF 06-0696). The degree of crystallinity is much higher on the shiny side, with peak intensity in XRD spectrum being about 8 times higher on the shiny side. The nanocrystalline phase can be identified as α-Fe on matte side and a mixture of α-Fe and Fe 3 Si on shiny side, with α-Fe being the major component. In addition to sharp crystalline peaks (37, 69 and 78 o ) in the XRD spectra of the as-prepared alloy, a broad spread halo around 53 o , corresponding to domains of short-range ordering in the sample, was also observed. The position of the spread halo corresponds to approximate position of Fe 3 Si peak and, using the Scherrer equation, we estimated the size of these domains to be 1-1.5 nm. The entire structure could best be described as a combination of nanocrystals and short-range ordered domains embedded in an amorphous matrix. Recent theoretical studies of ironbased binary systems predict existence of short-range ordering in iron-based amorphous alloys (Lass et al., 2010). Therefore, appearance of domains of short-range ordering in Fe 81 B 13 Si 4 C 2 amorphous alloy could be expected. Crystal structures of α-Fe and Fe 3 Si are closely related, as Fe 3 Si crystal system is cubic, same as α-Fe crystal system. Fe 3 Si lattice is composed of four sub-lattices: three composed of iron atoms and one of silicon atoms, and this leads to doubling of the unit cell (when compared to α-Fe). Fe 3 Si lattice is, also, slightly distorted, so the value of its lattice parameter is slightly higher than twice the value of lattice parameter of α-Fe.

Thermal stability of alloy
The thermal stability of the alloy was investigated using differential scanning calorimetry (DSC) in a nitrogen atmosphere (Minić, et al., 2009a). Typical DSC scan obtained during heating and cooling cycle is presented in Fig. 3. DSC scan involves a series of endothermic and exothermic peaks indicating a stepwise process of structural stabilization of the alloy in the temperature range 170-560 o C. A broad exothermic peak, indicated as (T sr ), in temperature range 170-400 o C, coresponding to structural relaxation, is followed by endothermic hump (temperature of glass transition T g ) and a short supercooled liquid region before the sharp exothermic crystallization peak (T k ) in temperature range 500-540 o C. The enthalpy of crystallization is 83.5 J/g as determined from area of corresponding peak obtained at heating rate of 20 o Cmin -1 . Fig. 3. DSC scan of heating and cooling cycle in nitrogen atmosphere; heating rate 10 o C/min.

Thermo-magnetic behavior
Thermally induced processes were also studied using the thermo-magnetic scan, where the sample is heated, annealed and then cooled in vacuum furnace at low magnetic field of 4 kA·m -1 while its magnetic moment is monitored (Minić, et al., 2011b). Both heating and cooling rate were 4 o C·min -1 , dwell time at maximum temperature of 800 o C was 30 minutes. The shape of the thermo-magnetic curve, Fig. 4, reflects changes in the magnetic moment of the sample, caused by phase or structural transitions.
Most pronounced change represents the Curie point (T c = 420 o C), where the magnetization of the respective phase drops to almost zero, because the thermal motion overcomes magnetic interaction. Annealing at the temperature near 200 o C is sometimes called stressrelieving (or structural relaxation). Temperatures marked by arrows in Fig. 4 have been identified as points of interest for further study of structural transformations.

Magnetic susceptibility
Measurements of relative magnetic susceptibility were performed using a modified Maxwell method, based on the action of an inhomogeneous field on the magnetic sample. The magnetic force measurements were performed with a sensitivity of 10 -6 N in an argon atmosphere (Minić, et al. 2009b).
The temperature dependence of the relative magnetic susceptibility of the as-prepared Fe 81 B 13 Si 4 C 2 amorphous alloy during three thermal treatments to different temperatures is presented in Fig. 5. During the first and second treatment, the decrease in the magnetic susceptibility in the temperature region from 320 o C to 380 o C is the result of proximity to the Curie temperature of the amorphous alloy. Before the start of the second treatment it was observed that magnetic susceptibility increased slightly. This was caused by the structural relaxation of an amorphous structure during the first treatment. During this process, internal strains and the free volume are reduced in the starting material. These changes are accompanied by subtle inter-atomic movements, causing the changes in the electron structure and leading to an increase in the number of electrons with unpaired spin in the direction of the outer magnetic field (Minić et al., 2009b). This also leads to a decrease in the number of electrons spinning in the reverse direction and causes an increase in the magnetic susceptibility upon cooling. At the same time, strains and decrease in the free volume enable greater mobility of the walls of the magnetic domains and this behavior further contributes to the increase in the magnetic susceptibility.
During the second treatment, the alloy loses its ferromagnetic properties in the temperature region from 400 o C to 470 o C. With further heating, the magnetic susceptibility starts to rise, and the alloy regains its ferromagnetic properties once the crystallization process starts at about 490 o C. After the second heating to 490 o C, the magnetic susceptibility decreases by 23 % when compared to the value of as-prepared alloy and to the value of the relaxed state of the lattice after the first heating cycle. During the third treatment, above the crystallization temperature, the alloy maintains its ferromagnetic features in the whole temperature region, whereas the maximum change in the magnetic susceptibility occurs at about 190 o C as a consequence of further phase transformation of the crystallized alloy.

Mössbauer spectra
In the Fig. 6, spectra illustrate the ability of Mössbauer effect to distinguish between individual iron-containing phases of different structure. Broad-line components (Fig. 6a) are Fig. 6. Mössbauer spectra of the as-prepared material (left) and of material after final annealing at 700°C (right), including components of iron containing phases. typical for the amorphous volume of the as-prepared sample, while the sharp lines ( Fig. 6b) characterize the crystalline structure with well defined position of atoms, which results from the thermally induced crystallization process during annealing of the sample (Minić, et al. 2011b). The computer processing of Mössbauer spectra yielded intensities, I, of components, their hyperfine inductions, B hf , isomer shifts, δ, and quadrupole splitting, σ (T. Žák & Y. Jirásková, 2006). The contents of the iron-containing phases are determined as proportional to the relative areas of the corresponding spectral components. The phase percentages correspond to the distribution of Mössbauer iron atoms among phases (Table 1) (Minić, et al. 2011e) In the as-prepared alloy, the amorphous structure, having a high-field and a low-field component is accompanied by a small amount of α-Fe(Si) solid solution and a FeB phase. Mössbauer phase analysis at higher temperatures reveals α-Fe(Si) solid solution and Fe 2 B phase to be the most important final crystallization products, although metastable phase Fe 3 B, was detected initially at 450 o C and in higher percentage at 500 o C. Amount of iron atoms in paramagnetic positions is almost below the sensitivity threshold. Content of silicon in the α-Fe(Si) solid solution seems to be about 9 at.%, which is close to 7 at.% published in (Saegusa & Morrish, 1982).

Electrical resistance
The electrical resistance of the ribbon was measured using the four-point method within a temperature interval of 20-630 o C in an argon atmosphere (Minić, et al., 2011d). Fig. 7a shows the temperature dependence of the electrical resistivity of the alloy in the temperature range of 25-630 o C. The dependence clearly shows each structural stabilization step which causes the change in the ordering of the investigated material. These changes are more obvious in the derivative curve (Fig. 7b).
The slow increase of electrical resistivity was caused by the structural relaxation process in the temperature range of 200-380 o C. This process is followed by an increase of electrical resistivity in the vicinity of Curie temperature T c at 420 o C, corresponding to the first maximum of the differential curve. At this point the effect that scattering of conductive electrons had on the magnons disappeared (I. Balberg & J. S. Helman, 1978;G. Bohnke et al., 1983) and the amorphous alloy loses its ferromagnetic properties (D. M. Minić et al. 2010). This is in excellent agreement with the results of the thermo-magnetic measurements (Fig.  4). The beginning of crystallization at about 520 o C caused a sharp decrease of electrical resistivity. The appearance of two clearly separated maxima, T k1 and T k2 (490 and 510 o C respectively) on the differential curve of electrical resistivity (Fig. 7b), suggests that crystallization of the amorphous alloy is a complex process, occurring in two steps, which appear as a single overlapping peak in the DSC scans, Fig. 3. The electrical resistivity of the crystalline alloy is lower than that of the amorphous alloy of the same composition, as a result of the increase in electron free path. The linear change of electrical resistivity with increasing temperature during the second treatment shows that crystallization was completed during the first heating cycle (Fig. 8).

Thermo-electromotor force
The structural relaxation processes, as well as the crystallization, in the temperature interval of 25-680 o C, were also investigated by measuring the thermo-electromotor force (TEMF) of a thermocouple made by coupling a copper conductor to the amorphous alloy (D.M. Minić, et al. 2009b). The alloy sample was mechanically attached to a copper conductor, forming the Cu -Fe 81 B 13 Si 4 C 2 thermocouple, which was placed into a specially designed furnace, while the other end of the sample was submerged into a mixture of water and ice. The TEMF produced by the thermocouple during the heating process was measured by a voltmeter with the sensitivity of 10 -5 V.
The temperature dependence of a thermo-electromotor force ( Fig. 9) shows three linear regions corresponding to the structural transformations of the alloy. Different slopes of these linear dependences correspond to structural changes involving a structural relaxation, the loss of ferromagnetic properties, and the crystallization, respectively. The temperature coefficient of TEMF is a function of the electron state density at the Fermi level: where k is Boltzmann's constant, e is electron charge, N 1(Ef) is the electron state density in copper and N 2(Ef) is the electron state density in the alloy.
The electron density of states in copper remained unchanged during heating to 680 o C, meaning that the change in the temperature coefficient during the heating of the thermocouple was caused only by the change of the electron density of states at the Fermi level of the alloy. Based on the slope of the first linear segment in Fig. 9, temperature coefficient α 1 = 9.4 µV/K, and the relative change in the electron density of states of the alloy caused by the structural relaxation process was determined to be The increase in the electron density of states at the Fermi level and the above mentioned increase in free path of the electron combine to diminish resistivity of the crystalline alloy (Fig. 8).

Structural transformations induced by thermal treatment
X-ray diffraction (XRD) spectra of the alloy ribbon samples (Fig. 10) show that thermal treatment (200-700 o C) caused a series of structural transformations of the amorphous alloy leading to formation of more than one crystalline phase: the stable α-Fe, Fe 3 Si and Fe 2 B as well as metastable Fe 3 B. Crystallization initially leads to formation of a nanocomposite structure of nanocrystals dispersed in the amorphous matrix. After thermal treatment at 700 o C, the alloy ribbon sample was fully crystallized and composed of interdispersed nanocrystals of three crystalline phases: α-Fe, Fe 3 Si and Fe 2 B. The analysis of crystallite orientation on the two sides of the ribbon showed that Fe 3 Si and α-Fe crystallites exhibit a degree of preferential orientation after thermal treatment. In the as-prepared alloy, Fe 3 Si and α-Fe crystallites on the shiny side are oriented preferentially in [100] direction, while those on the matte side are not. This is probably the reason for higher reflectivity of the shiny side of the ribbon and the consistently high intensity of peak around 78 o in the XRD spectra of the shiny side. After the treatment at 500 o C, the degree of preferential orientation increases on the shiny side and forms on the matte side, suggesting asymmetric growth of the crystallites. After treatment at 700 o C, amplitudes of the degree of preferential orientation on both sides decreased, suggesting that the crystallites were growing in more symmetrical manner. All of these changes are much more pronounced on the shiny side than on the matte side.
Rietveld analysis (Minić, et al. 2011c, Fig. 11) of XRD spectra yielded phase composition of the crystalline portion of the samples, average crystal sizes, dislocation density and microstrain for individual phases (Table 2). R 2 was greater than 0.98 for all XRD measurements. Individual phase contributions of α-Fe and Fe 3 Si could not be completely separated, because of the overlap of their peaks. The phase composition data shows that shiny side has higher percentage of metastable Fe 3 B phase, but lower percentage of Fe 2 B phase. After heating at 700 o C, the final phase composition on the matte side shows more crystalline Fe 2 B than on the shiny side. The phase content of Fe 3 B declines in two steps, with increase in heating temperature, to disappear completely after thermal treatment at 650 o C. The first sharp decrease in percentage of Fe 3 B (550 o C) coincides with an increase in percentage of combined α-Fe and Fe 3 Si. After thermal treatment at 650 o C, Fe 3 B phase disappears completely, and this coincides with increase in phase content of Fe 2 B. Phase content of Fe 2 B phase shows an increase after treatment at temperatures above 550 o C.
The evolution of average crystal sizes for the respective phases shows that while there is some difference between shiny and matte side of the ribbon at the onset of crystallization, those differences become almost negligible after heating at 700 o C. On closer examination, Fe 3 Si and α-Fe have lower crystal sizes on the matte side, at heating temperatures of 500 o C and below, and then grow faster than they do on the shiny side. As a consequence, crystal sizes of Fe 3 Si and α-Fe, respectively, at the end of crystallization are almost the same on both sides of the ribbon. The evolution of average crystal size of Fe 2 B shows the same trend of smaller initial crystal size and then faster crystal growth on the matte side, with increase in temperature. Metastable Fe 3 B phase shows the same trend with regards to growth, except, it has a higher average crystal size on the matte side, after the samples are treated at 500 o C. Fig. 10. XRD spectra of alloy samples after thermal treatment (left -shiny side; right -matte side) In order to further comprehend the changes in phase composition, unit cell volumes for individual phases were determined using XRD data, and compared with standard values in JCPDS database (Minić, et al. 2011c). This way, we estimated the lattice distortion caused by the presence of boron in α-Fe and Fe 3 Si lattices. The change in unit cell volume was negative for α-Fe and positive for Fe 3 Si phase. The diagram (Fig 12) shows that the distortion of the unit cell of α-Fe was the greatest before the crystallization started; it decreased during the crystallization and remained relatively stable after treatment at 500 o C and higher temperatures. The change in Fe 3 B percentage after treatment at 550 o C was accompanied by a decreased distortion of the unit cell of both α-Fe and Fe 3 Si and increase in their combined phase content. This indicates that iron from Fe 3 B was transformed into these two phases, with boron being incorporated back into the α-Fe/Fe 3 Si matrix. As Fe 3 Si showed greater change, Fe 3 B probably transformed more to Fe 3 Si than it did to α-Fe. Reduced lattice distortion in these phases is probably due to the fact that mass percentage of boron in Fe 3 B (6.06%) is lower than the mass percentage of boron in the as-prepared alloy (13%). Therefore, transformation of Fe 3 B would lead to dilution of boron in α-Fe/Fe 3 Si matrix, leading to stabilization of its crystal structure. The change in unit cell volumes of Fe 3 Si and α-Fe showed increase at higher heating temperatures and it is possible that the distortion of the lattice in α-Fe and Fe 3 Si is caused by the increasing occurrence of crystal/crystal interfaces between α-Fe and Fe 3 Si on one side and Fe 2 B on the other.

Mechanical properties
Microhardness was measured on the cross-section of the alloy ribbon samples, rather than on any of the sides, giving a good measure of the average properties of the alloy samples. This was done due to extreme brittleness of the ribbon samples after crystallization. In order to clearly show the influence of the change in microstructure, induced by thermal treatment, on microhardness, we presented the microhardness data in combination with DSC scan (Fig.  12). In terms of microstructure, the as-prepared alloy contains a small percentage (less than 5%) of crystalline α-Fe phase in form of nanocrystals dispersed in the amorphous matrix ( Fig. 1) and small domains (1-1.5 nm in size) of short-range crystalline ordering. This structure can be best described as a nanocomposite of nanocrystals and nanoclusters embedded in the amorphous matrix. The nanocomposite structure combines with chemical composition, involving significant percentage of boron, silicon and carbon, all of which are known to increase hardness in iron alloys, to produce high hardness of the as-prepared alloy sample (909HV).
The change of microhardness with respect to heating temperature showed three distinct temperature regions with completely different behavior. Before the onset of crystallization around 450 o C, microhardness exhibited slight growth, from 909 to 931HV, which corresponds to lattice relaxation, as shown by the broad exothermic peaks in 200-400 o C region in the DSC (Fig. 13a). In the second region, after thermal treatment at 450 o C, it increased to 951HV and then to 1250HV, after thermal treatment at 500 o C. This corresponds to crystallization involving formation of disordered Fe 3 Si crystal phase (around 450 o C), stable crystalline phases Fe 3 Si, Fe 2 B, α-Fe and metastable Fe 3 B (exothermic peak around 512 o C). These phases form a nanocomposite of nanocrystals dispersed in the amorphous matrix, which is the main cause of the increased microhardness. In the last temperature region, after the sample was treated at 700 o C, microhardness decreased to 908HV, as a consequence of further crystal growth, leading to the change in the nature of the interfaces. These changes in microhardness can also be correlated with change in microstructure, induced by thermal treatment, through evolution of average crystal size of the crystalline phases in the alloy (Fig. 13b) (Minić, et al. 2011c). It can be seen that average crystal size has a significant influence on the change in microhardness. When the formed nanocrystals are relatively small (below 50 nm), microhardness remains high, while appearance of larger nanocrystals (over 100 nm) leads to a sharp decrease in microhardness. After thermal treatment at temperatures from 200 to 700 o C, the alloy structure gradually transformed from a relatively homogeneous to a granulated structure with larger crystalline domains. This is consistent with creation, at the beginning of the crystallization, of a nanocomposite of small nanocrystals dispersed in amorphous matrix, which significantly increases microhardness, due to the fact that the dominant type of interface in the alloy is crystal/amorphous, as opposed to crystal/crystal in a completely crystalline alloy. As the average crystal size increased, crystal/crystal interfaces became dominant and microhardness decreased, due to increased interfacial energy and more successful propagation of shear bands and cracks along these interfaces. During the course of the observed structural transformations, the overall composition of the alloy did not change drastically, while microhardness showed significant fluctuation. This means that the changes of microhardness in our alloy were caused primarily by changes in its microstructure, rather than changes in its composition, with the main factor being the change in the average crystal size and creation of a granulated structure, as opposed to nanocrystals embedded in amorphous matrix ( Table 2). The granulated structure, in addition to containing larger crystals and more crystal/crystal interfaces, is also much more porous than the original nanocomposite crystal/amorphous structure of the as-prepared alloy.

Kinetics of crystallization
All kinetic studies assume that the isothermal rate of conversion dα/dt is a linear function of the temperature-dependent reaction rate constant, k (T), and a temperature-independent function of the conversion, f (α) (Vyazovkin, 2000)  where α is the fractional extent of reaction (conversion degree), t is time and the function f(α) depends on the particular crystallization mechanism.
The temperature dependence of the rate conversion is introduced by replacing k(T) with the Arrhenius equation, which gives the relation where A (pre-exponential factor) and E (activation energy) are the Arrhenius parameters and R is the gas constant.
Kinetic description of solid state transformations usually includes a kinetic triplet, involving Arrhenius parameters (activation energy, E and pre-exponential factor, A) as well as an algebraic expression of the conversion function, f(α) (presented in Table 3), which describes the dependence of the reaction rate on the conversion degree, α.
In solid state reactions, the constant value of activation energy can be expected only for a single-step reaction, therefore E in equation (3) becomes an apparent quantity (E a ), based on a quasi-single-step reaction. In non-isothermal measurements at constant heating rate, β, the equation (3) transforms to: The integral form of the reaction model can be obtained by integration of equation (4) where p(x) is the temperature integral for x=E a /RT which does not have analytical solution.

Fig. 14. DSC curves at different heating rates
The different algebraic expressions of conversion functions for solid state transformations are given in Table 3.
Fig. 14 shows the continuous DSC curves of Fe 81 B 13 Si 4 C 2 ribbon in temperature range 300-650 o C taken at four different heating rates. All curves contain a single well formed exothermic peak representing crystallization in the temperature range 500-560 o C.
Mechanism label f(α) g(α) Power law P4 4α 3/4 α 1/4 Power law P3 3α 2/3 α 1/3 Power law P2 2α 1/2 α 1/2 Power law P3/2 3/2α 1/3 (1-α) 2 (1-α) -1 F3 Third-order 1/2(1-α) 3 (1-α) -2 Table 3. Algebraic expressions of conversion functions for solid state transformations All values of initial (T i ), maximal (T p ) and final temperature (T f ) for both exo-peaks are shifted to higher values with increasing heating rate indicating the presence of kinetic effects ( Table 4). The shape of DSC curves depends on the heating rate, because, as the values of shape factor S show, asymmetry of peaks increases with the decrease of heating rates indicating that the heating rate has a strong influence on the crystallization process. The values of shape factor S is obtained as the ratio of half-widths (left and right) for individual crystallization peaks for particular heating rate. The fractional extent of the sample transformed into crystalline phase, α, has been obtained from the DSC curve as a function of temperature (T) (Minić & Adnađević, 2008). At any temperature T, α is defined as α = S T /S, where S is the total area of the exotherm, between the temperature T i , of the onset of crystallization and the temperature T f , of the end of crystallization, and S T is the area between the initial temperature T i and a generic temperature, T, ranging between T i and T f (Friedman, 1964). The plots of α versus T at different heating rates for the considered crystallization process are shown in Fig. 15.
The sigmoid shape of fractional conversion curves in Fig. 15 indicates, for all heating rates, a slow initial period corresponding to nucleation with increasing rate caused by the increase of specific surface area of nuclei. The saturation part that follows is the consequence of nuclei merging together causing a decrease in their surface area. Fig. 15. Fractional conversion (α) as a function of temperature (T) for the crystallization alloy at different heating rates (5, 10, 20 and 30 o Cmin -1 ).

The overall apparent activation energy of crystallization
The overall activation energy of crystallization of an amorphous alloy under linear heating condition can be determined by the Kissinger as well as by the Ozawa peak method relating to the dependence of exothermic peak temperature Tp on heating rate β.
Kissinger (Kissinger, 1957) proposed that the activation energy can be determined according the equation For the determination the activation energy in non-isothermal conditions Ozawa (Ozawa, 1970) proposed the equation: The values of peak temperatures together with values of kinetic parameters (E a and lnA) calculated by both methods and the symmetry factors (SF) are given in the Table 4.  The dependence of maximum of rate on rate of heating (Fig. 16) shows the growth of maximum of rate of crystallization with respect to the rate of heating. It indicates that the rate of crystallization reaches a saturation point at high heating rates.
The activation energy of crystallization process involving formation of nuclei and their growth, according to some opinions, has no physical meaning, just empirical significance and only establishes the dependence of the rate of conversion on temperature. This energy can be spent, not just for overcoming the activation barrier but also for its downturn due to cooperative displacement of groups of atoms. Finally, the crystallization of amorphous alloys is a very complicated process accompanied by nucleation and growth of various crystal phases under continuously varied conditions in the conversion zone. With the multitude of possible ways of conversions, only those mechanisms and activated complexes of the crystallization process will be realized that have the highest probability at a given temperature. Any change in crystallization conditions, such as heating rate, can result in change of the mechanism and main activation complex of the crystallization process. Thus, high values of activation energy of crystallization of amorphous alloys, first of all, indicate that a large number of atoms participate in an elementary act of structure reorganization, as well as high complexity of the transformation processes.

The dependence of apparent activation energy on range of conversion
On the basis of dynamic DSC measurements for different heating rates, isoconversional method of Kissinger-Akahira-Sunose method, also known as the "model-free method", enables the determination values of E a over a wide range of α without the determination of the conversion function (Kissinger, 1957;Akahira, Sunose, 1971). This model involves measuring the temperatures T α corresponding to fixed values of the crystallized volume fraction, α, for different heating rates, β, and application of the relation: where a subscript α designates values related to a given conversion degree, and i is a number of the non-isothermal experiment conducted at the heating rate β i .
The left-hand side of Eq. (8) is linear with respect to the inverse temperature, 1/T α , and enables the activation energy to be evaluated using a linear regression method. In case of a single step process, a constant value of E a(α) is obtained. On the other hand, the dependence of E a(α) on α indicates complex process involving more than one step with different activation energies. It can be observed that the apparent activation energy for the considered crystallization process (Fig. 17) is practically constant in the 0.05 ≤ α ≤ 0.7 range indicating the existence of a single-step reaction. (Vyazovkin, 2000;Opfermann & Flammersheim, 2003). The average value of apparent activation energy was determined as E a = 356.5  5.5 kJmol -1 . Fig. 17. Apparent activation energies (E a ) and the intercepts as function of the crystallized fraction α.

Preliminary determination of kinetic model
For the preliminary determination of kinetic model of the crystallization process, Dollimore's method was used (D. Dollimore, 1991Dollimore, , 1992Lee, 1998).  This model is based on the "sharpness" of initial and final temperature of differential rate curves, Fig. 18, as well as on its asymmetry. The "sharpness" of the initial and final temperatures is influenced by kinetic factors, and especially by the mechanism of the process. Certain kinetic models lead to an asymptotic or diffuse departure from the base line in the differential form of the thermal curve, while others produce a very sharp approach to the final plateau. The investigation of these parameters that describe geometry and asymmetry of the differential rate curves can be an indicator of the probable kinetic mechanism. According to these parameters, different types of kinetic mechanisms have been listed (Lee, 1998). So, when the crystallization process is not complex the qualitative approach to its kinetics may be obtained using parameters such as α max or (dα/dt) max , the shape of the initial and final temperatures as well as peak temperature (T p ) or half-width from differential rate curves. In this case, Dollimore's procedure was applied to the conversion and differential rate curves (Fig.18) whose slight asymmetry is observed between T i and T f . The other parameters such as the conversion at the rate of maximum crystallization, α max , peak temperature, T p , at (dα/dt) max , and the ratio ∆LoT/∆HiT (shape factor), which is the ratio between the low and high temperature points at half-width of the differential rate curve peak are presented in Table 5. It is clearly seen that the position of the broadening exotherm, which is connected with the crystallization, was shifted toward higher temperature with the increase of the heating rate as well as asymmetry of peaks. This suggests that the crystallization process should not be characterized by a definite critical temperature independent of the heating rate. The determined values of α max for different heating rates were in the range from 0.51 to 0.55. These results indicate that the non-isothermal crystallization mechanism of α-Fe in amorphous Fe 81 B 13 Si 4 C 2 alloy can not be fully described within the JMA (Johnson-Mehl-Avrami) models (A2, A3 and A4, Group A).

Determination of the kinetic model
General equation, enabling the analysis of conversion kinetics involving nucleation and growth in solid phase was proposed by Avrami (Avrami, 1939): where (t) is conversion degree, n is kinetic exponent, k=k o exp(-E a /RT).
Differentiation of this equation with respect to time gives the rate equation, known as the JMA equation: The JMA equation is based on assumptions of isothermal crystallization, homogenous or heterogeneous nucleation at randomly dispersed particles of the second phase. The growth rate of new phase is independent of time and controlled by temperature and low anisotropy of growing crystals. However if the entire nucleation process takes place during the early stage of transformation and becomes negligible afterwards, JMA equation can also be applied to non-isothermal conditions (Henderson, 1979).
The validity of listed assumptions is not given a priori and simple and reliable testing methods were developed (Málek, 1992(Málek, , 1995Gotor et al, 2000;Criado et al, 2003). Once the apparent activation energy has been determined, it is possible to find the kinetic model which best describes the measured set of thermoanalytical data. It can be shown that, for this purpose, it is useful to define two special functions y(α) and z(α), which can easily be obtained by simple transformation of the experimental data. The conversions, in which the y(α) and z(α) functions exhibit the maximum values are designated as y   and z   , respectively. Under non-isothermal conditions these functions can be expressed as follows (Málek, 1992(Málek, , 1995(Málek, , 2000: The maximum of the y(α) function for the JMA model depends on the value of the kinetic exponent: y   = 0 for n ≤ 1 y   =1-exp (n -1 -1) for n > 1 The value of y   is always lower than the maximum of value for z   . For JMA model, z   =0.632. This value is characteristic "fingerprint" of the JMA model and it can be used as a simple test of the applicability of this model.
The obtained normalized functions y(α) and z(α), Fig 19, are independent on the heating rate (β) of the system, and the both functions exhibits the well-defined maxima which were located at exactly defined value of α ( y   for the y(α) function and z   for z(α) function, respectively), Table 6. crystallization kinetics, where the crystallized phase further increases the rate of the crystallization. Such autocatalytic behavior can be well described using an empirical two parameter Šesták-Berggren's kinetic model (Šesták, Berggren, 1971;Málek, et al. 1989). This model is based on the equation: where M and N represents the kinetic exponents.
In this case the expression for reaction rate of the investigated crystallization process can be given as: For this model, the ratio of the kinetic exponents p = M / N can be calculated from the maximum of the y (α) function (Málek, 2000): Introducing this equation in equation this (13) gives:   (Gotor, et al. 2000).
In order to check the established kinetic model we applied the "Master-plot" method (Criado, et. al 2003;Gotor, 2000). Using the value at α = 0.5 as a reference point, the following differential master equation is easily derived from eq. (4): where (dα/dt) 0.5 , T 0.5 and f (0.5) are the reaction rate, the temperature reaction and the differential conversion function, respectively at α = 0.5.
The left side of Eq. (17) is a reduced theoretical curve which is characteristic of each kinetic function. The right side of the equation is associated with the reduced rate and can be obtained from experimental data if the apparent activation energy is known and remains constant throughout the reaction. Comparison of both sides of Eq. (17) tells us which kinetic model describes an experimental reaction process. It can be seen in Fig. 20, using the average value for the apparent activation energy determined from Kissinger-Akahira-Sunose isoconversional method, the suggested kinetic model works very well in the entire conversion range. Therefore, the Šesták-Berggren autocatalytic model represents the best reaction model for describing the crystallization process of α-Fe in the amorphous Fe 81 B 13 Si 4 C 2 alloy. The higher value of N exponent designates that the formed crystallized phase has the decisive influence on the kinetics of transformation and the rate of growth. In the propagation process, on account of overlapping of nuclei during growth, it acts to retard crystallization rate. Bearing this in mind, we can suppose that in the amorphous alloy, the α-Fe embryos already exist at T ≤ 500 o C and at T ≥ 500 o C these embryos are momentarily transformed into nuclei. The established acceleration of crystallization process is a consequence of significant increase of strain in alloy, which arises on account of formation of α-Fe.

Morphology of crystal growth
For non-isothermal crystallization, where the volume fraction of crystalline phase  precipitated in glass heated at a uniform heating rate  is related with the activation energy E a , Matusita et al.
proposed the following relation (Matusita & S. Sakka, 1979, 1980Matusita, et al. 1984 where m and n are constants with values between one and four depending on the morphology and kinetics of the growth nuclei, The values of n obtained from the slopes of linear plots ln[-ln (1-α)] versus -lnβ at different temperatures for considered crystallization process are given in Table 9. For all considered temperatures, the value of n is ≈ 4.0, within the limits of experimental errors. It follows, then, that the kinetics of crystallization process is independent from the temperature.
The crystallization exponent n is connected with the number of growth dimensions (m) and the number of nuclei forming stages (s) (Matusita, Sakka, 1979) where m is the number of growth dimensions as defined in Table 8, s is the number of the nuclei forming stages (s = 0 -at instantaneously nucleation; s = 1 -at constant nucleation rate and s > 1 at self-acceleratory nucleation rate).
In order to describe the crystallization process in detail, the value of parameter m should be determined from the plot of ln [-ln (1-α)], because a function of reciprocal temperature is linear with a slope of 1.051× (m + 1)E a /R, using the value of activation energy determined above. The values of parameters m and s obtained at the different heating rates for the investigated crystallization process of α-Fe in Fe 81 B 13 Si 4 C 2 amorphous alloy are given in Table 9. Based on the obtained values of parameters m and s, at the different heating rates, (Table 9), we asserted with high degree of reliability, that the nucleation process of α-Fe occurs within amorphous alloy with a constant rate, in three effective directions (three-dimensional growth) proceeding with constant nucleation rate.

Conclusion
Metallic amorphous alloy are a class of materials which has seen dramatic developments in recent times with design of materials stable enough to allow bulk production. While the functional properties of amorphous alloys allow for many possible fields of application, they can also be used as precursors in preparation of nanocomposite materials composed of nanocrystals in the amorphous matrix. These nanocomposites often exhibit superior mechanical, electrical and magnetic properties to both purely amorphous and purely crystalline materials. Thermal treatment of amorphous alloys can allow for controlled crystallization, leading to formation of nanocomposite materials with targeted properties.
Fe 81 B 13 Si 4 C 2 alloy undergoes multi-step crystallization process as a result of thermal treatment.
The changes in microstructure cause changes in electrical, magnetic and mechanical properties of the alloy as the alloy structure changes from predominantly amorphous to crystal/amorphous nanocomposite to nanocrystalline composite of α-Fe and Fe 2 B phases.