**2. Theory**

108 Advances in Crystallization Processes

In the past, small batches of amorphous metals have been produced through a variety of quick-cooling methods. For instance, amorphous metal wires have been produced by sputtering molten metal onto a spinning metal disk. The rapid cooling, of the order of millions of degrees a second, is too fast for crystals to form and the material is "locked in" a glassy state. Now-a-days number of alloys with critical cooling rates low enough to allow formation of amorphous structure in thick layers (over 1 millimetre) have been produced;

However, there are various methods in which amorphous metals can be produced, preventing the crystallization. Sputtering, glow discharge sputtering, chemical vapour deposition (CVD), gel desiccation, electrolyte deposition, reaction amorphization, pressure– induced amorphization, solid state diffusion amorphization, laser glazing, ion implantation,

The study of the thermally-activated phase transformations is of great significance in the field of materials science as the properties of materials change due to the change in the composition and/or microstructure. The properties of fully or partly crystalline materials are usually different from their amorphous counterparts. From the viewpoint of a materials scientist, the crystallization of amorphous or non-crystalline materials involves the nucleation and growth processes. The processes driven by nucleation and growth have attracted a lot of interest for tailoring technological applications. For example, the recrystallization of the deformed metals, controlling the nucleation and growth of islands on terraces in order to get large scale arrays of nanostructures in the manufacturing of thin-film transistors (Castro, 2003). Thus, the knowledge of the kinetics of crystallization would help to attain products with the required crystallized fraction and microstructure (e.g. nanocrystalline or quasicrystalline) or to avoid the degradation of materials at high

The kinetics of the crystallization process can be studied with the help of thermo-analytical techniques namely, differential scanning calorimetry (DSC) and differential thermal analyzer (DTA). The DSC/DTA experiments can be carried out in isothermal as well as nonisothermal (linear heating) conditions (Ligero et al., 1990; Moharram et al., 2001; Rysava et al., 1987; Giridhar & Mahadevan, 1982; Afify, 1991). Efforts made by the researchers in this field so far, to analyze the data obtained from DSC and hence to determine the kinetic parameters of the crystallization processes (say, activation energy, rate constant etc.), raise two important issues: (i) the selection of the mode of experiment (isothermal or nonisothermal) and, (ii) the choice of a sound method for the analysis of the experimental data. However, we are more concerned with the later issue due to the fact that several methods for the kinetic analysis are available in the literature. These methods are generally based on either the isokinetic hypothesis or the isoconversional principle and they can be accordingly categorized as: (1) isokinetic methods where the transformation mechanism is assumed to be the same throughout the temperature/time range of interest and, the kinetic parameters are assumed to be constant with respect to time and temperature; (2) isoconversional methods, which are generally used for non-isothermal analysis, assume that the reaction (transformation) rate at a constant extent of conversion (degree of transformation) is only a function of temperature (Lad et al, 2008; Patel & Pratap, 2012). The kinetic parameters, in this case, are considered to be dependent on the degree of transformation at different temperature and time. The use of isoconversional methods is widespread in the physical

thin-film deposition, melt quenching and melt spinning are some of them.

these are known as bulk metallic glasses (BMG).

processing (& operating) temperatures.

To study the phase transformation, which involves nucleation and growth, many methods are developed. Most of the methods depend on the transformation rate equation given by Kolmogorov, Johnson, Mehl and Avrami (Lesz & Szewieczek, 2005; Szewieczek & Lesz, 2005; Szewieczek & Lesz, 2004; Jones et al., 1986; Minic & Adnadevic, 2008), popularly known as KJMA equation, basically derived from experiments carried out under isothermal conditions. The KJMA rate equation is given by

$$\frac{d\alpha}{dt} = nk(1-\alpha)[-\ln(1-\alpha)]^{(n-1)/n} \tag{1}$$

where, *α* → degree of transformation at a given time t,

*n* → Avrami (growth) exponent

*k* → the rate constant

The Arrhenius form of the rate constant is given by

$$k(T) = k\_0 \exp\left(-\frac{E}{RT}\right) \tag{2}$$

where, *k0* → pre-exponential factor

E → activation energy, and

R → universal gas constant

KJMA rate equation is based on some important assumptions and it has been suggested that the KJMA kinetic equation is accurate for reactions with linear growth subject to several conditions (Minic et al., 2009).

The isoconversional methods are also known as model-free methods. Therefore, the kinetic analysis using these methods is more deterministic and gives reliable values of activation energy E, which depends on degree of transformation, *α*. However, only activation energy

Crystallization Kinetics of Metallic Glasses 111

crystallization and eutectic crystallization (Hsiao et al., 2002). In primary crystallization the primary phase of the alloy constituents crystallizes first. The dispersed primary crystallized phase coexists with the amorphous matrix and may serve as the nucleation site for secondary or tertiary crystallization. In Fe-based alloys α-Fe crystallizes first, which is a kind of primary crystallization. Polymorphous crystallization is a transition of the amorphous phase to a crystalline one without any change in the composition of that phase. There is no concentration difference across the reaction front because the concentration does not change. Eutectic crystallization is simultaneous crystallization of two crystalline phases by a discontinuous reaction. This reaction takes longer than polymorphous crystallization to proceed because the two components have to separate by diffusion into two separate phases

The DSC thermograms at four different heating rates are shown in Fig.1. The thermograms show three-stage crystallization. The first crystallization peak is evaluated for heating rates 4, 6, 8 and 10 deg/min. Glass transition becomes clear as we go for the higher heating rates, but the third crystallization peak becomes less prominent as we go to the higher heating rates. The onset and endset of first crystallization exotherms exhibit different levels of heat flow i.e. the crystallization ends at slightly higher level followed by the second and third crystallization peak. This difference of the level indicates that the phases at the start of crystallization and at the end of it are not same. The analysis of DSC data to evaluate the kinetic parameters can be obtained from non-isothermal rate laws by both isokinetic also

Fig. 1. DSC thermograms of the metallic glass Co66Si12B16Fe4Mo2 at different heating rates

within the crystallized region (Minic, 2006).

known as model fitting methods and isoconversional methods.

**3. Results and discussion** 

will not give a perfect picture of crystallization kinetics. The microstructural information (e.g. dimensionality of the growth) of the precipitating phase during the transformation is also very important for understanding the whole kinetics of crystallization. Microstructural information would be known to us when we take the isokinetic methods into account. Therefore, the complementary use of both the methods is more useful for understanding the kinetics of crystallization.

Differential Scanning Calorimetry (DSC) has become a convenient and widely used tool for studying the kinetics of phase transformations. The volume fraction (*x*) of the sample transformed in crystalline phase during the crystallization event has been obtained from the DSC curve as a function of temperature (*T*). The volume fraction of precipitated crystal can be obtained from the DSC curve by using

Where *S0* is the total area under the crystallization curve i.e. the area under the curve between the temperature at the onset of crystallization Ton and the end-set temperature Tend when the crystallization is completed. *S* is the area at any temperature T between Ton and T at which the fractional crystallization is required to be known.

There are three important modes of crystallization involving nucleation and growth processes, depending on the composition of a particular alloy: primary crystallization, polymorphous

$$\mathbf{x} = \bigvee\_{\mathbf{s}\_0} \mathbf{s}\_{\mathbf{s}\_0}$$

will not give a perfect picture of crystallization kinetics. The microstructural information (e.g. dimensionality of the growth) of the precipitating phase during the transformation is also very important for understanding the whole kinetics of crystallization. Microstructural information would be known to us when we take the isokinetic methods into account. Therefore, the complementary use of both the methods is more useful for understanding the

Differential Scanning Calorimetry (DSC) has become a convenient and widely used tool for studying the kinetics of phase transformations. The volume fraction (*x*) of the sample transformed in crystalline phase during the crystallization event has been obtained from the DSC curve as a function of temperature (*T*). The volume fraction of precipitated crystal can

= *s x*

Where *S0* is the total area under the crystallization curve i.e. the area under the curve between the temperature at the onset of crystallization Ton and the end-set temperature Tend when the crystallization is completed. *S* is the area at any temperature T between Ton and

There are three important modes of crystallization involving nucleation and growth processes, depending on the composition of a particular alloy: primary crystallization, polymorphous

T at which the fractional crystallization is required to be known.

0

*s*

kinetics of crystallization.

be obtained from the DSC curve by using

crystallization and eutectic crystallization (Hsiao et al., 2002). In primary crystallization the primary phase of the alloy constituents crystallizes first. The dispersed primary crystallized phase coexists with the amorphous matrix and may serve as the nucleation site for secondary or tertiary crystallization. In Fe-based alloys α-Fe crystallizes first, which is a kind of primary crystallization. Polymorphous crystallization is a transition of the amorphous phase to a crystalline one without any change in the composition of that phase. There is no concentration difference across the reaction front because the concentration does not change. Eutectic crystallization is simultaneous crystallization of two crystalline phases by a discontinuous reaction. This reaction takes longer than polymorphous crystallization to proceed because the two components have to separate by diffusion into two separate phases within the crystallized region (Minic, 2006).
