**1. Introduction**

106 Advances in Crystallization Processes

Okura, T.; Tanaka, M. & Sudoh, G. (1997). Conduction properties of Na+-ion implanted

Okura, T. & Yamashita, K. (2000). Ionic conductivities of Na+-ion implanted silico-phosphate

Okura, T.; Inami, Y.; Monma, H.; Nakamura, S. & Yamashita. K. (2002). Structure and

Okura, T.; Tanaka, M.; Monma, H.; Yamashita, K. & Sudoh, G. (2003). New superionic

crystallization and ionic conductivity. *J. Ceram. Soc. Jpn.*, 111/4, 257-261. Okura, T.; Monma, H. & K. Yamashita. (2004). Na+-superionic conductors of glass-ceramics in the system Na2O-Sm2O3-X2O3-P2O5-SiO2 (X=Al, Ga). *Solid State Ionics*, 172, 561-564. Okura, T.; Monma, H. & Yamashita, K. (2004). Effect of substitution of Si with Ge and Te on ionic conductivity of Na5SmSi4O12–type glass-ceramics. *J. Ceram. Soc. Jpn.*, 112/ 5, S685-S689.

Okura, T.; Monma, H. & Yamashita, K. (2006). Superionic conducting Na5SmSi4O12-type glass-

Okura, T.; Takahashi, T; Monma, H. & Yamashita. K. Effect of substitution of Si with V and

Okura, T.; Saimaru, M; Monma, H. & Yamashita. K. (2009). Ionic conductivities of Nasicon-

Okura, T; Monma, H & Yamashita, K. (2010). Na+-fast ionic conducting glass-ceramics of

Shannon, R. D.; Taylor, B. E.; Gier, T. E.; Chen, H. Y. & Berzins, T. (1978). Ionic conductivity

Shannon, R. D.; Gier, T. E.; Foris, C. M.; Nelen, J. A. & Appleman, D. E. (1980). Crystal data for some sodium rare earth silicates. *Phys. Chem. Minerals*, 5, 245-253. Yamashita, K.; Ohkura, S.; Umegaki, T. & Kanazawa, T. (1988). Synthesis and ionic conduction of C3A-type Nasicon Na3+3x-yY1-xSi3-yPyO9. *Solid State Ionics*, 26, 279-286. Yamashita, K.; Ohkura, S.; Umegaki, T. & Kanazawa, T. (1988). Synthesis, polymorphs and

Yamashita, K.; Nojiri, T.; Umegaki, T. & Kanazawa, T. (1989). New fast sodium-ion

Yamashita, K.; Nojiri, T.; Umegaki, T. & Kanazawa, T. (1990). Na+ superionic conductors of

Yamashita, K.; Tanaka, M. & Umegaki, T. (1992). Thermodynamic and kinetic study on the

Yamashita, K.; Tanaka, M.; Kakuta, T.; Matsuda, M. & Umegaki, T. (1993). Effects of rare

Yamashita, K.; Umegaki, T.; Tanaka, M.; Kakuta, T. & Nojiri, T. (1996). Microstructural

*Symp. Proc.*, Vol. 453, 611-616.

*Ionics*, 179, 1291-1295.

*State Ionics*, 40/41, 48-52.

xPySi3-yO9. *Solid State Ionics*, 58, 231-236.

Narpsio-V. *J. Alloys and Compounds*, 193, 283-285.

conductors. *J. Electrochem. Soc.*, 143, 2180-2186.

Ge, Te). *Solid State Ionics*, 180, 537-540.

silicophosphates. *J. Electroceram.*, 24, 83-90.

in Na5YSi4O12–type silicates. *Inorg. Chem.*, 17, 958-964.

conduction properties. *Solid State Ionics*, 35, 299-306.

composition Na3+3*<sup>x</sup>*-*<sup>y</sup>*Y1-*x*P*y*Si3-*y*O9. *J. Ceram. Soc. Jpn.*, 96, 967-972.

glass-ceramics. *Solid State Ionics*, 136-137, 1049-1054.

crystallization of glass. *Solid State Ionics*, 154-155, 361-366.

glass-ceramics in the system Na2O-R2O3P2O5-SiO2 (R = rare earth). *Mat. Res. Soc.* 

conduction properties of Na+ superionic conductor Narpsio synthesized by bias

conducting glass-ceramics in the system Na2O-Y2O3-Sm2O3-P2O5-SiO2:

ceramics: Crystallization condition and ionic conductivity. *J. Eur. Ceram. Soc.*, 26, 619.

Mo on ionic conductivity of Na5YSi4O12–type glass-ceramics. (2008). *Solid State* 

type glass-ceramic superionic conductors in the system Na2O-Y2O3-XO2-SiO2 (X=Ti,

sodium ionic conductivity of sodium yttrium silicophosphates with the

conducting glass-ceramics of silicophosphates: Crystallization, microstructure and

glass-ceramics in the system Na2O-Re2O3-P2O5-SiO2 (Re=rare-earth elements). *Solid* 

phase transformation of the glass-ceramic Na+ superionic conductors Na3+3x-yRe1-

earth elements on the crystallization of the glass-ceramic Na+ superionic conductor

effects on conduction properties of Na5YSi4O12-type glass-ceramic Na+-fast ionic

Metallic glasses are kinetically metastable materials. Metallic glass is defined as "A liquid, which has been cooled into a state of rigidity without crystallizing". Properties of metallic glasses differ form non metallic glasses. Ordinary glasses are made up of silica while metallic glasses are made of alloy metals. Ordinary glasses are transparent whereas metallic glasses are opaque. In ordinary glasses, covalent bond is observed while in metallic glasses metallic bond is observed. On the basis of internal arrangement of atoms or molecules and type of force acting between them, the material can be classified into the following two categories:

i. Crystalline solid: Those materials in which the constituent ions or atoms and molecules are arranged in regular pattern are called crystalline solids. Besides, crystalline solids have a definite external geometrical form.

e.g. Quartz, Calcite, Diamond, Sugar, and Mica

ii. Amorphous or glassy solid: Those materials do not have definite geometric pattern are called amorphous solids. In amorphous solid atoms, ions or molecules are not arranged in definite pattern.

e.g. Rubber, Glass, Plastic and Cement

Also, an amorphous solid is a solid in which there is no long range order of the positions of the atoms. Solids in which there is long-range atomic order are called crystalline solids.

At high cooling rate, any liquid can be made into an amorphous solid. Cooling reduces molecular mobility. If the cooling rate is faster, then molecules can not organize into a more thermodynamically favourable crystalline state and an amorphous solid will be formed. Materials in which such a disordered structure is produced directly from the liquid state during cooling are called "Glasses" and such amorphous metals are commonly referred to as "Metallic Glasses" or "Glassy Metals". The metallic glasses have a combination of amorphous structure and metallic bond. This combination provides a metallic glass a new and unique quality, which cannot be found in either pure metals or regular glass.

Crystallization Kinetics of Metallic Glasses 109

chemistry for the determination of the kinetics of the thermally activated solid-state reactions. The physicochemical changes during an exothermic or endothermic event in DSC (or DTA) are complex and involve multi-step (serial or parallel) processes occurring simultaneously at different rates. Therefore, the activation energies for such processes can logically not be same and it may vary with the degree of conversion. This is contrary to the isokinetic view assuming all the constituents of the material to react simultaneously at the same rate. The activation energy, in the isokinetic case, is thus constant and independent of the degree of conversion. A strong difference of opinion persists among the researchers in the field of thermal analysis about the concept of variable activation energy (Galwey, 2003; Vyazovkin, 2003). In the metallurgical branch of materials science, most of the thermal phase transformations (like crystallization, recovery) are morphological and are considered to be governed by the nucleation and growth processes. The transformation mechanisms in these processes are also complex e.g. interface-controlled, diffusion-controlled growth. Notwithstanding this, the kinetic analysis of the transformation process like crystallization is done according to isokinetic hypothesis. The isoconversional methods are scarcely used

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

<sup>1</sup> 1 1 *<sup>d</sup> (n ) n nk( )[ ln( )] dt*

 α

<sup>−</sup> = −− − (1)

= − (2)

α

0 *<sup>E</sup> k(T ) k exp RT*

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

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

for the study of the crystallization kinetics of metallic glasses.

α

where, *α* → degree of transformation at a given time t, *n* → Avrami (growth) exponent

The Arrhenius form of the rate constant is given by

conditions. The KJMA rate equation is given by

*k* → the rate constant

where, *k0* → pre-exponential factor E → activation energy, and R → universal gas constant

conditions (Minic et al., 2009).

**2. Theory** 

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; these are known as bulk metallic glasses (BMG).

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

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 processing (& operating) temperatures.

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 chemistry for the determination of the kinetics of the thermally activated solid-state reactions. The physicochemical changes during an exothermic or endothermic event in DSC (or DTA) are complex and involve multi-step (serial or parallel) processes occurring simultaneously at different rates. Therefore, the activation energies for such processes can logically not be same and it may vary with the degree of conversion. This is contrary to the isokinetic view assuming all the constituents of the material to react simultaneously at the same rate. The activation energy, in the isokinetic case, is thus constant and independent of the degree of conversion. A strong difference of opinion persists among the researchers in the field of thermal analysis about the concept of variable activation energy (Galwey, 2003; Vyazovkin, 2003). In the metallurgical branch of materials science, most of the thermal phase transformations (like crystallization, recovery) are morphological and are considered to be governed by the nucleation and growth processes. The transformation mechanisms in these processes are also complex e.g. interface-controlled, diffusion-controlled growth. Notwithstanding this, the kinetic analysis of the transformation process like crystallization is done according to isokinetic hypothesis. The isoconversional methods are scarcely used for the study of the crystallization kinetics of metallic glasses.
