**3. Kinetic investigation of crystallization**

The theory of crystallization in amorphous materials can be explained by considering the structure and the kinetics of the crystallization. Therefore, the investigation of crystallization kinetics is important since it quantifies the effect of the nucleation and growth rate of the resulting crystallites (Carter & Norton, 2007; Kashchiev, 2000). In this chapter, crystallization kinetics of amorphous materials was investigated by explaining the crystallization mechanism and the crystallization activation energy in terms of isothermal and nonisothermal methods with different approaches. Different thermal analysis techniques used in crystallization kinetic studies were presented and a correlation between kinetic and structural investigations was made to determine the crystallization mechanism (Araújo & Idalgo, 2009; Malek, 2000; Prasad & Varma, 2005).

The crystallization kinetics of amorphous materials can be investigated either isothermally or non-isothermally by using thermal analysis techniques. In the isothermal method, the sample is heated above the glass transition temperature and the heat absorbed during the crystallization process is measured as a function of time. On the other hand, in the nonisothermal method, the sample is heated at a fixed rate and then the change in enthalpy is recorded as a function of temperature. Thermal analysis techniques such as differential thermal analysis (DTA) and differential scanning calorimetry (DSC) are quite popular for kinetic analysis of crystallization processes in amorphous solids (Araújo & Idalgo, 2009; Malek, 2000; Prasad & Varma, 2005).

Many of the reactions of interest to materials scientists involve transformations in the solid state, reactions such as recrystallization of a cold-worked material, the precipitation of a crystalline polymer from an amorphous phase, or the growth of an equilibrium phase from

higher than the temperature at which the nucleation rate is highest. Therefore, when a glass is heated at a constant rate, crystal nuclei are formed at lower temperature and grow in size

at higher temperatures without any increase in number (Matusita & Sakka, 1981).

Fig. 10. The relationship between supersaturation and nucleation and growth

The theory of crystallization in amorphous materials can be explained by considering the structure and the kinetics of the crystallization. Therefore, the investigation of crystallization kinetics is important since it quantifies the effect of the nucleation and growth rate of the resulting crystallites (Carter & Norton, 2007; Kashchiev, 2000). In this chapter, crystallization kinetics of amorphous materials was investigated by explaining the crystallization mechanism and the crystallization activation energy in terms of isothermal and nonisothermal methods with different approaches. Different thermal analysis techniques used in crystallization kinetic studies were presented and a correlation between kinetic and structural investigations was made to determine the crystallization mechanism (Araújo &

The crystallization kinetics of amorphous materials can be investigated either isothermally or non-isothermally by using thermal analysis techniques. In the isothermal method, the sample is heated above the glass transition temperature and the heat absorbed during the crystallization process is measured as a function of time. On the other hand, in the nonisothermal method, the sample is heated at a fixed rate and then the change in enthalpy is recorded as a function of temperature. Thermal analysis techniques such as differential thermal analysis (DTA) and differential scanning calorimetry (DSC) are quite popular for kinetic analysis of crystallization processes in amorphous solids (Araújo & Idalgo, 2009;

Many of the reactions of interest to materials scientists involve transformations in the solid state, reactions such as recrystallization of a cold-worked material, the precipitation of a crystalline polymer from an amorphous phase, or the growth of an equilibrium phase from

**3. Kinetic investigation of crystallization** 

Idalgo, 2009; Malek, 2000; Prasad & Varma, 2005).

Malek, 2000; Prasad & Varma, 2005).

a non-equilibrium structure, the driving force for which is brought about by cooling from one temperature to another. Consider the initial phase to be α and the resulting phase to be β; the reaction is written:

$$
\mathfrak{a} \to \mathfrak{P}
$$

The total volume of the sample is the sum of the volumes of α and β:

$$V = V^{\mathfrak{a}} + V^{\mathfrak{b}} \tag{32}$$

The fraction transformed can be represented in the literature as *x*, α or F:

$$\mathbf{x} = \frac{V^{\theta}}{V} \tag{33}$$

Assume that the transformation from α to β is controlled by nucleation and growth, that is, the nucleation of phase β within α and then the rate of growth of β.

If we consider *N* is the nucleation rate per unit volume and *G* is the growth rate in one direction = *dr*/*dt* (assuming spherical form of β). Consider the time line from zero to a time, *t* (see Fig. 11*)*. We will consider another measure of time (τ), which starts when a nucleus is formed. The number of nuclei formed in differential time *d*τ is equal to

$$\begin{array}{c} N V^{a} \, d\mathfrak{r} \\ \to \mid d\mathfrak{r} \mid \lnot \, \begin{array}{c} \lnot \, \begin{array}{c} \lnot \, \begin{array}{c} \end{array} \\ \end{array} \end{array} \end{array} \tag{34}$$

Fig. 11. Definition of *t* and τ for derivation of solid state transformation equations (Ragone, 1994)

Assuming that the particles nucleated in this time *d*τ grow as spheres, the radius of the particles formed during *d*τ, after they have grown to time *t*, is:

$$\int\_0^\mathbf{r} \mathbf{d}\mathbf{r} = \int\_\mathbf{r}^\mathbf{t} \mathbf{G} \,\mathrm{d}\mathbf{t} \tag{35}$$

$$r = G(t - \mathbf{r})\tag{36}$$

The volume of the particle nucleated during *d*τ at time *t* is:

$$dV^{\beta} = \frac{4}{3} \text{m}G^{3} \text{(t - r)}^{3} (NV^{\alpha}) d\tau \tag{37}$$

$$dV^{\theta} = \frac{4}{3}\pi G^3 N (V - V^{\theta}) (t - \pi)^3 d\pi \tag{38}$$

Early in the transformations, when *V*β is small, *V*β can be considered negligible with respect to *V*. In this case, the fraction transformed may be calculated as follows:

$$\int\_0^{\mathbf{V}^\beta} \mathbf{d} \mathbf{V}^\beta = \int\_0^t \frac{4}{3} \mathbf{n} \mathbf{G}^3 \mathbf{N} \mathbf{V} \left(\mathbf{t} \cdot \mathbf{r}\right)^3 d\mathbf{r} \tag{39}$$

Crystallization Kinetics of Amorphous Materials 141

where *n* is called "the Avrami *n*". To determine the value of Avrami *n* from Equation 50, the

1 *– x* = exp ( -*kt*n) (53)

Thus the Avrami *n* is the slope of the plot of the logarithm of the logarithm of (1 – *x*) versus

The study of crystallization kinetics in glass-forming liquids has often been limited by the elaborate nature of the experimental procedures which are employed. The increasing use of thermoanalytical techniques such as differential thermal analysis (DTA) or differential scanning calorimetry (DSC) has, however, offered the promise of obtaining useful data with

When a reaction occurs in thermal analysis, the change in heat content and in the thermal properties of the sample is indicated by a deflection (Kissinger, 1957). It is conventional to represent an endothermic effect by a negative deflection and an exothermic effect by a positive deflection (see Fig. 12). The deflections, whether positive or negative, are called peaks (Kissinger, 1956). If the reaction proceeds at a rate varying with temperature, possesses an activation energy the position of the peak varies with the heating rate if other experimental conditions are maintained fixed. This variation in peak temperature could be

used to determine the energy of activation for first order reactions (Kissinger, 1957).

 - 

1 

*x* = 1 - exp ( -*kt*n) (52)

*– x*� = -*kt*n (54)

*n* ln *t* (55)

following mathematical manipulation is performed:

ln [ln(1 *– x*�� = ln *k*

the negative of the logarithm of *t* (Ragone, 1994).

simple methods (Yinnon & Uhlmann, 1983).

Fig. 12. A representative thermal analysis thermogram

**3.1 Thermal analysis techniques** 

ln(

$$V^{\not\beta} = V \frac{\text{m}}{3} \text{G}^3 \text{N}t^4 \tag{40}$$

$$\mathbf{x} = \frac{V^{\theta}}{V} = \frac{\mathbf{n}}{3}G^{3}N\mathbf{t}^{4} \tag{41}$$

To treat the regime beyond the early transformation the extended volume concept is adopted. In this case, the nucleation and growth rates are separated from geometrical considerations such as impingement. The extended volume (*V*e) is the volume that would have been formed if the entire volume had participated in nucleation and growth, even that portion transformed (*V*β). In this case,

$$d\boldsymbol{V}\_{\rm e}^{\emptyset} = \boldsymbol{V}\_{\frac{4}{3}}^{4} \boldsymbol{\rm{m}} \rm{G}^{3} \rm{N} \rm{(t} \rm{ - \, t)}^{3} \, d\boldsymbol{\tau} \tag{42}$$

$$V\_{\mathbf{e}}^{\beta} = \frac{4}{3} \text{tr} \, V \int\_{0}^{\mathbf{t}} \mathbf{G}^{3} \, \text{N} \{\mathbf{t} \cdot \mathbf{r}\}^{3} \, d\mathbf{r} \tag{43}$$

But the total volume is equal to the sum of the volumes of α and β:

$$V = V^{\alpha} + V^{\beta} \tag{44}$$

$$\frac{V^{\mu}}{V} = \mathbf{1} \cdot \frac{V^{\beta}}{V} = \mathbf{1} \cdot \mathbf{x} \tag{45}$$

where *x* = *V*β/*V* 

The amount of β formed, *dV*β, is the fraction of α times *dV*<sup>e</sup> β

$$\mathbf{d} \mathbf{V}^{\boldsymbol{\beta}} = \left( \mathbf{1} \cdot \frac{\mathbf{v}^{\boldsymbol{\beta}}}{\mathbf{V}} \right) \mathbf{d} \mathbf{V}\_{\mathbf{e}}^{\boldsymbol{\beta}} \tag{46}$$

Integrating Equation 46,

$$V\_{\rm e}^{\beta} = \text{-}V \ln\left(1 - \frac{\nu^{\beta}}{V}\right) = \text{-}V \ln\left(1 - x\right) \tag{47}$$

Combining Equations 43 and 47 yields,

$$-\ln\left(1\text{ - x}\right) = \frac{4}{3}\text{Tr}\int\_0^t \text{G}^3 \,\text{N}\left(t\text{ - }\text{t}\right)^3 \,d\text{\textdegree}\tag{48}$$

If *G* and *N* are constant,

$$-\ln\left(1-\chi\right) = \frac{4}{3}\pi G^3 N \int\_0^t \left(t-\tau\right)^3 \,d\tau = \frac{\pi}{3}G^3 N t^4 \tag{49}$$

$$\propto = 1 \cdot \exp\left(-\frac{n}{3}G^3Nt^4\right) \tag{50}$$

The resulting equation relating the fraction transformed to nucleation rate, growth rate and time is called Johnson-Mehl equation.

A similar treatment of the subject is given by Avrami. In general he expresses the fraction transformed as

$$\mathbf{x} = \mathbf{1} \text{ - } \exp\left(\cdot k t^{\mathbf{u}}\right) \tag{51}$$

3 *G*3 *Nt*<sup>4</sup>

(40)

(41)

<sup>3</sup> *d*τ (42)

 *d*τ (43)

(46)

<sup>3</sup> *d*τ (48)

) (50)

(49)

 *V* = *V*α + *V*β (44)

*<sup>V</sup>* <sup>=</sup> 1-*x* (45)

*<sup>V</sup>* � = -*V* ln (1 - *<sup>x</sup>*) (47)

*V*<sup>β</sup> = *V* <sup>π</sup>

<sup>β</sup> = *V* <sup>4</sup>

<sup>β</sup> = <sup>4</sup>

But the total volume is equal to the sum of the volumes of α and β:

The amount of β formed, *dV*β, is the fraction of α times *dV*<sup>e</sup>

*<sup>V</sup>* <sup>=</sup> <sup>π</sup> 3 *G*3 *Nt*<sup>4</sup>

<sup>3</sup> <sup>π</sup>*G*<sup>3</sup>

<sup>3</sup> <sup>π</sup>*<sup>V</sup>* � *<sup>G</sup>* t 3 <sup>0</sup> *N*(*t* - τ)

*<sup>V</sup>* <sup>=</sup> <sup>1</sup> - *V*<sup>β</sup>

dV<sup>β</sup> <sup>=</sup> �1- V<sup>β</sup>

<sup>3</sup> <sup>π</sup> � *<sup>G</sup>* t 3 <sup>0</sup> *N*(*t* - τ)

*<sup>N</sup>* � (*t* - τ) <sup>t</sup> <sup>3</sup>

The resulting equation relating the fraction transformed to nucleation rate, growth rate and

A similar treatment of the subject is given by Avrami. In general he expresses the fraction

*x* = 1 - exp ( -*kt*n) (51)

3 *G*3 *Nt*<sup>4</sup>

<sup>0</sup> *<sup>d</sup>*τ = <sup>π</sup>

3 *G*3 *Nt*<sup>4</sup>

<sup>β</sup> = -*V* ln �<sup>1</sup> - *V*<sup>β</sup>

<sup>3</sup> <sup>π</sup>*G*<sup>3</sup>



*x* = 1- exp ( - <sup>π</sup>

*N*(*t* - τ)

3

β

<sup>V</sup> � dVe β

To treat the regime beyond the early transformation the extended volume concept is adopted. In this case, the nucleation and growth rates are separated from geometrical considerations such as impingement. The extended volume (*V*e) is the volume that would have been formed if the entire volume had participated in nucleation and growth, even that

*<sup>x</sup>* <sup>=</sup> *<sup>V</sup>*<sup>β</sup>

portion transformed (*V*β). In this case,

*dV*<sup>e</sup>

*V*<sup>e</sup>

*<sup>V</sup>*<sup>α</sup>

where *x* = *V*β/*V* 

Integrating Equation 46,

If *G* and *N* are constant,

transformed as

*V*<sup>e</sup>

Combining Equations 43 and 47 yields,

time is called Johnson-Mehl equation.

where *n* is called "the Avrami *n*". To determine the value of Avrami *n* from Equation 50, the following mathematical manipulation is performed:

$$\alpha = 1 \text{ - } \exp \text{ (-}kt\text{")}\tag{52}$$

$$\mathbf{1} - \mathbf{x} \equiv \exp\left(\cdot k t^{0}\right) \tag{53}$$

$$
\ln(\mathcal{I} - \mathbf{x}) = -kt^{\mathrm{an}} \tag{54}
$$

$$
\ln\left[\ln(1-x)\right] = \ln k \cdot n \ln t \tag{55}
$$

Thus the Avrami *n* is the slope of the plot of the logarithm of the logarithm of (1 – *x*) versus the negative of the logarithm of *t* (Ragone, 1994).

#### **3.1 Thermal analysis techniques**

The study of crystallization kinetics in glass-forming liquids has often been limited by the elaborate nature of the experimental procedures which are employed. The increasing use of thermoanalytical techniques such as differential thermal analysis (DTA) or differential scanning calorimetry (DSC) has, however, offered the promise of obtaining useful data with simple methods (Yinnon & Uhlmann, 1983).

When a reaction occurs in thermal analysis, the change in heat content and in the thermal properties of the sample is indicated by a deflection (Kissinger, 1957). It is conventional to represent an endothermic effect by a negative deflection and an exothermic effect by a positive deflection (see Fig. 12). The deflections, whether positive or negative, are called peaks (Kissinger, 1956). If the reaction proceeds at a rate varying with temperature, possesses an activation energy the position of the peak varies with the heating rate if other experimental conditions are maintained fixed. This variation in peak temperature could be used to determine the energy of activation for first order reactions (Kissinger, 1957).

Fig. 12. A representative thermal analysis thermogram

Crystallization Kinetics of Amorphous Materials 143

Fig. 14. Plot of ln[-ln(1- *x*)] against lnt for determining the Avrami parameter, *n*

Fig. 15. Plot of ln*k* against 1/T for determining the activation energy, *EA*

2009; Çelikbilek et al., 2011; Prasad & Varma, 2005):

Fig. 15).

The activation energy can be evaluated by the Arrhenius-type equation (Araújo & Idalgo,

where *k* is the reaction rate constant, *k*0 is the frequency factor, *EA* is the activation energy, *R* is the gas constant. The activation energy for several isothermal temperatures is calculated from the slopes of the linear fits to the experimental data from a plot of ln*k* versus 1/*T* (see

ln (*k*) = ln *k*0 - *E*A/R*T* (56)

Differential thermal analysis (DTA) has been extensively used as a rapid and convenient means for detecting the reaction process. The rate of chemical reaction was analyzed quantitatively by DTA and the activation energies were obtained. Furthermore, this method was used to obtain the activation energy for the crystallization of glass, assuming that the process of crystallization is a first order reaction (Matusita & Sakka, 1981). DSC measurements are useful in obtaining kinetic parameters related to the glass crystallization process especially in non-isothermal method due to the rapidity of this thermoanalytical technique (Araújo & Idalgo, 2009; Cheng et al., 2007).

#### **3.2 Thermal analysis methods**

#### **3.2.1 Isothermal method**

In the isothermal method, kinetic parameters of amorphous materials crystallization are obtained by monitoring the shift in the crystallization peak as a function of time (Prasad & Varma, 2005). Crystallization peak temperatures, *Tp* and crystallized volume fractions, *x*, are determined from the thermal analysis curves with respect to time in isothermal method.

In isothermal analysis, the volume fraction crystallized, *x*, at any time *t* is given as *x* = *St* / *S* where *S* is the total area of the exothermic peak between the time *ti* at which the crystallization begins and the time *tf* at which the crystallization is completed. *St* is the area between *ti* and *t* (see Fig. 13) (Ray et al., 1991).

Fig. 13. Computation of the volume fraction crystallized, *x*, in isothermal method (Prasad & Varma, 2005)

Isothermal investigation of crystallization in amorphous materials can be described by the Johnson-Mehl-Avrami equation as given in Equation 55 (Araújo & Idalgo, 2009; Avrami, 1939, 1940; Çelikbilek et al., 2011; Prasad & Varma, 2005).

From the slopes of the linear fits to the experimental data from a plot of ln[−ln(1−*x*)] versus ln*t*, *k* and *n* values are calculated for different isothermal hold temperatures (see Fig. 14).

Differential thermal analysis (DTA) has been extensively used as a rapid and convenient means for detecting the reaction process. The rate of chemical reaction was analyzed quantitatively by DTA and the activation energies were obtained. Furthermore, this method was used to obtain the activation energy for the crystallization of glass, assuming that the process of crystallization is a first order reaction (Matusita & Sakka, 1981). DSC measurements are useful in obtaining kinetic parameters related to the glass crystallization process especially in non-isothermal method due to the rapidity of this thermoanalytical

In the isothermal method, kinetic parameters of amorphous materials crystallization are obtained by monitoring the shift in the crystallization peak as a function of time (Prasad & Varma, 2005). Crystallization peak temperatures, *Tp* and crystallized volume fractions, *x*, are determined from the thermal analysis curves with respect to time in isothermal method.

In isothermal analysis, the volume fraction crystallized, *x*, at any time *t* is given as *x* = *St* / *S* where *S* is the total area of the exothermic peak between the time *ti* at which the crystallization begins and the time *tf* at which the crystallization is completed. *St* is the area

Fig. 13. Computation of the volume fraction crystallized, *x*, in isothermal method (Prasad &

Isothermal investigation of crystallization in amorphous materials can be described by the Johnson-Mehl-Avrami equation as given in Equation 55 (Araújo & Idalgo, 2009; Avrami,

From the slopes of the linear fits to the experimental data from a plot of ln[−ln(1−*x*)] versus ln*t*, *k* and *n* values are calculated for different isothermal hold temperatures (see Fig. 14).

1939, 1940; Çelikbilek et al., 2011; Prasad & Varma, 2005).

technique (Araújo & Idalgo, 2009; Cheng et al., 2007).

between *ti* and *t* (see Fig. 13) (Ray et al., 1991).

**3.2 Thermal analysis methods** 

**3.2.1 Isothermal method** 

Varma, 2005)

Fig. 14. Plot of ln[-ln(1- *x*)] against lnt for determining the Avrami parameter, *n*

The activation energy can be evaluated by the Arrhenius-type equation (Araújo & Idalgo, 2009; Çelikbilek et al., 2011; Prasad & Varma, 2005):

$$\ln\left(k\right) = \ln k\_0 \ -E\_A/RT\tag{56}$$

where *k* is the reaction rate constant, *k*0 is the frequency factor, *EA* is the activation energy, *R* is the gas constant. The activation energy for several isothermal temperatures is calculated from the slopes of the linear fits to the experimental data from a plot of ln*k* versus 1/*T* (see Fig. 15).

Fig. 15. Plot of ln*k* against 1/T for determining the activation energy, *EA*

Crystallization Kinetics of Amorphous Materials 145

Fig. 16. Computation of the volume fraction crystallized, *x*, in non-isothermal method

**Method Approach** 

Table 2. Different methods for interpretation of non-isothermal kinetic data

Matusita et al. ln �

Afify ln (*β*/*T*<sup>p</sup>

et al. ln*β* = - 1.052 �

Some authors have applied the Johnson-Mehl-Avrami equation to the non-isothermal crystallization process, although it is not appropriate because the Johnson-Mehl-Avrami equation was derived for isothermal crystallization. Table 2 shows different approaches for the interpretation of the kinetic data obtained from thermal analysis measurements. The Kissinger equation was basically developed for studying the variation of the peak crystallization temperature with heating rate. According to Kissinger's method, the transformation under non-isothermal condition is represented by a first-order reaction. Moreover, the concept of nucleation and growth has not been included in Kissinger equation. Matusita et al. have developed a method on the basis of the fact that crystallization does not advance by an nth-order reaction but by a nucleation and growth process. They emphasized that crystallization mechanisms such as bulk crystallization or surface crystallization should be taken into account for obtaining avtivation energy. In addition to activation energy, Matusita's method provides information about the Avrami exponent and

Ozawa ln [- ln(1-*x*� ] =-*n*ln*β* + const (Equation 57)

� - �

ln�*β*� = - (*m*/*n*) (*E*A/*RT*p) + constant

ln[-ln(1-*x*��

<sup>2</sup> ) = - (*m*/*n*) (*E*A/*RT*p) + constant

+ const (Equation 58)

*<sup>n</sup>* � + const (Equation 59)

(Equation 61) (Equation 62)

*T*p 2 *<sup>β</sup>*n� <sup>=</sup> *mE*<sup>A</sup> R*T*<sup>P</sup>

*mE*<sup>A</sup> *nRT*<sup>p</sup>

Augis and Bennett ln(*β*/*T*p)= - *E*A/*RT*<sup>p</sup> + ln �� (Equation 60)

(Çelikbilek et al., 2011; Prasad & Varma, 2005)

Kissinger modified by

Ozawa modified by Matusita

In crystallization kinetic studies, as-cast (non-nucleated) and pre-nucleated samples can be used to recognize the effect of increasing and constant number of nuclei on the crystallization mechanism, respectively (Çelikbilek et al., 2011; Prasad & Varma, 2005). For the as-cast (non-nucleated) samples, when the nucleation takes place during thermal analysis, the number of nuclei of the as-cast sample is proportional to the heating rate. For the pre-nucleated samples, the number of nuclei of the pre-nucleated sample does not depend on the heating rate (Çelikbilek et al., 2011).

Crystallization mechanism of amorphous materials can be detected regarding to the following approaches; when the nucleation rate is zero during the thermal analysis experiment, *n* = *m*, when nucleation takes place during thermal analysis, *n* = *m* + 1 and when surface crystallization is the predominant mechanism, *n* = *m* = 1, where the parameters of *n* and *m* represent the values of the growth morphology depending on the crystallization mechanism (Çelikbilek et al., 2011; Matusita & Sakka, 1981; Prasad & Varma, 2005). As seen in Table 1, different crystallization mechanisms appear for different numerical factors, such as rod-like for one dimensional growth or surface crystallization (*m* = 1), disk-like for two-dimensional growth (*m* = 2) and spherical for three-dimensional growth (*m* = 3) (Çelikbilek et al., 2011; Matusita & Sakka, 1981; Matusita et al., 1984; Prasad & Varma, 2005; Ray et al. 1991).


Table 1. Values of *n* and *m* for different crystallization mechanisms (Matusita & Sakka, 1981)

#### **3.2.2 Non-isothermal method**

Non-isothermal measurements offer some advantages if compared with isothermal studies. The kinetics of crystallization of several amorphous materials has been extensively obtained from thermal analysis techniques in isothermal mode. However, non-isothermal measurements, using a constant heating rate until the complete crystallization, are usually applied to study the devitrification on different glasses since the rapidity with which this thermoanalytical technique can be performed (Araújo & Idalgo, 2009). In the non-isothermal method, crystallization peak temperatures, *Tp* and crystallized volume fractions, *x*, are determined from the thermal analysis curves with respect to temperature. The volume fraction crystallized, *x*, at any temperature *T* is given as *x* = *ST* / *S*, where *S* is the total area of the exothermic peak between the temperature, *Ti*, at which the crystallization begins and the temperature, *Tf*, at which the crystallization is completed and *ST* is the partial area of the exothermic peak up to the temperature *T* (see Fig. 16) (Ray et al., 1991).

In crystallization kinetic studies, as-cast (non-nucleated) and pre-nucleated samples can be used to recognize the effect of increasing and constant number of nuclei on the crystallization mechanism, respectively (Çelikbilek et al., 2011; Prasad & Varma, 2005). For the as-cast (non-nucleated) samples, when the nucleation takes place during thermal analysis, the number of nuclei of the as-cast sample is proportional to the heating rate. For the pre-nucleated samples, the number of nuclei of the pre-nucleated sample does not

Crystallization mechanism of amorphous materials can be detected regarding to the following approaches; when the nucleation rate is zero during the thermal analysis experiment, *n* = *m*, when nucleation takes place during thermal analysis, *n* = *m* + 1 and when surface crystallization is the predominant mechanism, *n* = *m* = 1, where the parameters of *n* and *m* represent the values of the growth morphology depending on the crystallization mechanism (Çelikbilek et al., 2011; Matusita & Sakka, 1981; Prasad & Varma, 2005). As seen in Table 1, different crystallization mechanisms appear for different numerical factors, such as rod-like for one dimensional growth or surface crystallization (*m* = 1), disk-like for two-dimensional growth (*m* = 2) and spherical for three-dimensional growth (*m* = 3) (Çelikbilek et al., 2011; Matusita & Sakka, 1981; Matusita et al., 1984; Prasad

**Crystallization mechanism** *n m* 

Three-dimensional growth of crystals 3 3 Two-dimensional growth of crystals 2 2 One-dimensional growth of crystals 1 1

Three-dimensional growth of crystals 4 3 Two-dimensional growth of crystals 3 2 One-dimensional growth of crystals 2 1 Surface crystallization 1 1 Table 1. Values of *n* and *m* for different crystallization mechanisms (Matusita & Sakka, 1981)

Non-isothermal measurements offer some advantages if compared with isothermal studies. The kinetics of crystallization of several amorphous materials has been extensively obtained from thermal analysis techniques in isothermal mode. However, non-isothermal measurements, using a constant heating rate until the complete crystallization, are usually applied to study the devitrification on different glasses since the rapidity with which this thermoanalytical technique can be performed (Araújo & Idalgo, 2009). In the non-isothermal method, crystallization peak temperatures, *Tp* and crystallized volume fractions, *x*, are determined from the thermal analysis curves with respect to temperature. The volume fraction crystallized, *x*, at any temperature *T* is given as *x* = *ST* / *S*, where *S* is the total area of the exothermic peak between the temperature, *Ti*, at which the crystallization begins and the temperature, *Tf*, at which the crystallization is completed and *ST* is the partial area of the

exothermic peak up to the temperature *T* (see Fig. 16) (Ray et al., 1991).

depend on the heating rate (Çelikbilek et al., 2011).

*Bulk crystallization with a constant number of nuclei* 

*Bulk crystallization with an increasing number of nuclei* 

& Varma, 2005; Ray et al. 1991).

**3.2.2 Non-isothermal method** 

Fig. 16. Computation of the volume fraction crystallized, *x*, in non-isothermal method (Çelikbilek et al., 2011; Prasad & Varma, 2005)

Some authors have applied the Johnson-Mehl-Avrami equation to the non-isothermal crystallization process, although it is not appropriate because the Johnson-Mehl-Avrami equation was derived for isothermal crystallization. Table 2 shows different approaches for the interpretation of the kinetic data obtained from thermal analysis measurements. The Kissinger equation was basically developed for studying the variation of the peak crystallization temperature with heating rate. According to Kissinger's method, the transformation under non-isothermal condition is represented by a first-order reaction. Moreover, the concept of nucleation and growth has not been included in Kissinger equation. Matusita et al. have developed a method on the basis of the fact that crystallization does not advance by an nth-order reaction but by a nucleation and growth process. They emphasized that crystallization mechanisms such as bulk crystallization or surface crystallization should be taken into account for obtaining avtivation energy. In addition to activation energy, Matusita's method provides information about the Avrami exponent and


Table 2. Different methods for interpretation of non-isothermal kinetic data

Crystallization Kinetics of Amorphous Materials 147

2/*β*n] against 1/*T*p for determining the activation energy, *EA*

ln�-ln�1-*x*��

*<sup>n</sup>* � <sup>+</sup> const (59)

=− + (60)

There exist also different approaches for the interpretation of the activation energy in the literature, such as the modified Ozawa equation by Matusita et al. (Equation 59) (Matusita &

where *x* is the crystallized volume fraction, *β* is the heating rate, *T*p is the peak temperature, *EA* is the activation energy, *R* is the gas constant, *n* is the Avrami parameter and *m* is the

The activation energy of crystallization can also be determined by an approximation proposed by Augis and Bennett (Augis & Bennett, 1978). The relation used by them is of the

where, *k*o is the frequency factor and *R* is gas constant. The plot of ln (*β*/*T*p) against 1/*T*<sup>p</sup>

This method has an extra advantage over the modifed Ozawa method employed in the literature for the determination of activation energy of crystallization that the intercept of

factor *k*o of Arrhenius equation (Eq. 56), which is defined as the number of attempts made by the nuclei per second to overcome the energy barrier. This also provides information for the calculation of number of nucleation sites, present in the material for crystal growth (Deepika

m*E*<sup>A</sup> *nRT*<sup>p</sup> � -�

Fig. 18. Plot of ln[*T*<sup>p</sup>

crystallization mechanism.

ln*β* = - 1.052 �

gives activation energy of crystallization (*E*A).

(/ ) / *<sup>p</sup> <sup>A</sup> <sup>p</sup> <sup>o</sup> ln T E RT ln k*

ln (*β*/*T*p) against 1000/*T*p gives the value of pre-exponential

β

Sakka, 1981):

form:

et al., 2009).

dimensionality of growth. Augis and Bennett method is helpful in obtaining kinetic parameters such as frequency factor (*k*o), rate constant (*k*) along with activation energy of crystallization and therefore preferred for the calculation of the kinetics over the other models (Deepika et al., 2009).

Ozawa and Kissinger plots are the most commonly used equations to calculate nonisothermal kinetic data, such as Avrami constant, *n* and crystallization activation energy, *EA,* respectively (Çelikbilek et al., 2011; Kissinger, 1956; Ozawa, 1971; Prasad & Varma, 2005).

In the non-isothermal method, the values of the Avrami parameter, *n*, are determined from the Ozawa equation (Çelikbilek et al., 2011; Ozawa, 1971; Prasad & Varma, 2005):

$$
\ln\left[\text{-}\ln(1\text{-}x)\right] = \text{-}n\ln\beta + \text{const}\tag{57}
$$

where *x* is the crystallized volume fraction at *T* for the heating rate of *β*. From the slopes of the linear fits to the experimental data from a plot of ln[−ln(1−*x*)] versus ln *β*, *n* values are calculated (see Fig. 17). Crystallization mechanism, *m*, of the glass samples can be detected from Matusita et al.'s approach (Matusita & Sakka, 1981) as shown in Table 1.

Fig. 17. Plot of ln[-ln(1- *x*)] against ln*β* for determining the Avrami parameter, *n*

The activation energy can be evaluated by the modified Kissinger equation by Matusita et al. (Araújo & Idalgo, 2009; Çelikbilek et al., 2011; Kissinger, 1956; Matusita & Sakka, 1981; Prasad & Varma, 2005):

$$\ln\left(\frac{T\_p^2}{\rho^\alpha}\right) = \frac{mE\_\Lambda}{RT\_P} + \text{const} \tag{58}$$

where *Tp* is the crystallization peak temperature for a given heating rate *β*, *EA* is the activation energy, *R* is the gas constant, *n* is the Avrami parameter and *m* is the numerical factor of crystallization mechanism. The activation energy is calculated from the slopes of the linear fits to the experimental data from a plot of ln(*Tp* 2/ *βn*) versus 1/*Tp* (see Fig. 18).

dimensionality of growth. Augis and Bennett method is helpful in obtaining kinetic parameters such as frequency factor (*k*o), rate constant (*k*) along with activation energy of crystallization and therefore preferred for the calculation of the kinetics over the other

Ozawa and Kissinger plots are the most commonly used equations to calculate nonisothermal kinetic data, such as Avrami constant, *n* and crystallization activation energy, *EA,* respectively (Çelikbilek et al., 2011; Kissinger, 1956; Ozawa, 1971; Prasad & Varma, 2005). In the non-isothermal method, the values of the Avrami parameter, *n*, are determined from

where *x* is the crystallized volume fraction at *T* for the heating rate of *β*. From the slopes of the linear fits to the experimental data from a plot of ln[−ln(1−*x*)] versus ln *β*, *n* values are calculated (see Fig. 17). Crystallization mechanism, *m*, of the glass samples can be detected

β 

+ const (57)

+ const (58)

2/ *βn*) versus 1/*Tp* (see Fig. 18).

the Ozawa equation (Çelikbilek et al., 2011; Ozawa, 1971; Prasad & Varma, 2005):

ln [- ln�1-*x*� ] =-*n*ln

from Matusita et al.'s approach (Matusita & Sakka, 1981) as shown in Table 1.

Fig. 17. Plot of ln[-ln(1- *x*)] against ln*β* for determining the Avrami parameter, *n*

*T*p 2 β<sup>n</sup>� <sup>=</sup> <sup>m</sup>*E*<sup>A</sup> R*T*<sup>P</sup>

The activation energy can be evaluated by the modified Kissinger equation by Matusita et al. (Araújo & Idalgo, 2009; Çelikbilek et al., 2011; Kissinger, 1956; Matusita & Sakka, 1981;

where *Tp* is the crystallization peak temperature for a given heating rate *β*, *EA* is the activation energy, *R* is the gas constant, *n* is the Avrami parameter and *m* is the numerical factor of crystallization mechanism. The activation energy is calculated from the slopes of

models (Deepika et al., 2009).

Prasad & Varma, 2005):

ln �

the linear fits to the experimental data from a plot of ln(*Tp*

Fig. 18. Plot of ln[*T*<sup>p</sup> 2/*β*n] against 1/*T*p for determining the activation energy, *EA*

There exist also different approaches for the interpretation of the activation energy in the literature, such as the modified Ozawa equation by Matusita et al. (Equation 59) (Matusita & Sakka, 1981):

$$1\text{n}\beta = -1.052\left(\frac{\text{m}E\_{\text{A}}}{nRT\_{\text{p}}}\right) \cdot \left\{\frac{\ln\left[\text{ln}\left[\text{ln}\left\{1\times\text{j}\right\}\right]\right]}{n}\right\} + \text{const} \tag{59}$$

where *x* is the crystallized volume fraction, *β* is the heating rate, *T*p is the peak temperature, *EA* is the activation energy, *R* is the gas constant, *n* is the Avrami parameter and *m* is the crystallization mechanism.

The activation energy of crystallization can also be determined by an approximation proposed by Augis and Bennett (Augis & Bennett, 1978). The relation used by them is of the form:

$$\ln(\mathcal{B} \mid \mathcal{T}\_p) = -E\_A \nmid RT\_p + \ln k\_o \tag{60}$$

where, *k*o is the frequency factor and *R* is gas constant. The plot of ln (*β*/*T*p) against 1/*T*<sup>p</sup> gives activation energy of crystallization (*E*A).

This method has an extra advantage over the modifed Ozawa method employed in the literature for the determination of activation energy of crystallization that the intercept of ln (*β*/*T*p) against 1000/*T*p gives the value of pre-exponential

factor *k*o of Arrhenius equation (Eq. 56), which is defined as the number of attempts made by the nuclei per second to overcome the energy barrier. This also provides information for the calculation of number of nucleation sites, present in the material for crystal growth (Deepika et al., 2009).

Crystallization Kinetics of Amorphous Materials 149

Fig. 19. The crystallized fraction *x* as a function of isothermal time. Inset shows the isothermal differential thermal analyses trace obtained at 537 °C (Prasad & Varma, 2005)

Fig. 20. Plot of ln[-ln(1-*x*)] against ln *t* for determining the *n* (Prasad & Varma, 2005)

A plot of ln *k* versus 1/*T* shown in Fig. 21 yielded the activation energy, *E*A, to be 293

Non-isothermal crystallization kinetics of the (1−*x*)TeO2–*x*WO3 (where *x*=0.10, 0.15 and 0.20, in molar ratio) glass system was studied by Çelikbilek et al. and DSC curves of the as-cast (non-nucleated) sample recorded at different heating rates, *β*, are shown in Fig. 22. The *T* value for the calculation of the volume fraction crystallized of 0.90TeO2–0.10WO3 glass was

kJ/mol (Prasad & Varma, 2005).

determined at 420 °C (Çelikbilek et al., 2011).

The value of activation energy can also be calculated also by using the variation of *T*p with the heating rate *β* for both crystallization phases (Afify, 1990; Afify et al., 1991). By using similar relations to Kissinger equation, the relations can be written in the form:

$$\ln\left(\left\|\left<\boldsymbol{\mathcal{T}}\right|\right\|^2\right) = -\left(m/n\right)\left(E\_{\rm A}/RT\_{\rm p}\right) + \text{const}\tag{61}$$

$$\ln\left\{\mathcal{J}\right\} = \text{-}\left(m/n\right)\left(E\_{\text{A}}/RT\_{\text{p}}\right) + \text{const}\tag{62}$$

where, *R* is gas constant, *T*p is the peak temperature, *n* is the Avrami parameter and *m* is the crystallization mechanism.

#### **3.3 Crystallization kinetic studies in amorphous materials**

Numerous studies exist on crytallization kinetics of amorphous materials, such as glasses (Araújo, & Idalgo, 2009; Araújo, 2009; Cheng, 2007; Çelikbilek, 2010; Çelikbilek, 2011; El-Mallawany, 1997; Idalgo, 2006; Jeong, 2007; Prasad & Varma 2005; Ray, C.S., Huang, W.H. & Day, 1991; Shaaban, 2009; Yukimitu, 2005), amorphous alloys (Abu El-Oyoun, 2009; Afify, 1990; Afify, 1991; Al-Ghamdi, 2010; Al-Ghamdi, 2011; Aly, 2009; Dahshan, 2010; Deepika, 2009; Elabbar, 2008; Huang, 2008; Mehta, 2004; Yahia, 2011, Zhang, 2008), amorphous thin films (Abdel-Wahaba, 2005; Bhargava, 2010; Chen & Wu, 1999; Hajiyev, 2009; Lei, 2010; Liu & Duh, 2007; Seeger & Ryder, 1994), amorphous nanomaterials (Ahmadi, 2011; Gridnev, 2008; Qin, 2004; Tomasz, 2010), etc. In this section, selected studies reported in the literature on crystallization kinetics of different types of amorphous materials were given to make the theory of kinetics more understandable for the reader.

#### **3.3.1 Glasses**

Isothermal crystallization kinetic studies in the glass system (100-*x*) LiBO2-*x*Nb2O5 (5≤*x*≤20, in molar ratio) have been realized using differential thermal analyses by Prasad et al. (Prasad & Varma, 2005). The isothermal experiments were carried out by heating the samples to the desired temperature at a rate of 50 °C/min. After attaining the required temperature, the run was held for a period of about 30 seconds to reach the equilibrium. The temperature range of 527–547 °C with an interval of 5 °C was selected for isothermal experiments because the glass shows reasonable peak shapes in this range, which is recommended for accurate data analyses.

The crystallized fraction *x* as a function of time at all the holding temperatures is shown in Fig. 19. It reveals that the time taken to complete the crystallization peak is indirectly proportional to the isothermal holding temperature. A typical isothermal DTA trace obtained at 537 °C (holding temperature) is shown in the inset of Fig. 19.

Plots of ln[-ln(1-*x*)] against ln *t* are shown in Fig. 20. Values of Avrami exponent *n* and the reaction rate constants *k* were determined by least square fits of the experimental data. The average value of *n* is 2.62. Since *n* takes only integer values from 1 to 4, the *n* (close to 3) value observed in the present study indicates the near three-dimensional growth of LiNbO3 (see Table 1). The values of ln *k* are determined for all the temperatures from the plots of ln[ ln(1-*x*)] against ln *t* (Fig. 20).

The value of activation energy can also be calculated also by using the variation of *T*p with the heating rate *β* for both crystallization phases (Afify, 1990; Afify et al., 1991). By using

 ln�*β*� = - (*m*/*n*) (*E*A/*RT*p) + const (62) where, *R* is gas constant, *T*p is the peak temperature, *n* is the Avrami parameter and *m* is the

Numerous studies exist on crytallization kinetics of amorphous materials, such as glasses (Araújo, & Idalgo, 2009; Araújo, 2009; Cheng, 2007; Çelikbilek, 2010; Çelikbilek, 2011; El-Mallawany, 1997; Idalgo, 2006; Jeong, 2007; Prasad & Varma 2005; Ray, C.S., Huang, W.H. & Day, 1991; Shaaban, 2009; Yukimitu, 2005), amorphous alloys (Abu El-Oyoun, 2009; Afify, 1990; Afify, 1991; Al-Ghamdi, 2010; Al-Ghamdi, 2011; Aly, 2009; Dahshan, 2010; Deepika, 2009; Elabbar, 2008; Huang, 2008; Mehta, 2004; Yahia, 2011, Zhang, 2008), amorphous thin films (Abdel-Wahaba, 2005; Bhargava, 2010; Chen & Wu, 1999; Hajiyev, 2009; Lei, 2010; Liu & Duh, 2007; Seeger & Ryder, 1994), amorphous nanomaterials (Ahmadi, 2011; Gridnev, 2008; Qin, 2004; Tomasz, 2010), etc. In this section, selected studies reported in the literature on crystallization kinetics of different types of amorphous materials were given to make the

Isothermal crystallization kinetic studies in the glass system (100-*x*) LiBO2-*x*Nb2O5 (5≤*x*≤20, in molar ratio) have been realized using differential thermal analyses by Prasad et al. (Prasad & Varma, 2005). The isothermal experiments were carried out by heating the samples to the desired temperature at a rate of 50 °C/min. After attaining the required temperature, the run was held for a period of about 30 seconds to reach the equilibrium. The temperature range of 527–547 °C with an interval of 5 °C was selected for isothermal experiments because the glass shows reasonable peak shapes in this range, which is

The crystallized fraction *x* as a function of time at all the holding temperatures is shown in Fig. 19. It reveals that the time taken to complete the crystallization peak is indirectly proportional to the isothermal holding temperature. A typical isothermal DTA trace

Plots of ln[-ln(1-*x*)] against ln *t* are shown in Fig. 20. Values of Avrami exponent *n* and the reaction rate constants *k* were determined by least square fits of the experimental data. The average value of *n* is 2.62. Since *n* takes only integer values from 1 to 4, the *n* (close to 3) value observed in the present study indicates the near three-dimensional growth of LiNbO3 (see Table 1). The values of ln *k* are determined for all the temperatures from the plots of ln[-

obtained at 537 °C (holding temperature) is shown in the inset of Fig. 19.

<sup>2</sup> ) = - (*m*/*n*) (*E*A/*RT*p) + const (61)

similar relations to Kissinger equation, the relations can be written in the form:

**3.3 Crystallization kinetic studies in amorphous materials** 

theory of kinetics more understandable for the reader.

recommended for accurate data analyses.

ln(1-*x*)] against ln *t* (Fig. 20).

ln (*β*/*T*<sup>p</sup>

crystallization mechanism.

**3.3.1 Glasses** 

Fig. 19. The crystallized fraction *x* as a function of isothermal time. Inset shows the isothermal differential thermal analyses trace obtained at 537 °C (Prasad & Varma, 2005)

Fig. 20. Plot of ln[-ln(1-*x*)] against ln *t* for determining the *n* (Prasad & Varma, 2005)

A plot of ln *k* versus 1/*T* shown in Fig. 21 yielded the activation energy, *E*A, to be 293 kJ/mol (Prasad & Varma, 2005).

Non-isothermal crystallization kinetics of the (1−*x*)TeO2–*x*WO3 (where *x*=0.10, 0.15 and 0.20, in molar ratio) glass system was studied by Çelikbilek et al. and DSC curves of the as-cast (non-nucleated) sample recorded at different heating rates, *β*, are shown in Fig. 22. The *T* value for the calculation of the volume fraction crystallized of 0.90TeO2–0.10WO3 glass was determined at 420 °C (Çelikbilek et al., 2011).

Crystallization Kinetics of Amorphous Materials 151

The values of the Avrami parameter, *n*, were calculated from the linear fits to the experimental data based on the Ozawa equation (Equation 57), as shown in Fig. 23. The *n*  value was determined as 1.14 for 0.90TeO2–0.10WO3 glasses. On the basis of the determination about the non-integer value of the Avrami parameter, in this study the *n* value was determined as 1, indicating the formation of surface crystallization during the

Fig. 23. The Ozawa plot for determining *n* associated with the first exotherm of the

Fig. 24. The Kissinger plots for determining *E*A associated with the first exotherm of the pre-

nucleated 0.90TeO2–0.10WO3 sample (Çelikbilek et al., 2011)

0.90TeO2–0.10WO3 sample (Çelikbilek et al., 2011)

crystallization process (see Table 1).

Fig. 21. Plot of ln*k* versus 1/*T* from which the values of crystallization activation energy are obtained (Prasad & Varma, 2005)

Fig. 22. DSC curves with heating rate *β* (5, 7.5, 10, 15, 20, 25, 30, 35 and 40 °C/min) for 0.90TeO2–0.10WO3 sample (Çelikbilek et al., 2011)

Fig. 21. Plot of ln*k* versus 1/*T* from which the values of crystallization activation energy are

Fig. 22. DSC curves with heating rate *β* (5, 7.5, 10, 15, 20, 25, 30, 35 and 40 °C/min) for

0.90TeO2–0.10WO3 sample (Çelikbilek et al., 2011)

obtained (Prasad & Varma, 2005)

The values of the Avrami parameter, *n*, were calculated from the linear fits to the experimental data based on the Ozawa equation (Equation 57), as shown in Fig. 23. The *n*  value was determined as 1.14 for 0.90TeO2–0.10WO3 glasses. On the basis of the determination about the non-integer value of the Avrami parameter, in this study the *n* value was determined as 1, indicating the formation of surface crystallization during the crystallization process (see Table 1).

Fig. 23. The Ozawa plot for determining *n* associated with the first exotherm of the 0.90TeO2–0.10WO3 sample (Çelikbilek et al., 2011)

Fig. 24. The Kissinger plots for determining *E*A associated with the first exotherm of the prenucleated 0.90TeO2–0.10WO3 sample (Çelikbilek et al., 2011)

Crystallization Kinetics of Amorphous Materials 153

Fig. 26. The Kissinger plot for determining *E*A associated with the first exotherm of the 0.90TeO2–0.10WO3 sample pre-nucleated at 350 °C for 2 hours (Çelikbilek et al., 2011)

 Fig. 27. SEM micrographs of the 0.90TeO2–0.10WO3 sample heat-treated at 410 °C (a) surface,

Fig. 27a-b represents the SEM micrographs taken from the surface and the cross-section of the 0.90TeO2–0.10WO3 sample heat-treated at 410 °C, above the first crystallization onset temperature, respectively. Fig. 27a exhibits the presence of dendritic leaf-like crystallites differently oriented on the surface. However, in the cross-sectional micrograph (see Fig. 27b), a typical amorphous structure without any crystallization on bulk structure can be clearly observed following the crystallites on the surface. Based on the SEM investigations, it was determined that the crystallites formed on the surface and did not diffuse into the bulk

structure proving the surface crystallization mechanism (Çelikbilek et al., 2011).

(b) cross-section (Çelikbilek et al., 2011)

Using the modified Kissinger equation (Equation 58), activation energy, *EA*, for the first crystallization reaction of 0.90TeO2–0.10WO3 glass was determined from the linear fits of ln(*Tp* 2/*βn*) versus 1 */ Tp* plots, as shown in Fig. 24. The activation energy of the first exotherm was calculated as 379 kJ/mol. Comparing with this result, in another study realized by Çelikbilek et al., the *E*A value was determined as 372 kJ/mol by applying the modified Ozawa method from the slopes of the linear fits to the experimental data from a plot of ln *β* versus 1/*T*p for 0.90TeO2–0.10WO3 sample.

To determine the crystallization mechanism of the glasses with fixed nuclei number, 0.90TeO2–0.10WO3 sample was pre-nucleated by heat-treating for 2 hours at 350 °C. The prenucleation temperature was determined by nucleating the as-cast glass sample for 2 hours at three different temperatures between *Tg* and *Tp*. The nucleation temperature which corresponds to the maximum peak temperature was selected as the pre-nucleation temperature (350 °C) (Çelikbilek et al., 2011).

The value of the Avrami constant of 0.90TeO2–0.10WO3 glass pre-nucleated at 350 °C is shown in Fig. 25. The *n* value of the pre-nucleated sample was calculated as 1 from the Ozawa equation (Equation 57) and while the number of nuclei do not depend on the heating rate for pre-nucleated samples, according to the approach *n* = *m*, the mechanism was determined as one-dimensional growth of the crystals (see Table 1). The *n* value calculated for the as-cast 0.90TeO2–0.10WO3 glass was also calculated as 1, indicating the surface crystallization.

The activation energy, *EA*, of the first crystallization reaction of pre-nucleated 0.90TeO2– 0.10WO3 glass was calculated using the Kissinger equation, as shown in Fig. 26. The *EA* value of the pre-nucleated sample was determined as 382 kJ/mol, very close to the calculated *EA* value of the as-cast sample (379 kJ/mol) and from the obtained data it was concluded that constant or increasing number of nuclei does not have a significant effect on crystallization activation energy.

Fig. 25. The Ozawa plot for determining *n* associated with the first exotherm of the 0.90TeO2–0.10WO3 sample pre-nucleated at 350 °C for 2 hours (Çelikbilek et al., 2011)

Using the modified Kissinger equation (Equation 58), activation energy, *EA*, for the first crystallization reaction of 0.90TeO2–0.10WO3 glass was determined from the linear fits of

To determine the crystallization mechanism of the glasses with fixed nuclei number, 0.90TeO2–0.10WO3 sample was pre-nucleated by heat-treating for 2 hours at 350 °C. The prenucleation temperature was determined by nucleating the as-cast glass sample for 2 hours at three different temperatures between *Tg* and *Tp*. The nucleation temperature which corresponds to the maximum peak temperature was selected as the pre-nucleation

The value of the Avrami constant of 0.90TeO2–0.10WO3 glass pre-nucleated at 350 °C is shown in Fig. 25. The *n* value of the pre-nucleated sample was calculated as 1 from the Ozawa equation (Equation 57) and while the number of nuclei do not depend on the heating rate for pre-nucleated samples, according to the approach *n* = *m*, the mechanism was determined as one-dimensional growth of the crystals (see Table 1). The *n* value calculated for the as-cast 0.90TeO2–0.10WO3 glass was also calculated as 1, indicating the surface

The activation energy, *EA*, of the first crystallization reaction of pre-nucleated 0.90TeO2– 0.10WO3 glass was calculated using the Kissinger equation, as shown in Fig. 26. The *EA* value of the pre-nucleated sample was determined as 382 kJ/mol, very close to the calculated *EA* value of the as-cast sample (379 kJ/mol) and from the obtained data it was concluded that constant or increasing number of nuclei does not have a significant effect on

Fig. 25. The Ozawa plot for determining *n* associated with the first exotherm of the 0.90TeO2–0.10WO3 sample pre-nucleated at 350 °C for 2 hours (Çelikbilek et al., 2011)

2/*βn*) versus 1 */ Tp* plots, as shown in Fig. 24. The activation energy of the first exotherm was calculated as 379 kJ/mol. Comparing with this result, in another study realized by Çelikbilek et al., the *E*A value was determined as 372 kJ/mol by applying the modified Ozawa method from the slopes of the linear fits to the experimental data from a plot of ln *β*

ln(*Tp*

crystallization.

crystallization activation energy.

versus 1/*T*p for 0.90TeO2–0.10WO3 sample.

temperature (350 °C) (Çelikbilek et al., 2011).

Fig. 26. The Kissinger plot for determining *E*A associated with the first exotherm of the 0.90TeO2–0.10WO3 sample pre-nucleated at 350 °C for 2 hours (Çelikbilek et al., 2011)

Fig. 27. SEM micrographs of the 0.90TeO2–0.10WO3 sample heat-treated at 410 °C (a) surface, (b) cross-section (Çelikbilek et al., 2011)

Fig. 27a-b represents the SEM micrographs taken from the surface and the cross-section of the 0.90TeO2–0.10WO3 sample heat-treated at 410 °C, above the first crystallization onset temperature, respectively. Fig. 27a exhibits the presence of dendritic leaf-like crystallites differently oriented on the surface. However, in the cross-sectional micrograph (see Fig. 27b), a typical amorphous structure without any crystallization on bulk structure can be clearly observed following the crystallites on the surface. Based on the SEM investigations, it was determined that the crystallites formed on the surface and did not diffuse into the bulk structure proving the surface crystallization mechanism (Çelikbilek et al., 2011).

Crystallization Kinetics of Amorphous Materials 155

The activation energy for crystallization was determined to be 411 and 315 kJ/mol by Kissinger (Equation 58) and Augis & Bennett (Equation 60) method, respectively (Fig. 28a-b) (Liu & Duh, 2007). Comparing with this study, previous works on near equiatomic NiTi films showed that the activation energy was 370–419 kJ/mol (Chen & Wu 1999; Seeger &

Fig. 28. Plot of the (a) Kissinger and (b) Augis & Bennett equations for the crystallization in

The isothermal crystallization kinetics of amorphous materials is described by the Johnson– Mehl–Avrami (JMA) equation. The Avrami exponent n for different temperature ranges from 2.63 to 3.12 between 793 and 823 K (Fig. 29a), which indicates that the isothermal annealing was governed by diffusion-controlled three-dimensional growth for Ni50.54Ti49.46

Fig. 29. Plots of the (a) Avrami and (b) Arrhenius equations for the isothermal crystallization

Ryder, 1994).

Ni50.54Ti49.46 thin films (Liu & Duh, 2007)

of Ni50.54Ti49.46 thin films(Liu & Duh, 2007)
