**3. Results and discussion**

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 known as model fitting methods and isoconversional methods.

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

Crystallization Kinetics of Metallic Glasses 113

( ) 1 0516 *<sup>E</sup> ln . const RT*α

The plot of *lnβ* vs *1000/Tα* (Fig.2) gives the slope *–1.0516 E(α)/R* from which the activation

Kissinger, Akahira and Sunose (Kissinger, 1957; Ozawa, 1965; Augis & Bennett, 1978; Boswell, 1980; Flynn & Wall, 1966; Akahira & Sunose, 1971) used the approximation given by Coats & Redfern (Coats & Redfern, 1964) to evaluate the integral in the rate Eq. (4). KAS

> ( ) <sup>2</sup> *AR E ln ln T Eg RT*

The activation energy can be evaluated from the slope of plot *ln(β/T2)* vs *1000/T* for constant conversion, α (Fig. 3) Values of *E* are given in Table 1. The discussion given ahead describes some of the methods available in the literature which are basically special cases of the KAS

*i) Kissinger method:* This well-known method assumes that the reaction rate is maximum at the peak temperature (*Tp*). This assumption also implies a constant degree of conversion (

=− +

*2)* vs *1000/Tp* gives an approximate straight line and the activation energy *E*

= −

α

β

2 *p p E AR ln ln T RT E*

β

α

β

energy has been evaluated (Table 1). At *T*

using Eq. (5) is given in Table 2.

Fig. 2. OFW plot for peak-1

equation (6).

A plot of *ln(β /Tp*

method is based on the expression

at *Tp*. The equation used by Kissinger is

is calculated using the slope (Table 2).

b. Kissinger-Akahira-Sunose (KAS) method

α

= − + (5)

 *= Tp*, (Ozawa method) the value of *E* determined

(7)

(6)

α)

#### **3.1 Isoconversional analysis**

The isoconversional methods require the knowledge of temperatures *T*α*(*β*)* at which an equivalent stage of reaction occurs for various heating rates. The equivalent stage is defined as the stage at which a fixed amount is transformed or at which a fixed fraction, α of the total amount is transformed (Starink, 1997). These methods are further categorized as linear and non-linear isoconversional methods. The linear integral isoconversional methods (Kissinger, 1957; Ozawa, 1965; Augis & Bennett, 1978; Boswell, 1980; Flynn & Wall, 1966; Akahira & Sunose, 1971; Li & Tang, 1999) depend on the approximation of the temperature integral and require the data on *T*α*(*β*)*. The differential isoconversional methods depend on the rate of transformation at *T*α*(*β*)* and the data on *T*α*(*β*)* (Gupta et al., 1988; Friedman, 1964; Gao & Wang, 1986). Vyazovkin (Vyazovkin & Wight, 1997) introduced a non-linear isoconversional method to increase the accuracy of evaluating the activation energy. The isoconversional methods are based on the basic kinetic equation (Paulik, 1995)

$$\frac{d\alpha}{dT} = \frac{1}{\beta}k(T)f(\alpha) = \frac{A}{\beta}\exp\left(-\frac{E}{RT}\right)f(\alpha) \tag{3}$$

where *k(T)* is the rate constant, β is the heating rate, α is the conversion fraction and *f(*α*)* is the reaction model which in case of KJMA formalism gives the Eq. (1). Eq. (3) can also be expressed in the integral form as

$$\log(\alpha) = \prod\_{0}^{\alpha} [f(\alpha)]^{-1} d\alpha = \frac{A}{\mathcal{B}} \Big| \exp\left(-\frac{E}{RT}\right) dT \tag{4}$$

As mentioned earlier, exact solution of the temperature integral is not available and various approximations made for this has resulted into different methods. We have selected a few most commonly used methods. The accuracy of various isoconversional methods and, the experimental and analytical errors associated with these methods are discussed in detail by Starink (Starink, 2003). Roura and Farjas (Roura & Farjas, 2009) have proposed an analytical solution for the Kissinger equation. Rotaru and Gosa (Rotaru & Gosa, 2009) describe their recently developed software that implements a number of known techniques such as various isoconversional methods, a method of invariant kinetic parameters, master plots methods, etc. Cai and Chen (Cai & Chen, 2009) have proposed a new numerical routine for a linear integral isoconversional method that allows one to obtain accurate values of the activation energy in the cases when the latter varies strongly with the extent of conversion. Criado et al. (Criado et al., 2008) provide a critical overview of isoconversional methods, putting the focus on establishing whether the observed variations in the activation energy are real or apparent (Vyazovkin, 2010).

#### **Linear integral isoconversional methods**

#### a. Ozawa-Flynn-Wall (OFW) method

In this method (Ozawa, 1965; Augis & Bennett, 1978; Boswell, 1980; Flynn & Wall, 1966) the temperature integral in Eq. (4) is simplified by using the Doyle's approximation (Doyle, 1961, 1962, 1965) and hence we obtain the following equation:

equivalent stage of reaction occurs for various heating rates. The equivalent stage is defined

total amount is transformed (Starink, 1997). These methods are further categorized as linear and non-linear isoconversional methods. The linear integral isoconversional methods (Kissinger, 1957; Ozawa, 1965; Augis & Bennett, 1978; Boswell, 1980; Flynn & Wall, 1966; Akahira & Sunose, 1971; Li & Tang, 1999) depend on the approximation of the temperature

Gao & Wang, 1986). Vyazovkin (Vyazovkin & Wight, 1997) introduced a non-linear isoconversional method to increase the accuracy of evaluating the activation energy. The

*d AE* <sup>1</sup> *k(T ) f ( ) exp f ( ) dT RT*

= =−

 β

the reaction model which in case of KJMA formalism gives the Eq. (1). Eq. (3) can also be

β

<sup>−</sup> = =−

As mentioned earlier, exact solution of the temperature integral is not available and various approximations made for this has resulted into different methods. We have selected a few most commonly used methods. The accuracy of various isoconversional methods and, the experimental and analytical errors associated with these methods are discussed in detail by Starink (Starink, 2003). Roura and Farjas (Roura & Farjas, 2009) have proposed an analytical solution for the Kissinger equation. Rotaru and Gosa (Rotaru & Gosa, 2009) describe their recently developed software that implements a number of known techniques such as various isoconversional methods, a method of invariant kinetic parameters, master plots methods, etc. Cai and Chen (Cai & Chen, 2009) have proposed a new numerical routine for a linear integral isoconversional method that allows one to obtain accurate values of the activation energy in the cases when the latter varies strongly with the extent of conversion. Criado et al. (Criado et al., 2008) provide a critical overview of isoconversional methods, putting the focus on establishing whether the observed variations in the activation energy

In this method (Ozawa, 1965; Augis & Bennett, 1978; Boswell, 1980; Flynn & Wall, 1966) the temperature integral in Eq. (4) is simplified by using the Doyle's approximation (Doyle,

α*(*β

α

as the stage at which a fixed amount is transformed or at which a fixed fraction,

*)* and the data on *T*

α

is the heating rate,

1 0 0 *<sup>T</sup> A E g( ) [ f ( )] d exp dT RT*

isoconversional methods are based on the basic kinetic equation (Paulik, 1995)

α*(*β

*)* (Gupta et al., 1988; Friedman, 1964;

(3)

is the conversion fraction and *f(*

*)*. The differential isoconversional methods depend on

 α

(4)

*)* at which an

αof the

> α*)* is

The isoconversional methods require the knowledge of temperatures *T*

α*(*β

α*(*β

α

α

1961, 1962, 1965) and hence we obtain the following equation:

β

α

 αα

β

**3.1 Isoconversional analysis** 

integral and require the data on *T*

the rate of transformation at *T*

where *k(T)* is the rate constant,

expressed in the integral form as

are real or apparent (Vyazovkin, 2010). **Linear integral isoconversional methods**  a. Ozawa-Flynn-Wall (OFW) method

$$
\ln \beta = -1.0516 \frac{E(\alpha)}{RT\_{\alpha}} + const \tag{5}
$$

The plot of *lnβ* vs *1000/Tα* (Fig.2) gives the slope *–1.0516 E(α)/R* from which the activation energy has been evaluated (Table 1). At *T*α *= Tp*, (Ozawa method) the value of *E* determined using Eq. (5) is given in Table 2.

Fig. 2. OFW plot for peak-1

#### b. Kissinger-Akahira-Sunose (KAS) method

Kissinger, Akahira and Sunose (Kissinger, 1957; Ozawa, 1965; Augis & Bennett, 1978; Boswell, 1980; Flynn & Wall, 1966; Akahira & Sunose, 1971) used the approximation given by Coats & Redfern (Coats & Redfern, 1964) to evaluate the integral in the rate Eq. (4). KAS method is based on the expression

$$\ln\left(\frac{\mathcal{B}}{T^2}\right) = \ln\left(\frac{AR}{Eg(\alpha)}\right) - \frac{E}{RT} \tag{6}$$

The activation energy can be evaluated from the slope of plot *ln(β/T2)* vs *1000/T* for constant conversion, α (Fig. 3) Values of *E* are given in Table 1. The discussion given ahead describes some of the methods available in the literature which are basically special cases of the KAS equation (6).

*i) Kissinger method:* This well-known method assumes that the reaction rate is maximum at the peak temperature (*Tp*). This assumption also implies a constant degree of conversion (α) at *Tp*. The equation used by Kissinger is

$$\ln\left(\frac{\mathcal{B}}{T\_p^2}\right) = -\frac{E}{RT\_p} + \ln\left(\frac{AR}{E}\right) \tag{7}$$

A plot of *ln(β /Tp 2)* vs *1000/Tp* gives an approximate straight line and the activation energy *E* is calculated using the slope (Table 2).

Crystallization Kinetics of Metallic Glasses 115

Boswell method determines the activation energy at peak temperature (Table 2) using the

*p p <sup>E</sup> ln const T RT*

Li and Tang (Li & Tang, 1999) have developed an isoconversional integral method which does not make any assumption about the kinetic model and involves no approximation in

( )

 α

 α= − (11)

> 1 *d T*

α, for a set of

α

 α β

's at constant

*dT T* α β

(13)

α

= + has the same value for a given reaction under study

αvs

0

 α== − (12)

*)*, i.e. it is a so-called model-free method. However, being a

α

*d E* <sup>1</sup> *ln d G <sup>d</sup> dt R T*

> *<sup>d</sup> ln d dt*

The method suggested by Friedman (Friedman, 1964) sometimes known as transformation rate-isoconversional method, utilizes the differential of the transformed fraction and hence it is called differential isoconversional method. Substituting value of *k(T)* in Eq. (3) and taking

( ) ( ) *dd E ln ln ln A f dt dT RT*

by taking logarithm on both sides of Eq.(3). For a constant α, the plot of *<sup>d</sup>* <sup>1</sup> *ln vs*

Since this method does not take any mathematical approximation for the temperature integral, it is considered to give accurate estimate of *E*. Thus the method does not require

A method suggested by Gao and Wang (Gao & Wang, 1986) is a special case of the Friedman

*p p d E ln const dT RT*

=− +

differential method, its accuracy is limited by the signal noise (Dhurandhar et al, 2010).

method. This method uses the following expression to determine the activation energy.

α β

 α β

α

0 0

0

logarithm, Friedman derived a linear differential isoconversional expression as

αα

should be a straight line (Fig. 4) whose slope gives us the value of *E* (Table 1).

α

αα

α

 αα

β

α

α

. A plot of

=− +

(10)

β

following equation

c. Li-Tang Method

Where ( ) ( ) ( )

and a given α irrespective of

α

any assumption on *f(*

α α

0 *G ln A ln f d* α

will have the slope *–E/R*.

α

**Linear differential isoconversional method** 

the Eq. (3) as

conversion

Fig. 3. KAS plot for peak-1

*ii) Augis & Bennett's method:* This method was suggested by Augis and Bennett (Augis & Bennett, 1978) and is an extension of Kissinger method showing its applicability to heterogeneous reaction described by Avrami expression. Apart from the peak crystallization temperature it also incorporates the onset temperature of crystallization, *To* and it is supposed to be a very accurate method of determining *E* through the equation

$$\ln\left(\frac{\beta}{\left(T\_p - T\_o\right)}\right) = -\frac{E}{RT\_p} + \ln A \tag{8}$$

where *Tp* and *To* are the peak and the onset temperatures of crystallization respectively. The values of *E* obtained from the plot *(ln(β /(Tp-To))* vs *1000/Tp* is given in Table 2.

Further,

$$n = 2.5 \frac{T\_p^2}{\Delta T \left(\frac{E}{R}\right)}\tag{9}$$

where ΔT is the full width at half maximum of the DSC curve. *n* can be derived using Eq. (9).

*iii) Boswell method:* Boswell (Boswell, 1980) has found a limitation in the Augis & Bennett method that if

$$\frac{T\_p - T\_o}{T\_p} = 1$$

then Augis & Bennett gives crude results. However, it may be noted that this condition may not apply to the present case.

Boswell method determines the activation energy at peak temperature (Table 2) using the following equation

$$\ln\left(\frac{\mathcal{B}}{T\_p}\right) = -\frac{E}{RT\_p} + const\tag{10}$$

#### c. Li-Tang Method

114 Advances in Crystallization Processes

*ii) Augis & Bennett's method:* This method was suggested by Augis and Bennett (Augis & Bennett, 1978) and is an extension of Kissinger method showing its applicability to heterogeneous reaction described by Avrami expression. Apart from the peak crystallization temperature it also incorporates the onset temperature of crystallization, *To* and it is

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

> > 2 5 *Tp n . <sup>E</sup> <sup>T</sup>*

<sup>=</sup> <sup>Δ</sup>

where ΔT is the full width at half maximum of the DSC curve. *n* can be derived using Eq. (9). *iii) Boswell method:* Boswell (Boswell, 1980) has found a limitation in the Augis & Bennett

> 1 *p o p T T T* <sup>−</sup> <sup>≈</sup>

then Augis & Bennett gives crude results. However, it may be noted that this condition may

=− + <sup>−</sup>

where *Tp* and *To* are the peak and the onset temperatures of crystallization respectively. The

2

*R*

(8)

(9)

supposed to be a very accurate method of determining *E* through the equation

β

values of *E* obtained from the plot *(ln(β /(Tp-To))* vs *1000/Tp* is given in Table 2.

Fig. 3. KAS plot for peak-1

Further,

method that if

not apply to the present case.

Li and Tang (Li & Tang, 1999) have developed an isoconversional integral method which does not make any assumption about the kinetic model and involves no approximation in the Eq. (3) as

$$\int\_0^\alpha \left(\ln \frac{d\alpha}{dt}\right) d\alpha \quad = G(\alpha) - \frac{E\_\alpha}{R} \int\_0^\alpha \left(\frac{1}{T}\right) d\alpha \tag{11}$$

Where ( ) ( ) ( ) 0 *G ln A ln f d* α α α αα= + has the same value for a given reaction under study

and a given α irrespective ofβ. A plot of 0 *<sup>d</sup> ln d dt* α α α vs 0 1 *d T* α α , for a set of β's at constant

conversion αwill have the slope *–E/R*.

#### **Linear differential isoconversional method**

The method suggested by Friedman (Friedman, 1964) sometimes known as transformation rate-isoconversional method, utilizes the differential of the transformed fraction and hence it is called differential isoconversional method. Substituting value of *k(T)* in Eq. (3) and taking logarithm, Friedman derived a linear differential isoconversional expression as

$$\ln\left(\frac{d\alpha}{dt}\right)\_{\alpha} = \ln\beta \left(\frac{d\alpha}{dT}\right)\_{\alpha} = \ln\left(A\,f\left(\alpha\right)\right) - \frac{E\_{\alpha}}{RT\_{\alpha}}\tag{12}$$

by taking logarithm on both sides of Eq.(3). For a constant α, the plot of *<sup>d</sup>* <sup>1</sup> *ln vs dT T* α β should be a straight line (Fig. 4) whose slope gives us the value of *E* (Table 1).

Since this method does not take any mathematical approximation for the temperature integral, it is considered to give accurate estimate of *E*. Thus the method does not require any assumption on *f(*α*)*, i.e. it is a so-called model-free method. However, being a differential method, its accuracy is limited by the signal noise (Dhurandhar et al, 2010).

A method suggested by Gao and Wang (Gao & Wang, 1986) is a special case of the Friedman method. This method uses the following expression to determine the activation energy.

$$\ln\left(\beta \frac{d\alpha}{dT\_p}\right) = -\frac{E}{RT\_p} + const \tag{13}$$

Crystallization Kinetics of Metallic Glasses 117

Vyazovkin and Wight (Vyazovkin & Wight, 1997) described an advanced isoconversional method. Similar to other isoconversional methods, this method is also based on the

two experiments conducted at different heating rates the ratio of the temperature integral I

performed under different heating rates, the activation energy can be determined at any

α

( ) ( )

 

β

β

*n Const*

*n n i j*

*I E ,T I E ,T* α α

*i j j i*

α α

Matusita and Sakkka (Matusita & Sakka, 1979) suggested the following equation specifically

where *m* is an integer depends on the dimensionality of the crystal and the Avarami exponent *n* depends on the nucleation process. For a constant temperature, the plot of ln[-

*)* is independent of the heating program. So, for any

(15)

α

*RT* (16)

to find the

is a constant. For a given conversion and a set of n experiments

for which the function

Fig. 5. Local Activation energy E at different α from different methods

α

**Non-linear isoconversional method** 

assumption that the reaction model, *g(*

α

β

by finding the value of *E*

dependence of activation energy on the extent of conversion.

ln[ ln(1 )] ln − −α =− β− + *mE*

≠

is a minimum. The minimization procedure is repeated for each value of

) to the heating rate

particular value of

**3.2 Isokinetic methods** 

for the non-isothermal data

a. Matusita and Sakka method

(E, Tα

$$K\_p = \frac{\beta \mathcal{E}}{RT\_p^2} \tag{14}$$

where,

$$K\_p = A \exp\left(\frac{-E}{RT\_p}\right) \\ \text{and} \\ \left(\frac{d\alpha}{dt}\right)\_p = 0.37nK\_p$$

Fig. 4. Friedman plot for peak-1


Table 1. Local Activation energy (E) at different conversion for different methods.


Table 2. Activation energy (E) using various methods.

*p* 2 *p*

*p p* 0 37 *p p E d K Aexp and . nK RT dt* −

= =

α

KAS OFW Friedman

= (14)

*<sup>E</sup> <sup>K</sup> RT* β

where,

Fig. 4. Friedman plot for peak-1

<sup>α</sup> E (kJ/mol)

Table 2. Activation energy (E) using various methods.

0.1 602 ± 2 584 ± 2 555 ± 1 0.2 597 ± 1 580 ± 1 626 ± 1 0.3 603 ± 1 586 ± 2 648 ± 1 0.4 615 ± 1 597 ± 1 687 ± 1 0.5 635 ± 1 616 ± 1 725 ± 1 0.6 654 ± 1 634 ± 1 702 ± 3 0.7 648 ± 1 629 ± 1 522 ± 5 0.8 606 ± 1 589 ± 1 398 ± 5 0.9 549 ± 1 534 ± 1 318 ± 2

Table 1. Local Activation energy (E) at different conversion for different methods.

Method E ( kJ/mol ) Kissinger 553 ± 2 Ozawa 546 ± 2 Augis & Bennett 532 ± 2 Boswell 443 ± 7

Fig. 5. Local Activation energy E at different α from different methods

#### **Non-linear isoconversional method**

Vyazovkin and Wight (Vyazovkin & Wight, 1997) described an advanced isoconversional method. Similar to other isoconversional methods, this method is also based on the assumption that the reaction model, *g(*α*)* is independent of the heating program. So, for any two experiments conducted at different heating rates the ratio of the temperature integral I (E, Tα) to the heating rate β is a constant. For a given conversion and a set of n experiments performed under different heating rates, the activation energy can be determined at any particular value of α by finding the value of *E*αfor which the function

$$\sum\_{i \neq j}^{n} \sum\_{j}^{n} \frac{\left[ I\left(E\_{\alpha}, T\_{\alpha i}\right) \beta\_{j} \right]}{\left[ I\left(E\_{\alpha}, T\_{\alpha j}\right) \beta\_{i} \right]} \tag{15}$$

is a minimum. The minimization procedure is repeated for each value of α to find the dependence of activation energy on the extent of conversion.

#### **3.2 Isokinetic methods**

a. Matusita and Sakka method

Matusita and Sakkka (Matusita & Sakka, 1979) suggested the following equation specifically for the non-isothermal data

$$
\ln[-\ln(1-\alpha)] = -n\ln\beta - \frac{mE}{RT} + \text{Const} \tag{16}
$$

where *m* is an integer depends on the dimensionality of the crystal and the Avarami exponent *n* depends on the nucleation process. For a constant temperature, the plot of ln[-

Crystallization Kinetics of Metallic Glasses 119

(17)

(Fig. 8) gives the

*p p mE ln Const T RT*

Where *E* is the activation energy for crystallization, *Tp* is the peak temperature and *R* is the universal gas constant. *m* is known as the dimensionality of growth and here *m = n.* In order to derive *E* from this equation, one must know the value of *n*. The *n* value can be obtained from the slope of the plot of ln[-ln(1-α)] Vs. lnβ at constant temperature. In order to evaluate

A general trend of decrease in the values of n with increasing heating rate can be observed. Such trend has been also seen by Matusita and Sakka (Matusita & Sakka, 1979) and in other

*Heating rate Matusita & Sakka* 

*4 1.81 6 1.76 8 1.75 10 1.79*  *n*

Vs. <sup>1</sup> *Tp*

*p*

*ln T* β

=− +

2 *n*

β

*E*, the values of *n* are substituted in Eq. (17). Then plots of 2

Fig. 8. Mod. Kissinger plot for n = 1.33 and n = 1.36

Fe-based (Raval et al., 2005) metallic glasses.

Table 3. Values of Avrami exponent (n)

values of activation energy *E*, and the average *E* obtained is 549.80 kJ/mol.

ln(1-α)] versus lnβ gives a straight line (Fig.6) and the slope gives the value of *n*, which come out to be n = 1.33 and *n* = 1.36 for temperatures T = 775 K and *T* = 778 K respectively. The plot of ln[-ln(1-α)] versus 1/T at constant heating rate should be a straight line and the value of mE is obtained from the slope (Fig.7).

Fig. 6. Plot of ln[-ln(1- α)] Vs. lnβ for diff. Temp.

Fig. 7. Plot of ln[-ln(1- α)] Vs. 1000/T for diff. heating rates.

a) Modified Kissinger method

The modified Kissinger equation (Matusita & Sakka, 1980) given below can be utilized to derive the activation energy (*E*).

ln(1-α)] versus lnβ gives a straight line (Fig.6) and the slope gives the value of *n*, which come out to be n = 1.33 and *n* = 1.36 for temperatures T = 775 K and *T* = 778 K respectively. The plot of ln[-ln(1-α)] versus 1/T at constant heating rate should be a straight line and the value

of mE is obtained from the slope (Fig.7).

Fig. 6. Plot of ln[-ln(1- α)] Vs. lnβ for diff. Temp.

Fig. 7. Plot of ln[-ln(1- α)] Vs. 1000/T for diff. heating rates.

The modified Kissinger equation (Matusita & Sakka, 1980) given below can be utilized to

a) Modified Kissinger method

derive the activation energy (*E*).

$$\ln\left(\frac{\mathcal{B}^n}{T\_p^2}\right) = -\frac{mE}{RT\_p} + \text{Const} \tag{17}$$

Where *E* is the activation energy for crystallization, *Tp* is the peak temperature and *R* is the universal gas constant. *m* is known as the dimensionality of growth and here *m = n.* In order to derive *E* from this equation, one must know the value of *n*. The *n* value can be obtained from the slope of the plot of ln[-ln(1-α)] Vs. lnβ at constant temperature. In order to evaluate

*E*, the values of *n* are substituted in Eq. (17). Then plots of 2 *n p ln T* β Vs. <sup>1</sup> *Tp* (Fig. 8) gives the

values of activation energy *E*, and the average *E* obtained is 549.80 kJ/mol.

Fig. 8. Mod. Kissinger plot for n = 1.33 and n = 1.36

A general trend of decrease in the values of n with increasing heating rate can be observed. Such trend has been also seen by Matusita and Sakka (Matusita & Sakka, 1979) and in other Fe-based (Raval et al., 2005) metallic glasses.


Table 3. Values of Avrami exponent (n)

Crystallization Kinetics of Metallic Glasses 121

Fig. 9. Bright field TEM image of Ti20Zr20Cu60 metallic glass after annealing at 673 K for

Fig. 10. SAD pattern of Ti20Zr20Cu60 metallic glass after annealing at 673 K for 4 hours

4 hours

#### b. Coats & Redfern method

One of the most popular model-fitting methods is the Coats and Redfern method (Coats & Redfern, 1964). This method is based on the equation

$$
\ln \frac{g\_i(\alpha)}{T^2} = \ln \left[ \frac{A\_i R}{\beta E\_i} \left( 1 - \frac{2RT}{E\_i} \right) \right] - \frac{E\_i}{RT} \tag{18}
$$

$$
\equiv \ln \frac{A\_i R}{\beta E\_i} - \frac{E\_i}{RT}
$$

The graph of 2 *<sup>i</sup> g( ) ln T* α Vs. <sup>1</sup> *T* gives a straight line whose slope and intercept allow us to calculate *E* and *A* for a particular reaction model. For the different kinetic models and for 0.1≤α≤0.9, the straight lines corresponding to CR method are characterized by correlation coefficients (r). The general practice in this method to determine *E* is to look for the model corresponding to maximum r. In some cases, the so-obtained value of *E* is significantly different from those obtained from other methods.

#### c. The invariant kinetic parameter (IKP) method

It has been observed that the same experimental curve α*=*α*(T)* can be described by different function of conversion *(f(*α*))*. Further, the values of the activation energy obtained for various *f(*α*)* for single non-isothermal curve are correlated through the compensation effect (Galwey, 2003). These observations form the basis of the IKP method. In order to apply this method, α *=* α*(T)* curves are obtained at different heating rates (βν = 4, 6, 8, 10) using DSC. For each heating rate the pairs (*A*ν*j, E*ν*<sup>j</sup>*), where *j* corresponds to a particular degree of conversion, are determined using the following equation:

$$
\ln \frac{\mathbf{g}(\alpha)}{T^2} = \ln \frac{AR}{\beta \mathbf{E}} - \frac{\mathbf{E}}{RT} \tag{19}
$$

For constant β, a plot of 2 *g( ) ln T* α Vs. <sup>1</sup> *T* is a straight line whose slope allows the evaluation

of activation energy *E*ν and intercept, pre-exponential factor, *A*ν for different reaction models *g(*α*)*. The same procedure is repeated to obtain the pairs (*E*ν*, A*ν) for different heating rates. Now, the calculation of invariant activation parameters is done using the compensation relation (Budrugeac, 2007)

$$
\ln A\_{\upsilon} = \alpha^\* + \beta^\* E\_{\upsilon} \tag{20}
$$

The Eq. (20) represents a linear relationship between *lnA* and *E*; any increase in the magnitude of one parameter is offset, or compensated, by appropriate increase of the other. Plotting *lnA*ν Vs. *E*ν for different heating rates, the compensation effect parameters α\* and β\* are obtained. These parameters follow an equation

$$
\alpha^\* = \ln A - \beta^\* E \tag{21}
$$

One of the most popular model-fitting methods is the Coats and Redfern method (Coats &

<sup>2</sup> <sup>1</sup> *<sup>i</sup> i i i i*

calculate *E* and *A* for a particular reaction model. For the different kinetic models and for 0.1≤α≤0.9, the straight lines corresponding to CR method are characterized by correlation coefficients (r). The general practice in this method to determine *E* is to look for the model corresponding to maximum r. In some cases, the so-obtained value of *E* is significantly

(Galwey, 2003). These observations form the basis of the IKP method. In order to apply this

*g( ) AR E ln ln T E RT*

Now, the calculation of invariant activation parameters is done using the compensation

*ln A E* υ

α β

The Eq. (20) represents a linear relationship between *lnA* and *E*; any increase in the magnitude of one parameter is offset, or compensated, by appropriate increase of the other.

 υ

for different heating rates, the compensation effect parameters α\* and β\*

β

*(T)* curves are obtained at different heating rates (

α

and intercept, pre-exponential factor, *A*

ν*j, E*ν

Vs. <sup>1</sup>

= −−

*T* gives a straight line whose slope and intercept allow us to

α*=*α

*)* for single non-isothermal curve are correlated through the compensation effect

*))*. Further, the values of the activation energy obtained for

βν

*<sup>j</sup>*), where *j* corresponds to a particular degree of

= − (19)

∗ ∗ = + (20)

\* ln *A E* <sup>∗</sup> α= − β (21)

*T* is a straight line whose slope allows the evaluation

ν

ν*, A*ν (18)

*(T)* can be described by different

= 4, 6, 8, 10) using DSC.

for different reaction models

) for different heating rates.

*g( ) A R RT E ln ln T E E RT*

β

*i i i AR E ln* β

*E RT* <sup>−</sup>

b. Coats & Redfern method

The graph of 2

function of conversion *(f(*

For each heating rate the pairs (*A*

For constant β, a plot of 2

of activation energy *E*

relation (Budrugeac, 2007)

ν Vs. *E*ν

2

ν

are obtained. These parameters follow an equation

α

α *=* α

various *f(*

method,

*g(*α

Plotting *lnA*

*<sup>i</sup> g( ) ln T* α

Redfern, 1964). This method is based on the equation

Vs. <sup>1</sup>

different from those obtained from other methods. c. The invariant kinetic parameter (IKP) method

It has been observed that the same experimental curve

α

conversion, are determined using the following equation:

*g( ) ln T* α

*)*. The same procedure is repeated to obtain the pairs (*E*

2

≅

α

Fig. 9. Bright field TEM image of Ti20Zr20Cu60 metallic glass after annealing at 673 K for 4 hours

Fig. 10. SAD pattern of Ti20Zr20Cu60 metallic glass after annealing at 673 K for 4 hours

Crystallization Kinetics of Metallic Glasses 123

superior to the isokinetic methods as far as the determination of *E* is concerned (Pratap et al, 2007). Nonetheless, accurate determination of E is not the only issue in the kinetic analysis of crystallization process in metallic glasses. The micro-structural evolution during the nonisothermal heating of the metallic glasses is also important. For the determination of the dimensionality of the growth and the grain size, one needs to know a precise reaction model that closely follows the crystallization process. A reaction model independently proposed by John-Mehl- Avrami-Kolmogorov (JMAK) is found to be the most suitable for describing the nucleation and growth process during the non- isothermal crystallization of metallic glasses. This model does help to determine of the kinetic parameters, like the dimensionality of growth (apart from *E* and *A*). The model-free isoconversional methods are definitely superior to the isokinetic methods for the accurate determination of kinetic parameters like E and A. However, the knowledge of accurate E and A is not sufficient for the detailed investigations of the dimensionality of the growth and the grain size using thermal analysis. A precise reaction model accounting for the phase transformations during the crystallization process is a prerequisite for deriving such micro-structural information. This could be a valid proposition if it is explicitly related to the phase transformations involving significant chemical changes. One can find numerous publications where JMAK formalism has been found to be the most appropriate for the description of kinetics of nucleation and growth processes in metallic glasses. Therefore, in our opinion, isokinetic methods (despite its limited applicability) are important and useful for the analysis of non-isothermal crystallization data. So, as far as the study of thermally activated phase transformation in metallic glasses is concerned, both the types of methods are complementary and provide not

only useful data, but also pave way into the insight of the crystallization process.

Afify, S. (1991). Differential scanning calorimetric study of chalcogenide glass Se0.7Te0.3, *Journal of Non Crystalline Solids,* Vol. 128, pp. 279-284; ISSN: 0022-3093 Akahira, T. & Sunose, T. (1971). Joint convention of four electrical institutes, *Research Report, Chiba. Institute of Technology (Science and Technology)*, Vol. 16, pp. 22-31 Augis, J.A. & Bennett, J.E. (1978). Calculation of the Avrami parameters for heterogeneous

Boswell, P.G. (1980). On the calculation of activation energies using modified Kissinger

Budrugeac, P. (2007). The Kissinger law and the IKP method for evaluating the non-

Cai, J.M. & Chen, S.Y. (2009).A new iterative linear integral isoconversional method for the

*of Computational Chemistry*, Vol. 30, pp. 1986-1991; Online ISSN: 1096-987X Castro, M. (2003). Phae-field approach to heterogeneous nucleation, *Physical Review B*, Vol. 67, pp. 035412-035419; ISSN: 0163-1829 (Print), 1095-3795 (electronic version)

pp. 143-151 ISSN:1388-6150 (Print), 1572-894 (electronic version)

solid state reactions using a modification of the Kissinger method, *Journal of Thermal Analysis and Calorimetry*, Vol. 13, pp. 283-292; ISSN:1388-6150 (Print), 1572-

method, *Journal of Thermal Analysis and Calorimetry*, Vol. 18, pp. 353-358 ISSN:1388-

isothermal kinetic parameters, *Journal of Thermal Analysis and Calorimetry*, Vol. 89,

determination of the activation energy varying with the conversion degree *Journal* 

**5. References** 

894 (electronic version)

6150 (Print), 1572-894 (electronic version)

The plot of α\* and β\* gives the true values of *E* and *A*.

Nano-structures can be synthesized by controlled crystallization of metallic glasses also known as de-vitrification method.

The selected area diffraction (SAD) pattern shows characteristic rings with discontinuity. The phases can also be idetified as seen from fig.11.

Fig. 11. Nano-phases present in Ti20Zr20Cu60 metallic glass after annealing at 673 K for 4 hours
