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Department of Sciences & Humanities, M.V.S.R. Engineering College, Nadergul, Hyderabad, Hyderabad, India

- *Corresponding Author:
- Bhanu Prasad B

Department of Sciences & Humanities

M.V.S.R. Engineering College

Nadergul, Hyderabad, India

**Tel:**40-24530846

**E-mail:**[email protected]

**Received Date**: 04/01/2017; **Accepted Date:** 04/03/2017; **Published Date**: 14/03/2017

**Visit for more related articles at** Research & Reviews: Journal of Material Sciences

The activation energy of crystallization (Ea), Frequency factor(k0) and Avrami exponent (n) of amorphous Fe76Pr4B20 and Fe76Dy4B20 alloys have been calculated by three methods namely Kissinger, Augis- Bennet and Matusita-Sakka on the basis of crystallization kinetics using Differential Scanning Calorimetry (DSC). The average activation energy for primary crystallization of amorphous Fe76Pr4B20 and Fe76Dy4B20 alloys using the above three methods is determined as 389.67 kJouls/ mole and 578.59 kJouls/mole, respectively. The frequency factor of amorphous Fe76Pr4B20 and Fe76Dy4B20 alloys using Kissinger method is found to be 4.542 × 1019 (sec)-1 and 8.5 × 1031 (sec)-1, respectively. Similarly, the Avrami exponent of amorphous Fe76Pr4B20 and Fe76Dy4B20 alloys is found to be 1.67 and 0.831, respectively.

Differential scanning calorimetry; Activation energy; Crystallization temperature; Frequency factor

The Thermal stability of amorphous alloys is a subject of considerable interest, since the properties of these engineering
materials may be significantly changed by the onset of crystallization. The crystallization is associated with nucleation and growth
process. Activation energy (Ea) is one of the important parameters describing the transformation kinetics. Ea values are found to
be larger at higher temperatures in crystallization. These features probably originate from the crystallization micro-mechanism.
Thus, crystallization kinetics of amorphous materials was investigated by explaining the crystallization mechanism and the
crystallization activation energy in terms of isothermal and non-isothermal methods with different approaches. Different thermal
analysis techniques used in crystallization kinetics were reported and a correlation between kinetic and structural investigations
were made to determine the crystallization mechanism in these materials [1,2]. Therefore, the investigation of crystallization kinetics
is important since it quantifies the effect of the nucleation and growth rate of the resulting crystallites [1]. In this paper, we present
the Calculation of activation energy, frequency factor and Avrami exponent of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys on the
basis of crystallization kinetics using Differential Scanning Calorimetry (DSC).

Specimens of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} ribbons prepared by single roller melt spinning technique under inert
atmosphere were procured from our other researchers. The alloy ribbons were about 1 mm wide and about 30 μm thick. The
amorphous nature of ribbons was confirmed by X-ray diffraction (XRD). The as-quenched samples of Fe_{76}Pr_{4}B_{20} and Fe_{76}Pr_{4}B_{20} ribbons were heated in DSC (DSC-50, Shimadzu, Japan) at four linear heating rates (10, 20, 30 and 40 Kelvin/min) from room
temperature to 1000 K. The DSC scans were recorded by a thermal analyzer interfaced to a computer.

The DSC curves of as-quenched samples of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys at four heating rates of crystallization
are shown in **Figure 1**.

The activation energy for crystallization of an amorphous alloy under a linear heating rate can be estimated using Kissinger’s peak shift method [3], which relates the peak temperature, Tp, with heating rate (β) through the equation.

ln (β/T_{p}^{2})=-(E_{a}/RT_{p}) +ln(k0_{R}/E_{a}) (1)

Where E_{a} is the activation energy for crystallization, k0 the frequency factor which is defined as the number of attempts made
by the nuclei per second to overcome the energy barrier and and R is the universal gas constant. **Figure 2** shows the graph of ln
(β/T_{p}^{2}) vs 1000/Tp of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys, which is a straight line with a slope (–Ea/R) and an intercept
of ln (k0_{R}/E_{a}). The activation energy and the frequency factor k0 for crystallization peak calculated using Kissinger’s peak shift
method for the given samples are given in **Table 1**.

**Table 1:** Composition, Activation Energy, Ea (kJouls/mole) and Frequency factor, ko (sec)^{-1} of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys.

Composition | Activation Energy, Ea (kJouls/mole) | Frequency factor, ko (sec)-1 | ||||
---|---|---|---|---|---|---|

Kissinger’s Method | Augis-Bennet’s Method | Matusita-Sakka’s Method | Average (kJouls/mole) | Kissinger’s Method | Augis-Bennet’s Method | |

Fe_{76}Pr_{4}B_{20} |
372.625 | 409.315 | 387.09 | 389.67 | 4.542 × 10^{19} |
2.37 × 10^{21} |

Fe_{76}Dy_{4}B_{20} |
571.3 | 578.488 | 586.08 | 578.59 | 8.5 × 10^{31} |
2.92 × 10^{33} |

The activation energy for crystallization of an amorphous alloy under a linear heating rate can also be estimated using
Matusita-Sakka’s peak shift method [4], which relates the peak temperature, T_{p}, with heating rate (β) through the equation.

ln (β)=-(E_{a}/RT_{p}) + constant (2)

where E_{a} is the activation energy for crystallization and R is the universal gas constant. **Figure 3** shows the graph of ln (β)
vs 1000/T_{p} of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys, which is a straight line with a slope (–E_{a}/R). The activation energy
calculated using Matusita-Sakka’s peak shift method for the given samples, is given in **Table 1**.

The activation energy for crystallization of an amorphous alloy under a linear heating rate can be estimated using Augis &
Bennett method [5], which relates the peak temperature, T_{p}, with heating rate (β) through the equation.

ln(β/T_{p})=−E_{a}/RT_{p} + ln k_{o} (3)

Where E_{a} is the activation energy for crystallization, R is universal gas constant and k_{0} the frequency factor. **Figure 4** shows the graph of ln (β/T_{p}^{2}) vs. 1000/T_{p} of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys, which is a straight line with a slope (–E_{a}/R) and an intercept of ln k_{0}. The activation energy E_{a} and the frequency factor k_{0} for crystallization peak using Augis & Bennett method for the given samples are also given in **Table 1**.

**Table 1** also gives the average value of activation energy of the samples. Thus, the average activation energy for primary
crystallization of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys using a Kissinger’s Method, Augis-Bennet’s Method and Matusita-
Sakka’s Method is determined as 389.67 kJouls/mole and 578.59 kJouls/mole, respectively. The activation energy increases as
the atomic number increases. It is observed that the activation energies of amorphous alloys calculated by means of the different
theoretical models differ slightly from each other which may be attributed to the different approximations used in the models. The
frequency factor ko of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys using Kissinger method is found to be 4.542 × 10^{19} (sec)^{-1} and
8.5 × 10^{31} (sec)^{-1}, respectively. Also, the frequency factor ko of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys using Augis-Bennet’s
method is found to be 2.37 × 10^{21} (sec)^{-1} and 2.92 × 10^{33} (sec)^{-1}, respectively.

Avrami [6] expresses that x is the volume fraction transformed after time t as:

x=1 - exp (-kt^{n}) (4)

Where n is called “the Avrami ‘n’ or a dimensionless quantity called the kinetics exponent”.

The kinetics exponent n is described by the equation

n=(dx/dt)_{p} RT_{p}^{2} (0.37βE_{a})^{-1} (5)

Values of Avrami exponent, n and the reaction rate constants k can be determined by least square fits of the experimental
data. The Avrami kinetics exponent n was calculated from the above equation developed by Gao and Wang [7]. The crystallization
rates, (dx/dt) versus temperature (T) of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys are presented in **Figure 5**. The maximum crystallization (dx/dt)_{p} for each heating rate gives n, according to the equation n=(dx/dt)_{p} RT_{p}^{2} (0.37βE_{a})^{-1}. Thus, mean values of the kinetics exponent,
_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys are included in **Table 2**. Form **Table 2**, the Avrami exponent (or) mean values of the kinetics exponent,
_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys is found to be 1.67 and 0.831, respectively. The Avrami exponent decreases as the atomic number increases.

**Table 2.** Avrami (or) kinetics exponent (<n>) values calculated from (dx/dt) vs Temp(K) curves.

Composition | 10 K/min | 20 K/min | 30 K/min | 40 K/min | <n> |
---|---|---|---|---|---|

Fe_{76}Pr_{4}B_{20} |
2.59 | 1.52 | 1.16 | 1.42 | 1.67≈2 |

Fe_{76}Dy_{4}B_{20} |
0.6314 | 0.6644 | 0.92 | 1.11 | 0.831≈1 |

The average activation energy (Ea) for primary crystallization of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys using the a
Kissinger’s Method, Augis-Bennet’s Method and Matusita-Sakka’s Method is found to be 389.67 kJouls/mole and 578.59 kJouls/
mole, respectively. The activation energy increases as the atomic number increases. The frequency factor (ko) of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys using Kissinger method is found to be 4.542 × 10^{19} (sec)^{-1} and 8.5 × 10^{31} (sec)^{-1}, respectively.
Also, the frequency factor(ko) of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys using Augis-Bennet’s method is found to be 2.37 ×
10^{21} (sec)^{-1} and 2.92 × 10^{33} (sec)^{-1}, respectively. The Avrami exponent of the kinetics exponent, <n> of amorphous Fe_{76}Pr_{4}B_{20} and Fe_{76}Dy_{4}B_{20} alloys is found to be 1.67 and 0.831, respectively. The Avrami exponent decreases as the atomic number increases.

The authors B. Bhanu Prasad and N. Rajya Lakshmi acknowledge the encouragement given by the management, the Principal, HOD of Sciences & Humanities and the staff of M.V.S.R. Engineering College, Nadergul, Hyderabad.

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