Calculation of Activation Energy, Frequency Factor and Avrami Exponent of Amorphous Fe76Pr4B20 and Fe76Dy4B20 Alloys on the Basis of Crystallization Kinetics

Abstract

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.

Keywords

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

Introduction

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 Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys on the basis of crystallization kinetics using Differential Scanning Calorimetry (DSC).

Experimental Section

Specimens of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ 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 Fe₇₆Pr₄B₂₀ and Fe₇₆Pr₄B₂₀ 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.

Results and Discussion

The DSC curves of as-quenched samples of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys at four heating rates of crystallization are shown in Figure 1.

Figure 1: The DSC curves of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys at four heating rates of crystallization in the temperature range 800 K–940 K (Blue - 10 K/min, Red - 20 K/min, Green - 30 K/min and Violet-40 K/min).

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_p²)=-(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_p²) vs 1000/Tp of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ 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.

Figure 2: ln (β/Tp2) vs. (1000/Tp) of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys.

Table 1: Composition, Activation Energy, Ea (kJouls/mole) and Frequency factor, ko (sec)^-1 of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys.

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 Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ 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.

Figure 3: ln (β) vs. (1000/Tp) of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys.

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 k₀ the frequency factor. Figure 4 shows the graph of ln (β/T_p²) vs. 1000/T_p of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys, which is a straight line with a slope (–E_a/R) and an intercept of ln k₀. The activation energy E_a and the frequency factor k₀ for crystallization peak using Augis & Bennett method for the given samples are also given in Table 1.

Figure 4: ln (β) vs. (1000/Tp) of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys.

Table 1 also gives the average value of activation energy of the samples. Thus, the average activation energy for primary crystallization of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ 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 Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys using Kissinger method is found to be 4.542 × 10¹⁹ (sec)^-1 and 8.5 × 10³¹ (sec)^-1, respectively. Also, the frequency factor ko of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys using Augis-Bennet’s method is found to be 2.37 × 10²¹ (sec)^-1 and 2.92 × 10³³ (sec)^-1, respectively.

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

x=1 - exp (-ktⁿ) (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_p² (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 Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ 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_p² (0.37βE_a)^-1. Thus, mean values of the kinetics exponent, of Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys are included in Table 2. Form Table 2, the Avrami exponent (or) mean values of the kinetics exponent, of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys is found to be 1.67 and 0.831, respectively. The Avrami exponent decreases as the atomic number increases.

Figure 5: Crystallization rate, dx/dt, vs. temperature, T at different heating rates for Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys.

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

Conclusions

The average activation energy (Ea) for primary crystallization of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ 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 Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys using Kissinger method is found to be 4.542 × 10¹⁹ (sec)^-1 and 8.5 × 10³¹ (sec)^-1, respectively. Also, the frequency factor(ko) of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys using Augis-Bennet’s method is found to be 2.37 × 10²¹ (sec)^-1 and 2.92 × 10³³ (sec)^-1, respectively. The Avrami exponent of the kinetics exponent, <n> of amorphous Fe₇₆Pr₄B₂₀ and Fe₇₆Dy₄B₂₀ alloys is found to be 1.67 and 0.831, respectively. The Avrami exponent decreases as the atomic number increases.

Acknowledgements

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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