Determine optimal annealing temperature of Fe based nanocrystalline alloys from their melting point

Introduction

In recent decades, widespread attention has been paid to Fe based nanocrystalline alloys in the fields of both physics and material science because of their excellent soft magnetic properties.1^–5 Among these magnetic properties, permeability is one of the most important physical quantities to characterise soft magnetic materials, which are known to depend on their microstructure, especially the grain size.6 Moreover, it is also known that Fe based nanocrystalline alloys have special two-phase structures (amorphous matrix and inserted nanocrystals), generally obtained from their amorphous precursors after annealing. Furthermore, the grain size D greatly depends on the annealing conditions, especially the temperature;7 therefore, it is predicted that the permeability would definitely have a close relationship with the annealing temperature. Generally speaking, the grain size increases with the annealing temperature;8^–10 thus, we can infer from it that the permeability should decrease with increasing annealing temperature, according to the random anisotropy theory.6 In fact, a maximum permeability always appears at a certain annealing temperature,11,12 and a surprising phenomenon that the grain size will first decrease and then increase with increasing annealing temperature was found,13^–18 and it may be very important to the soft magnetic properties of Fe based nanocrystalline alloys. However, these surprising phenomena did not attract enough attention, and little investigation has been conducted to interpret this variation. In addition, it is known that these alloys did not begin to nanocrystallise when the annealing temperature was below the primary crystallisation temperature (T_x1), which led to imperfect soft magnetic properties, and when the annealing temperature is higher than the secondary crystallisation temperature (T_x2), the precipitated impurities further annealed caused the deterioration of the soft magnetic properties. Therefore, when Fe based nanocrystalline alloys are synthesised, the annealing temperature is generally controlled between T_x1 and T_x2, and the optimal annealing temperature (T_opt) is then determined according to the soft magnetic properties, especially permeability.19,20 If the relationship between permeability and annealing temperature can be established and the optimal annealing temperature can be predetermined during this temperature range (T_x1<T_a<T_x2), the experimental procedures will be simplified further.

In order to determine the optimal annealing temperature and explain the experimental phenomena of Fe based nanocrystalline alloys during annealing, a theoretical model concerning the variations of grain size and permeability with annealing temperature for Fe based nanocrystalline alloys was proposed based on the random anisotropy theory21 and the nucleation and growth theory.22

Theoretical model

Amorphous alloys are thermodynamically unstable and have a tendency to transit into crystallites after the annealing treatment. This crystallisation process consists of two elementary steps: nucleation of embryos (crystallisation centres) and the following growth of crystallites from these embryos. According to the nucleation and growth theory,23^–26 the nucleation rate I and the grain growth rate U can be expressed as (1) (2) where N_v is the Avogadro constant, k is Boltzmann's constant, γ is a constant, R is the ideal gas constant, T_a is the annealing temperature, a₀ is the average atomic diameter, η is the viscosity, T_m is the melting point of nanocrystalline soft magnetic alloys, ΔG is the free energy difference between crystalline phases and residual amorphous matrix, E_n is the nucleation activation energy and E_g is the growth activation energy.

In the nanocrystallisation process during annealing, the grain size D of Fe based nanocrystalline alloys was determined by both the nucleation rate I and the growth rate U,22 which can be written as (3) Assuming (4) (5) Then the grain size of Fe based nanocrystalline alloys in the annealing process can be expressed as (6) It was reported that the nucleation activation energy E_n and the growth activation energy E_g are very close to each other when T_a>T_x127 and ΔG>RT_a for these alloys28; then, . Therefore, equation (6) can be further simplified as (7) It can be seen from equation (7) that the grain size mainly depends on the annealing temperature. According to the random anisotropy model,21 the relationship between permeability μ and grain size D can be expressed as (8) where p_u is the material constant, μ₀ is the vacuum magnetic permeability, J_s is the saturation magnetisation, A is the exchange stiffness constant and K₁ is the crystalline anisotropy constant.

By substituting equation (8) with equation (7), the relationship between permeability and annealing temperature can be obtained as (9)

Results and discussion

In the proposed model discussed above, the heating process, cooling process and other factors were not taken into account, so it is applicable only for different annealing temperatures in isochronal annealing. In the practice of nanocrystalline alloy preparation, the annealing temperature was generally controlled as T_x1<T_a<T_x2, and the annealing time is usually 1 h,29 during which the values of α, T_m, J_s, K₁ and A would show very little change.8,21 Therefore, the grain size and permeability of Fe based nanocrystalline alloys are mainly determined by the annealing temperature in these cases.

Relationship between grain size and annealing temperature

It can be obtained from equation (7) that the change of grain size with annealing temperature is (10) From equation (10), it can be drawn that the grain size decreases with increasing annealing temperature below 0·6T_m (∂D/∂T_a<0), while it increases with increasing annealing temperature above 0·6T_m (∂D/∂T_a>0), which means that the grain size has a minimum value when annealed near 0·6 times that of the melting point T_m.

It has been reported that the primary crystallisation temperature T_x1 and the secondary crystallisation temperature T_x2 for Fe_73·5Cu₁Nb₃Si_13·5B₉ alloy are about 760 and 900 K respectively.30,31 Moreover, previous experiments13 also showed that the grain size of this alloy has a clear valley value when annealed for 1 h at a temperature range of T_x1<T_a<T_x2. The grain size first decreased with increasing annealing temperature, but when the annealing temperature reached a minimum of ∼823 K, the grain size then began to increase. This experimental result has been confirmed by transmission electron microscopy.13 Similarly, the investigation on the grain size of Fe_77·5Si_15·5B₇14 and Fe_76·5Si_15·5B₇Cu₁14 also showed the same changing tendency, and the annealing temperatures corresponding to the minimum grain size were 875 and 775 K respectively. The theoretical curves of grain size as a function of annealing temperature were fitted based on equation (7) for Fe_73·5Cu₁Nb₃Si_13·5B₉, Fe_74·5Nb₃Si_13·5B₉ and Fe_76·5Si_15·5B₇Cu₁ alloys during the temperature range of T_x1<T<T_x2, as shown in Figs. 1–3 respectively.

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Figure 1

Comparisons of experimental data13 and calculations of grain size with annealing temperature for Fe_73·5Cu₁Nb₃Si_13·5B₉

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Figure 2

Comparisons of experimental data14 and calculations of grain size with annealing temperature for Fe_77·5Si_15·5B₇

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Figure 3

Comparisons of experimental data14 and calculations of grain size with annealing temperature for Fe_76·5Si_15·5B₇Cu₁

In addition, it was found that the same changing tendency exists for many nanocrystalline alloys, e.g. Fe₇₈B₁₃Si₉,15 Fe_74·5Nb₃Si_13·5B₉,16 Fe_73·5Cu₁Ta₃Si_13·5B₉,17 and Fe_73·5Cu₁Nb₃Si_13·5B₉,18 that is, the grain size first decreases and then increases with annealing temperature, and minimum grain sizes were presented. The theoretical model is in good agreement with the experimental data.

Relationship between permeability and annealing temperature

The change of permeability with annealing temperature for Fe based nanocrystalline alloys during the crystallization process can be obtained from equation (9), which can be expressed as (11) Equation (11) reveals that permeability increases with increasing annealing temperature below about 0·6T_m (∂μ/∂T_a>0) while decreases with increasing annealing temperature above about 0·6T_m (∂μ/∂T_a<0), which means that permeability would have a maximum value when annealed near 0·6 times that of the melting point T_m.

Experiments concerning the permeability and annealing temperature of Fe_73·5Cu₁Nb₃Si_13·5B₉,32 Fe_74·5Nb₃Si_13·5B₉16 and Fe_73·5Cu₁Ta₃Si_13·5B₉33 have been reported previously. Theoretical curves of permeability changing with annealing temperature for these alloys can be fitted using equation (9). By comparing theoretical curves with experimental data, the surprising phenomenon was found that permeability first increases when the annealing temperature is above the primary crystallisation temperature (T_x1), while it began to decrease when the annealing temperature is higher than the optimal annealing temperature. A clear peak value presents in this annealing process, as shown in Figs. 4 6 respectively.

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Figure 4

Comparisons of experimental data32 and calculations of permeability with annealing temperature for Fe_73·5Cu₁Nb₃Si_13·5B₉

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Figure 5

Comparisons of experimental data16 and calculations of permeability with annealing temperature for Fe_74·5Nb₃Si_13·5B₉

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Figure 6

Comparisons of experimental data33 and calculations of permeability with annealing temperature for Fe_73·5Cu₁Ta₃Si_13·5B₉

In addition, based on the experiment data of the famous Fe_73·5Cu₁Nb₃Si_13·5B₉ (Ref. 12), the same changing tendency was also found, that is, that permeability will first increase and then decrease with the annealing temperature. The theoretical model is in excellent agreement with the experimental data during annealing.

Determination of optimal annealing temperature

As discussed above, the minimum grain size corresponds to the maximum permeability present when Fe based nanocrystalline alloys are annealed at a certain temperature during the annealing range T_x1<T_a<T_x2, and the optimal annealing temperature should be T_a≈0·6T_m as analysed in equations (10) and (11), that is, the optimal annealing temperature for Fe based nanocrystalline soft magnetic alloys is ∼0·6 times that of the melting point. Then, theoretical values of the optimal annealing temperature were calculated and subsequently compared with the existing experiments for several Fe based nanocrystalline alloys, which are found in good agreement with each other, with a maximum error of <6·1, as shown in Table 1.

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Table 1

Comparisons of theoretical optimal annealing temperature with experiments for Fe based nanocrystalline alloys

Fe based nanocrystalline alloys	Tm (Exp)/K	Topt (Cal)/K	Topt (Exp)/K	Relative error/
Fe65Co20Nb5B8P2*	1250	750	773	2·98
Fe43Co42Nb5B8P2*	1250	750	798	6·02
Fe73·5Nb3Cu1Si13·5B9†	1420	852	825	3·27
(Fe0·99Mo0·01)78B13Si9‡	1330	798	813	1·85
Fe67Co17Nb7B9§	1310	786	823	4·50

*Taken from Ref. 34.

†Taken from Ref. 35.

‡Taken from Ref. 36.

§Taken from Refs. 37 and 38.

Table 1 shows that permeability first increases and then decreases with the annealing temperature during T_x1<T_a<T_x2, which is mainly due to the fact that the grain size has an inverse changing tendency in this annealing range. Moreover, when the annealing temperature reaches ∼0·6 times that of the melting point, the grain size becomes the smallest, and the permeability becomes the largest.

Conclusions

A theoretical model that related grain size and permeability to annealing temperature during annealing between the primary crystallisation temperature and the secondary crystallisation temperature was proposed, and the optimal annealing temperature was pointed out to be 0·6 times that of the melting point. The physical mechanism concerning the presence of the minimum grain size and the maximum permeability during this annealing range was explained based on this model. The theoretical values are in good agreement with the existing experimental data.