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This paper considers that the crystal grains of HDDR Pr2Fe14B permanent magnetic material are cubic, the size is 0.3 μm, and the crystal grains are in simple cubic accumulation. It is considered that there are boundary phases between grains. It is assumed that the boundary phases are non-magnetic phases with the thickness of d, and evenly distributed between grains. The anisotropy expression of single grain boundary is given considering structure defect and intergranular exchange coupling interaction. Based on micro-magnetic simulation calculation, the variation of the average anisotropy of a single grain with the structural defects and boundary phases was calculated. The results show that when the thickness of structural defects is constant, the average anisotropy of a single grain decreases with increasing of grain boundary phase thickness, and while the thickness of grain boundary phase is constant, it also decreases with increasing of structural defect thickness.

HDDR (Hydrogenation, Disproportionation, Desorption, Recombination) process is now well established as an effective process for preparing anisotropic NdFeB magnetic powders [_{2}Fe_{14}B and Nd_{2}Fe_{14}B are very close [_{2}Fe_{14}B-type alloy are comparable to those of Nd_{2}Fe_{14}B-type alloy, recently researchers had attempted the HDDR process to prepare Pr_{2}Fe_{14}B-type magnetic powders with additives such as Co, Zr, Ga and Nb [_{2} nanorods embedded in Fe matrix, and the highly ordered rod-like structure is responsible for the high degree of texture orientation of HDDR Pr_{13}Fe_{79.4}B_{7}Nb_{0.3}Ga_{0.3} magnetic powders. Zhong [_{2}Fe_{14}B-type magnetic powders. At present, there is much experimental research work on PrFeB permanent magnetic materials, and no theoretical research work has been seen. This paper attempted to establish the anisotropy theoretical model of PrFeB permanent magnet material, and further investigated the anisotropy variation of magnetic powders with structural defects and grain boundary phase. It hopes that these results of this paper can provide theoretical guidance for the experimental preparation of highly anisotropic magnetic powders.

Assumed that the HDDR Pr_{2}Fe_{14}B grain is a cube with size of 0.3 μm, and these grains are stacked in simple cubic form, as shown in

Arcas [_{1}(r) = K_{1}/N^{1/2} to describe anisotropy variation of grain surface. Based on the special microstructure of HDDR Nd_{2}Fe_{14}B grains, the grain surface is affected by both exchange coupling interaction and structural defects. When the grain surface structure defect thickness r_{0} is less than the exchange coupling interaction length lex/2, Liu

[

indicate the surface anisotropy change of Nd_{2}Fe_{14}B grain. When the surface structure defect thickness r_{0} of the grain is larger than the exchange coupling interaction length lex/2, Liu [

K 1 ( r ) = { K 1 { 1 − exp [ − ( 2 r 2 r 0 l e x ) 2 ] } , 0 ≤ r ≤ l e x 2 K 1 { 1 − exp [ − ( r r 0 ) 2 ] } , l e x 2 ≤ r ≤ r 0 to represent the surface anisotropy change of Nd_{2}Fe_{14}B grain.

Not only the intrinsic magnetic properties of Pr_{2}Fe_{14}B-type alloy are comparable to those of Nd_{2}Fe_{14}B-type alloy, but also the microstructure of the Pr_{2}Fe_{14}B magnetic powder grains is similar to that of the Nd_{2}Fe_{14}B magnetic powder grains [_{2}Fe_{14}B grains is similar to that of surface anisotropy of Nd_{2}Fe_{14}B grains. Since the grains of the Pr_{2}Fe_{14}B magnetic powder are stacked in a simple cubic structure, The face of a single grain directly contacts with one grain (record as N = 1), the ridge of a single grain directly contacts with three grains (record as N = 3), and corner regions of a single grain directly contact with seven grains (record as N = 7), the anisotropy change of the three regions is related to N, thus, the surface anisotropy K(r) of Pr_{2}Fe_{14}B grain can be rewritten as:

When the structure defect thickness r_{0} of grain surface is smaller than the exchange coupling interaction length lex/2

K ( r ) = { 0 0 ≤ r ≤ d 2 K 1 N 1 2 { 1 − exp [ − ( 2 ( r − d 2 ) 2 ( r 0 − d 2 ) ( l e x − d ) ) 2 ] } d 2 < r ≤ r 0 K 1 N 1 2 { 1 − exp [ − ( 2 ( r − d 2 ) l e x − d ) 2 ] } r 0 < r ≤ l e x 2 (1)

When the structure defect thickness r_{0} of grain surface is larger than the exchange coupling interaction length lex/2

K ( r ) = { 0 0 ≤ r ≤ d 2 K 1 N 1 2 { 1 − exp [ − ( 2 ( r − d 2 ) 2 ( r 0 − d 2 ) ( l e x − d ) ) 2 ] } d 2 < r ≤ l e x 2 K 1 N 1 2 { 1 − exp [ − ( 2 ( r − d 2 ) r 0 − d 2 ) 2 ] } l e x 2 < r ≤ r 0 (2)

where K_{1} is the normal magnetocrystalline anisotropy constant, r_{0} is the structure defect thickness of grain surface, r is the distance to the grain intergranular center, lex is the exchange coupling length between grains, d is the boundary phase thickness.

When r 0 ≤ l e x 2 , the average anisotropy 〈 K i n 〉 , 〈 K p 1 〉 , 〈 K p 2 〉 , 〈 K p 3 〉 of the interior, face center, ridge and corner region of a single grain can be respectively represented as:

〈 K i n 〉 = 2 l e x ( ∫ d / 2 r 0 K 1 N 1 / 2 { 1 − exp [ − ( 2 r 2 r 0 l e x ) 2 ] } d r + ∫ r 0 l e x / 2 K 1 N 1 / 2 { 1 − exp [ 1 − ( 2 r l e x ) 2 ] } d r )

〈 K p 1 〉 = 2 l e x ( ∫ d / 2 r 0 { 1 − exp [ − ( 2 r 2 r 0 l e x ) 2 ] } d r + ∫ r 0 l e x / 2 K 1 N 1 / 2 { 1 − exp [ 1 − ( 2 r l e x ) 2 ] } d r )

〈 K p 2 〉 = 2 l e x ( ∫ d / 2 r 0 K 1 3 1 / 2 { 1 − exp [ − ( 2 r 2 r 0 l e x ) 2 ] } d r + ∫ r 0 l e x / 2 K 1 N 1 / 2 { 1 − exp [ 1 − ( 2 r l e x ) 2 ] } d r )

〈 K p 3 〉 = 2 l e x ( ∫ d / 2 r 0 K 1 7 1 / 2 { 1 − exp [ − ( 2 r 2 r 0 l e x ) 2 ] } d r + ∫ r 0 l e x / 2 K 1 N 1 / 2 { 1 − exp [ 1 − ( 2 r l e x ) 2 ] } d r )

When r 0 > l e x 2 , the average anisotropy 〈 K i n 〉 , 〈 K p 1 〉 , 〈 K p 2 〉 , 〈 K p 3 〉 of the interior, face center, ridge and corner region of a single grain can be respectively represented as:

〈 K i n 〉 = 1 r 0 ( ∫ d / 2 l e x / 2 K 1 N 1 / 2 { 1 − exp [ − ( 2 r 2 r 0 l e x ) 2 ] } d r + ∫ l e x / 2 r 0 K 1 N 1 / 2 { 1 − exp [ 1 − ( 2 r l e x ) 2 ] } d r ) (7)

〈 K p 1 〉 = 1 r 0 ( ∫ d / 2 l e x / 2 K 1 3 1 / 2 { 1 − exp [ − ( 2 r 2 r 0 l e x ) 2 ] } d r + ∫ l e x / 2 r 0 K 1 N 1 / 2 { 1 − exp [ 1 − ( 2 r l e x ) 2 ] } d r ) (8)

〈 K p 2 〉 = 1 r 0 ( ∫ d / 2 l e x / 2 K 1 7 1 / 2 { 1 − exp [ − ( 2 r 2 r 0 l e x ) 2 ] } d r + ∫ l e x / 2 r 0 K 1 N 1 / 2 { 1 − exp [ 1 − ( 2 r l e x ) 2 ] } d r ) (9)

〈 K p 3 〉 = 1 r 0 ( ∫ d / 2 l e x / 2 K 1 { 1 − exp [ − ( 2 r 2 r 0 l e x ) 2 ] } d r + ∫ l e x / 2 r 0 K 1 N 1 / 2 { 1 − exp [ 1 − ( 2 r l e x ) 2 ] } d r ) (10)

Boundary defect zone anisotropy of Pr_{2}Fe_{14}B grain K ′ 1 can be expressed as:

K ′ 1 = 〈 K p 1 〉 V 1 + 〈 K p 2 〉 V 2 + 〈 K p 3 〉 V 3 V t o t − V i n (11)

The average anisotropy 〈 K 〉 of a single grain can be expressed as:

〈 K 〉 = 6 * ( 〈 K p 1 〉 V 1 + 〈 K p 2 〉 V 2 + 〈 K p 3 〉 V 3 ) + K 1 V i n V t o t (12)

where, V t o t = ( D + d ) 2 .

If r 0 ≤ l e x 2 , V 1 = ( D + d − l e x ) 2 * l e x 2 , V 2 = ( D + d − l e x ) * l e x 2 , V 3 = 4 3 ( l e x 2 ) 2 , V i n = ( D + d − l e x ) 3 .

If r 0 > l e x 2 , V 1 = ( D + d − 2 r 0 ) 2 * r 0 , V 2 = ( D + d − 2 r 0 ) * r 0 2 , V 3 = 4 3 r 0 3 , V i n = ( D + d − 2 r 0 ) 3 .

V t o t and V i n indicate the volume of a single grain and that of a grain not affected by structural defects and exchange coupling effect, respectively. V 1 , V 2 and V 3 represents the volume of the face center, ridge and corner region of a single grain, respectively. The intrinsic magnetic parameter of Pr_{2}Fe_{14}B is: K_{1} = 5.6 MJ/m^{3}, A = 7.7 × 10^{−12} J/m, δ_{B} = 3.7 nm, Grain size D = 0.3 μm, N_{eff} = 0.6, J_{s} = 1.56 T, lex = 3.7 nm.

When the boundary phase thickness d is 1 nm and the structure defect thickness r_{0} takes different values, _{0} is constant, K p 1 ( r ) , K p 2 ( r ) , K p 3 ( r ) all increase with increasing of r. This illustrates that the closer to the grain center, the bigger anisotropy of the grain face center region, the ridge region and the corner region. It also shows that with decreasing of r_{0}, the faster decrease rate of K p 1 ( r ) , K p 2 ( r ) , K p 3 ( r ) with r, this belongs to with decrease of r_{0}, the change range of anisotropy from K_{1} to zero is narrowing, so the variation rate of K p 1 ( r ) , K p 2 ( r ) , K p 3 ( r ) with r is faster.

When the grain structure defect thickness is 4 nm and the grain boundary phase thickness d takes different values,

defects in 0 < r < 1.85 , But in 1.85 < r < 4 , the anisotropy is only affected by structural defects. _{0} taking different values, 〈 K 〉 decrease with increasing of d.

This paper investigates the effects of exchange coupling interactions and structural defects on the anisotropy of a single grain. The results show that both structure defects and exchange coupling interactions affect the anisotropy of single grains. When the thickness of structural defects is constant, the average anisotropy of a single grain decreases with increasing of grain boundary phase thickness, and while the thickness of grain boundary phase is constant, it also decreases with increasing of structure defect thickness.

The work is supported by the National Natural Science Foundation of China (Grant No. 51602376, 51602121), Guangdong Nature Science Foundation (Grant No. 2017A030310665), Natural Science Foundation of Huizhou College (Grant No. 2015167, hzuxl201626).

The authors declare no conflicts of interest regarding the publication of this paper.

Liu, M., Cai, X.H., Gong, W.P., Li, Y.J. and Cheng, L.X. (2020) Model Building and Anisotropy of PrFeB Permanent Magnetic Materials. Materials Sciences and Applications, 11, 757-766. https://doi.org/10.4236/msa.2020.1111051