Model Building and Anisotropy of PrFeB Permanent Magnetic Materials

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1. Introduction

HDDR (Hydrogenation, Disproportionation, Desorption, Recombination) process is now well established as an effective process for preparing anisotropic NdFeB magnetic powders [1] [2] [3]. Theoretically, the structure, lattice constant, magnetocrystalline anisotropy constant, exchange integral constant and saturation magnetization of Pr_{2}Fe_{14}B and Nd_{2}Fe_{14}B are very close [4]. Since the intrinsic magnetic properties of Pr_{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 [5] [6] [7]. Han’s [8] [9] investigation shows that as long as disproportionation time is reasonably controlled, the disproportionation products display a rod-like microstructure with self-organized hexagonal PrH_{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 [10] adopted the modified HDDR process to prepare pure ternary anisotropy Pr_{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.

2. Theory Model of Pr_{2}Fe_{14}B-Type 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 Figure 1. represents a single grain, represents the boundary phase. The grain is affected by the exchange coupling interaction between adjacent grains. For a single crystal grain, due to its face center, the rib and the apex angle own to different contact conditions with adjacent grain, therefore, the anisotropy of three regions is also different. The plane-centered region of grains is only affected by the exchange coupling of adjacent single grains, denoted as *N* = 1. The edge regions are affected by adjacent three grains, denoted by *N* = 3. The top Angle regions are affected by adjacent seven grains, denoted by *N* = 7.

Arcas [11] considered that a single crystal of a nano-magnet is directly connected to the surrounding N-grain, and used this expression *K*_{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

Figure 1. Grains stacking schematic diagram of Pr_{2}Fe_{14}B-type magnetic powders, represents a single grain, represents the boundary phase.

[12] used this expression ${K}_{1}\left(r\right)=\{\begin{array}{l}{K}_{1}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le r\le {r}_{0}\\ {K}_{1}\left\{1-\mathrm{exp}\left[-{\left(\frac{2r}{lex}\right)}^{2}\right]\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}_{0}\le r\le \frac{lex}{2}\end{array}$ to

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 [12] used this expression

${K}_{1}\left(r\right)=\{\begin{array}{l}{K}_{1}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le r\le \frac{lex}{2}\\ {K}_{1}\left\{1-\mathrm{exp}\left[-{\left(\frac{r}{{r}_{0}}\right)}^{2}\right]\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{lex}{2}\le r\le {r}_{0}\end{array}$ 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 [9], thus, this paper considered that the change of surface anisotropy of Pr_{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\left(r\right)=\{\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le r\le \frac{d}{2}\\ \frac{{K}_{1}}{{N}^{\frac{1}{2}}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{\left(r-\frac{d}{2}\right)}^{2}}{\left({r}_{0}-\frac{d}{2}\right)\left(lex-d\right)}\right)}^{2}\right]\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{d}{2}<r\le {r}_{0}\\ \frac{{K}_{1}}{{N}^{\frac{1}{2}}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2\left(r-\frac{d}{2}\right)}{lex-d}\right)}^{2}\right]\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}_{0}<r\le \frac{lex}{2}\end{array}$ (1)

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

$K\left(r\right)=\{\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le r\le \frac{d}{2}\\ \frac{{K}_{1}}{{N}^{\frac{1}{2}}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{\left(r-\frac{d}{2}\right)}^{2}}{\left({r}_{0}-\frac{d}{2}\right)\left(lex-d\right)}\right)}^{2}\right]\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{d}{2}<r\le \frac{lex}{2}\\ \frac{{K}_{1}}{{N}^{\frac{1}{2}}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2\left(r-\frac{d}{2}\right)}{{r}_{0}-\frac{d}{2}}\right)}^{2}\right]\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{lex}{2}<r\le {r}_{0}\end{array}$ (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.

3. Anisotropy of Pr_{2}Fe_{14}B Grain

When ${r}_{0}\le \frac{lex}{2}$, the average anisotropy $\langle {K}_{in}\rangle $, $\langle {K}_{p1}\rangle $, $\langle {K}_{p2}\rangle $, $\langle {K}_{p3}\rangle $ of the interior, face center, ridge and corner region of a single grain can be respectively represented as:

$\begin{array}{c}\langle {K}_{in}\rangle =\frac{2}{lex}({\displaystyle {\int}_{d/2}^{{r}_{0}}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\}\text{d}r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle {\int}_{{r}_{0}}^{lex/2}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[1-{\left(\frac{2r}{lex}\right)}^{2}\right]\right\}\text{d}r})\end{array}$

$\begin{array}{c}\langle {K}_{p1}\rangle =\frac{2}{lex}({\displaystyle {\int}_{d/2}^{{r}_{0}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\}\text{d}r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle {\int}_{{r}_{0}}^{lex/2}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[1-{\left(\frac{2r}{lex}\right)}^{2}\right]\right\}\text{d}r})\end{array}$

$\begin{array}{c}\langle {K}_{p2}\rangle =\frac{2}{lex}({\displaystyle {\int}_{d/2}^{{r}_{0}}\frac{{K}_{1}}{{3}^{1/2}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\}\text{d}r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle {\int}_{{r}_{0}}^{lex/2}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[1-{\left(\frac{2r}{lex}\right)}^{2}\right]\right\}\text{d}r})\end{array}$

$\begin{array}{c}\langle {K}_{p3}\rangle =\frac{2}{lex}({\displaystyle {\int}_{d/2}^{{r}_{0}}\frac{{K}_{1}}{{7}^{1/2}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\}\text{d}r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle {\int}_{{r}_{0}}^{lex/2}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[1-{\left(\frac{2r}{lex}\right)}^{2}\right]\right\}\text{d}r})\end{array}$

When ${r}_{0}>\frac{lex}{2}$, the average anisotropy $\langle {K}_{in}\rangle $, $\langle {K}_{p1}\rangle $, $\langle {K}_{p2}\rangle $, $\langle {K}_{p3}\rangle $ of the interior, face center, ridge and corner region of a single grain can be respectively represented as:

$\begin{array}{c}\langle {K}_{in}\rangle =\frac{1}{{r}_{0}}({\displaystyle {\int}_{d/2}^{lex/2}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\}\text{d}r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle {\int}_{lex/2}^{{r}_{0}}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[1-{\left(\frac{2r}{lex}\right)}^{2}\right]\right\}\text{d}r})\end{array}$ (7)

$\begin{array}{c}\langle {K}_{p1}\rangle =\frac{1}{{r}_{0}}({\displaystyle {\int}_{d/2}^{lex/2}\frac{{K}_{1}}{{3}^{1/2}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\}\text{d}r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle {\int}_{lex/2}^{{r}_{0}}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[1-{\left(\frac{2r}{lex}\right)}^{2}\right]\right\}\text{d}r})\end{array}$ (8)

$\begin{array}{l}\langle {K}_{p2}\rangle =\frac{1}{{r}_{0}}({\displaystyle {\int}_{d/2}^{lex/2}\frac{{K}_{1}}{{7}^{1/2}}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\}\text{d}r}\\ +{\displaystyle {\int}_{lex/2}^{{r}_{0}}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[1-{\left(\frac{2r}{lex}\right)}^{2}\right]\right\}\text{d}r})\end{array}$ (9)

$\begin{array}{c}\langle {K}_{p3}\rangle =\frac{1}{{r}_{0}}({\displaystyle {\int}_{d/2}^{lex/2}{K}_{1}\left\{1-\mathrm{exp}\left[-{\left(\frac{2{r}^{2}}{{r}_{0}lex}\right)}^{2}\right]\right\}\text{d}r}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\displaystyle {\int}_{lex/2}^{{r}_{0}}\frac{{K}_{1}}{{N}^{1/2}}\left\{1-\mathrm{exp}\left[1-{\left(\frac{2r}{lex}\right)}^{2}\right]\right\}\text{d}r})\end{array}$ (10)

Boundary defect zone anisotropy of Pr_{2}Fe_{14}B grain
${{K}^{\prime}}_{1}$ can be expressed as:

${{K}^{\prime}}_{1}=\frac{\langle {K}_{p1}\rangle {V}_{1}+\langle {K}_{p2}\rangle {V}_{2}+\langle {K}_{p3}\rangle {V}_{3}}{{V}_{tot}-{V}_{in}}$ (11)

The average anisotropy $\langle K\rangle $ of a single grain can be expressed as:

$\langle K\rangle =\frac{6*\left(\langle {K}_{p1}\rangle {V}_{1}+\langle {K}_{p2}\rangle {V}_{2}+\langle {K}_{p3}\rangle {V}_{3}\right)+{K}_{1}{V}_{in}}{{V}_{tot}}$ (12)

where, ${V}_{tot}={\left(D+d\right)}^{2}.$

If ${r}_{0}\le \frac{lex}{2}$, ${V}_{1}={\left(D+d-lex\right)}^{2}*\frac{lex}{2}$, ${V}_{2}=\left(D+d-lex\right)*le{x}^{2}$, ${V}_{3}=\frac{4}{3}{\left(\frac{lex}{2}\right)}^{2}$, ${V}_{in}={\left(D+d-lex\right)}^{3}.$

If ${r}_{0}>\frac{lex}{2}$, ${V}_{1}={\left(D+d-2{r}_{0}\right)}^{2}*{r}_{0}$, ${V}_{2}=\left(D+d-2{r}_{0}\right)*{r}_{0}^{2}$, ${V}_{3}=\frac{4}{3}{r}_{0}^{3}$, ${V}_{in}={\left(D+d-2{r}_{0}\right)}^{3}.$

${V}_{tot}$ and
${V}_{in}$ 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.

4. Result and Discussion

When the boundary phase thickness *d* is 1 nm and the structure defect thickness *r*_{0} takes different values, Figure 2 shows the dependence of anisotropy
${K}_{p1}\left(r\right)$ of the face center region with the distance *r* to the center of boundary phase, and Figure 3 shows that the dependence of ridge region anisotropy
${K}_{p2}\left(r\right)$ on *r*, the variation of the corner region anisotropy
${K}_{p3}\left(r\right)$ with *r* is shown in Figure 4. When the structure defect thickness *r*_{0} is constant,
${K}_{p1}\left(r\right),{K}_{p2}\left(r\right),{K}_{p3}\left(r\right)$ 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
${K}_{p1}\left(r\right),{K}_{p2}\left(r\right),{K}_{p3}\left(r\right)$ 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
${K}_{p1}\left(r\right),{K}_{p2}\left(r\right),{K}_{p3}\left(r\right)$ with *r* is faster.

When the grain structure defect thickness is 4 nm and the grain boundary phase thickness d takes different values, Figure 5 indicates that the face center region anisotropy
${K}_{p1}\left(r\right)$ varies with *r*. and Figure 6 shows that the dependence of ridge region anisotropy
${K}_{p2}\left(r\right)$ on *r*, the variation of the corner region anisotropy
${K}_{p3}\left(r\right)$ with *r* is shown in Figure 7. The figures show that when d takes different values,
${K}_{p1}\left(r\right),{K}_{p2}\left(r\right),{K}_{p3}\left(r\right)$ all decrease with reducing of r. This indicates that the closer to the center of grain boundary phase, the smaller the anisotropy. The data in the figure further shows that the decrease rate of *K* with *r* in
$0<r<1.85$ is greater than that of *K* with *r* in
$1.85<r<4$, because the anisotropy is influenced by exchange-coupled affect and structure

Figure 2. For the boundary phase thickness *d* of 1nm and the structure defect thickness takes different values, the dependence of anisotropy
${K}_{p1}\left(r\right)$ of the face center region with the distance *r* to the center of grain boundary phase.

Figure 3. For the boundary phase thickness *d* of 1 nm and the structure defect thickness takes different values, the dependence of ridge region anisotropy
${K}_{p2}\left(r\right)$ on *r*.

Figure 4. For the grain boundary phase thickness *d* of 1 nm and the structure defect thickness takes different values, the variation of the corner region anisotropy
${K}_{p3}\left(r\right)$ with *r*.

Figure 5. For the grain structure defect thickness of 4 nm and the grain boundary phase thickness *d* takes different values, the change of the face center region anisotropy
${K}_{p1}\left(r\right)$ with *r*.

defects in
$0<r<1.85$, But in
$1.85<r<4$, the anisotropy is only affected by structural defects. Figure 8 shows the variation of material average anisotropy
$\langle K\rangle $ with structure defect thickness *d*. This figure indicates that for *r*_{0} taking different values,
$\langle K\rangle $ decrease with increasing of *d. *

Figure 6. For the grain structure defect thickness of 4 nm and the grain boundary phase thickness *d* takes different values, the dependence of ridge region anisotropy
${K}_{p2}\left(r\right)$ on *r*.

Figure 7. For the grain structure defect thickness of 4 nm and the grain boundary phase thickness *d* takes different values, the variation of the corner region anisotropy
${K}_{p3}\left(r\right)$ with *r.*

Figure 8. For different values of r_{0}, the variation of average anisotropy
$\langle K\rangle $ of a single grain with structure defect thickness *d*.

5. Conclusion

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.

Acknowledgements

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).

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