Coercivity Theory of Permanent Magnet Material NdFeB
1. Nucleation theory
Kronmuller et al. proposed the theory of demagnetization nucleation in the soft magnetic defect region of grain boundaries. The mathematical model expressions they use are:
K1(Z ) = K1-△K/ch2(Z/r0)
In the formula, K1(Z) and K1 represent the anisotropic constants in the defect region and the grain, respectively, ΔK represents a decrease in the anisotropy constant of the defect region, r0 is the thickness of the defect region, and Ζ represents the depth of the surface layer.
The formula determines the defect model with continuous variation of univariate. According to this formula, the nucleation field is determined by applying the principle of total free energy minimum, and the nucleation field is determined to determine the coercive force:
Hc = Hn = 2K1(Z)/Js-2πMs +2K1δB/Msπr0
Where δB is the domain wall thickness in the grain, and Js and Ms represent the magnetic polarization and the saturation magnetization, respectively.
According to the theory, the coercive force of NdFeB magnets is determined by the demagnetization nucleation field of the grain boundary soft magnetic defect region. When the nucleation field is high, the coercive force of the magnet is high, and conversely, the coercive force of the magnet is low.
2. Thermal activation theory
Givord et al. proposed the theory of thermal activation of demagnetization, which advocates the formation and expansion of the magnetization nucleus at the grain boundary activation volume to control the coercivity. The difference from the nucleation theory is that the anisotropy constant at the activation volume is not significantly smaller than the corresponding value inside the hard magnetic grains, and the formation of the magnetization nucleus is due to the influence of the thermal fluctuation. The energy E0 forming the demagnetization field can be expressed as:
E0 = μ0VMSHC +μ0VNeffMS2 +25KT
Where V is the activation volume, Neff is the effective demagnetization factor, K is the anisotropy constant, and T is the temperature. The first term is the external magnetic field energy, the second term is the dipole interaction energy, and the third term is the energy barrier of the thermal fluctuation. E0 should also be equal to the interaction energy between the remagnetized core and other parts of the grain:
E0 = αγV2/3
Where γ is the density of the grain domain wall energy and α is the proportional coefficient. Comparing the above two formulas, the coercive force is:
HC = Hn = αγ/NeffMSV1/3-NeffMs-25KT/μ0MSV
Givord pointed out that the influence of thermal fluctuations leads to the formation of a magnetized nucleus, which is controlled by the formation and expansion of the magnetization nucleus at the grain boundary activation volume.
3, pinning theory
According to the observation of the magnetic domain structure and the measurement of macroscopic magnetic properties, Li D et al. proposed a pinning theory for controlling the coercive force of NdFeB magnets, and believed that the grain boundaries have a strong pinning effect on the domain walls. Hadjipanayis proposed that the thin Nd-rich phase layer at the grain boundary has the function of attracting domain walls, which becomes the pinning site of domain wall motion. Zhou Shouzeng et al. systematically studied that the defects of metal such as grain boundaries, vacancies, and dislocations are strong pinning centers of domain walls, and their existence will limit the displacement of domain walls, thereby increasing the coercive force of magnets.
4, launch field theory
Gao Weiwei and others systematically studied the theory of the launching field of NdFeB magnets in combination with experimental facts. According to this theory, the volume of the magnetized nucleus is small, only has the order of magnitude of the domain wall, and it is necessary to grow the domain and reversibly deform the entire grain from the grain surface to the internal irreversible domain wall. During the growth of the magnetization nucleus, it is necessary to overcome the resistance caused by the change of the density of the domain wall energy, and also provide the energy required for the increase of the volume and surface area of the core. The corresponding critical field H0 and the expansion field Hε are:
H 0 =γ/2Jsr0, Hε=πγ/4J S r0
Where γ is the domain wall energy density inside the grain. The starting field Hs required for the growth (expansion) of the demagnetization nucleus should be equal to the sum of H0 and Hε. Considering the effect of the effective demagnetizing field, the launching field can be expressed as:
Hs = γ/2Js r0(1+2/π)-Neff MS
The coercive force required to completely remagnetize the grains should be determined by the larger of the nucleation field and the launch field. After comparison, the launch field is larger than the nucleation field, so the coercive force of the magnet should be determined by the launch field.