1. Introduction
The magnetic shielding device is used to reduce the ambient magnetic interfering signals (including the geomagnetic field and the ambient magnetic field fluctuations) significantly, by which, an extreme weak, high uniform, small time drifts, and low noisy magnetic field could be obtained [
1,
2,
3]. The demands for the magnetic shielding device come from a variety of fields such as fundamental physics [
4,
5], biomedical science [
6,
7], underground survey [
8], space research [
9], and industrial applications [
10]. One typical application of a magnetic shielding device in industry is the high sensitivity magnetic detection based on the effect of spin change relaxation free (SERF). With the advantages of non-radiation, non-contact, high sensitivity, small volume, and low cost [
11,
12], the high sensitivity magnetic detection brings broad application prospect and high commercial value, especially in the areas of magnetocardiogram (MCG) and magnetoencephalography (MEG) [
13]. The magnetic shielding device plays a key role to ensure the accuracy and sensitivity of the magnetic detection.
Magnetic shielding device is usually constructed by the high-permeability materials based on the effect of flux shunting [
14], and high-conductivity materials based on the effect of eddy current [
15]. Among them, the property of the high-permeability materials (a kind of soft magnetic material) always determines the performance of the magnetic shielding device in most applications [
16,
17]. In order to achieve a better magnetic shielding performance, it is important to obtain the complete property of the high-permeability material before the construction of the magnetic shielding device.
The most concerned magnetic properties of the high-permeability material related to magnetic shielding performance are the initial permeability, the coercivity, the maximal permeability, the saturation magnetic induction, and the remanence [
18], which could be obtained by a standard soft magnetic testing instrument such as MATS-3000S. The measuring principle by a standard soft magnetic testing instrument is based on the electromagnetic induction, and the magnetic properties could be obtained by the hysteresis loop [
19]. The test ring needs to be wound evenly by coils to generate the excitation magnetic field and obtain induced voltage for the calculation of magnetic induction intensity [
20], which brings the testing process inconvenience and low efficiency. In some recent studies, the probe and H-coil without winding were used to obtain the magnetic induction intensity and magnetic field intensity, respectively [
17,
21]. However, the structure used to form the closed magnetic loop is still complex and the accuracy in low magnetic field intensity is hard to maintain.
According to the research on the soft magnetic materials, the magnetic properties could be explained by the magnetic domain theory [
22,
23,
24,
25]. The minimum free energy principle (MFE) [
26] based on magnetic domain theory is usually used to calculate the magnetic properties of soft magnetic materials, such as the initial permeability and the coercivity. The initial permeability and the coercivity could be reflected by the anisotropy and the magnetostriction (related to the material composition and texture), the grain structure, the defect, the stress, etc. [
18,
27]. The MFE method is already used to calculate the influence of stress and defects, but rarely used to analyze the pinning effect of the grain boundary. Since the measurement of the microstructure is more direct and efficient, we are motivated to explain the mechanism of microstructure in theory and explore a new test method for the high-permeability material focusing on the microstructure.
In this paper, a test method of high-permeability material’s microstructure including the material composition, the texture, and the grain structure is proposed, and the relationship between the microstructure and the magnetic properties of the high-permeability material is analyzed and calculated through the magnetic domain theory, which is verified by the experiment. The main contributions are listed as follows: (1) a novel test method of the high-permeability material’s microstructure to reflect the high-permeability material’s magnetic properties is proposed for the first time. (2) MFE based on magnetic domain theory to explain the pinning effect of the grain boundary is firstly introduced to analyze the relationship between the microstructure and the magnetic properties. (3) An effective formulation of etchant for permalloy corrosion is obtained. (4) Compared with typical magnetic testing methods, the proposed method of microstructure to obtain the magnetic properties will promote the detection efficiency and reduce the detection complexity.
This paper is organized as follows. The theory reflecting the relationship between the microstructure and the magnetic properties of the high-permeability material is described in
Section 2. The test of magnetic properties and microstructure of the high-permeability material is recorded in
Section 3. The discussion of the test results when compared with the theory is arranged in
Section 4. Through the experiment results, the relationship between the microstructure and the magnetic properties is proved to be clear and credible, which provides a new approach to evaluate the property of the high permeability material.
2. Theory of Microstructure’s Magnetic Properties
The magnetic domain theory is widely applied to explain the magnetic properties’ mechanism of the high-permeability material [
24,
28,
29], which follows the minimum free energy principle. For a ferromagnet with unit volume, the total free energy
could be described as follows:
where
is the external magnetic field energy,
is the magnetoelastic energy and
is the domain wall energy. The most commonly used high-permeability material in the magnetic shielding device is the large-grained material such as permalloy, and the grain size
D follows the relation
, where
is the ferromagnetic exchange length,
A is the exchange energy constant and
is the anisotropy constant. The magnetic properties of large-grained material could be explained by the domain wall extension theory. During the process of domain wall expansion, the change of domain distribution follows the minimum free energy principle:
Ignoring the change of magnetoelastic energy near the domain wall, the magnetization equation could be written as:
For the typical 180° domain wall, the variation of the external magnetic field driving energy
and the domain wall energy
could be expressed as:
where
represents the permeability of vacuum,
is the saturated magnetization and
H is the external magnetic field intensity,
is the domain wall energy and
x is the displacement of domain wall expansion in one dimension. Then, the displacement equilibrium equation for the 180° domain wall could be obtained from Equation (
2) as follows:
The differential of the 180° domain wall energy
could be expressed as [
29]:
where
is the rotation angle of the domain and
The typical parameters to evaluate the magnetic properties of the magnetic shielding device include the coercivity and the initial permeability. The coercivity is produced when the variation of the domain wall energy exceeds the peak value, which makes the magnetization process irreversible. The coercivity in the domain wall extension model
from Equation (
6) could be expressed as follows:
For material with large grains, the magnetization process is determined by domain wall pinning at the grain boundaries, so the coercivity for material with large grains
is obtained as follows:
where
D is the grain size and
is the correction factor for the domain size and the rotation angle of the adjacent magnetic domains.
The initial permeability is related to the magnetic susceptibility of the material. When the external magnetic field intensity varies
, domain wall moves a distance of
. According to Equation (
6), the relationship of
and
in the 180° domain wall could be written as:
where
is the initial position of the domain wall, and the magnetization varies along with the domain wall displacement as:
where
represents the variation of the magnetization and
S is the area of the domain wall. Then, the initial susceptibility
could be obtained as:
It is not easy to determine the initial position in a domain, so the average variation rate is used as the approximate substitution as follows:
where
is the width of the domain. In the unit volume, the area of domain wall could be expressed as
. According to Equation (
13), the initial permeability of the domain
could be written as:
Considering the material with large grains pinning at the grain boundaries, the initial permeability
could be obtained as follows:
where
is the correction factor for the domain size and the rotation angle of the adjacent magnetic domains.
The relationship between the magnetic properties and the microstructure is established according to Equations (
10) and (
16), where the parameters
A,
, and
are related to the material composition and the texture, and
D is related to the grain structure.
4. Discussion
The test method of permalloy including the material composition, the texture, and the grain structure is proposed are described in the previous section of this paper. According to the test result of different samples, the grain structures are closely related to the magnetic properties. The theory shows that the initial permeability is proportional to the grain size (
) and the coercivity is inversely proportional to the grain size (
) according to Equations (
10) and (
16). To explore the relation between the theory and the experiments, the D fitting curve and 1/D fitting curve are introduced for comparison, which are shown in
Figure 10. Considering the saturation magnetic induction for each sample is different, which is proportional to the saturated magnetization (
), the saturated magnetization correction for the fitting curve is also drawn in
Figure 10.
It could be seen that the fitting curve could clearly describe the variation trend of the grain size with the initial permeability and the coercivity from test results, which proves the consistency between theory and experiment. Meanwhile, after saturated magnetization correction, the fitting points are closer to the test data, which verifies the correctness of Equations (
10) and (
16). Other magnetic properties such as the saturation magnetic induction, the maximum permeability and the remanence are related to the inherent attributes of the material, which are comprehensive and hard to observe. So the change patterns of the other magnetic properties are difficult to obtain, which are not discussed in the manuscript. Since the initial permeability and the coercivity could directly reflect the performance of the permalloy, which could be observed directly from the grain size, the method of evaluating the property of the high-permeability material has been broadened. Meanwhile, the procedure of polishing and chemical corrosion could be easily completed by automated devices, which brings the promotion of the efficiency comparing with the soft magnetic testing instruments.