Modeling of magnetic Barkhausen noise for layered ferromagnetic materials based on extended Ising model with double Boltzmann function

Abstract

Ising model is a promising tool for predicting magnetic Barkhausen noise (MBN) signal on electromagnetic nondestructive evaluation (ENDE). However, theoretical prediction of MBN of composite ferromagnetic materials using Ising model is seldom reported. In this study, we incorporate double Boltzmann transition function with the exchange coupling efficiency (J) of Ising model to achieve simulation of MBN in layered materials. The transition behavior of J was assumed to represent the changes in magnetic properties from hard layer to soft layer. Monte Carlo algorithm is used to solve the Ising model to obtain the MBN of layered material. The influence of the volume fraction (p) of hard layer in the double Boltzmann transition function on the shape of MBN profiles was investigated through both simulation and experiments. In the experiments, two-layer ferromagnetic materials was laminated using SAE 1065 carbon steel films of different thickness and a base strip of SAE 1045 carbon steel with a thickness of 20 mm. Both simulation and experimental results show that the MBN peak of soft layer gradually descends with the increase in thickness of SAE 1065 (or the volume fraction of hard layer). The second-order Gaussian function was used to fit the MBN envelop for extract the peaks of hard and soft layers from both experimental and simulation results. The ratio of the MBN peak of hard layer to soft layer demonstrates monotonously increasing trend as the value of p increases. Therefore, ratio of the MBN peak can act as good indicator for qualitative characterization of the changes of thickness or volume fraction of hard layer in two-layer ferromagnetic materials.

Keywords

magnetic Barkhausen noise Ising model layered ferromagnetic materials Monte Carlo algorithm

1. Introduction

Surface-hardened or ferromagnetic material-coated steels are often used in parts manufacturing. Layered features in the parts’ surface lead to complexity in microstructures and in-depth profile of element content, both of which are related with the mechanical properties of the surface material [1]. The evaluation of the micro structural changes in the surface material is of importance to the ferromagnetic parts for quality assurance. Nondestructive testing (NDT) techniques [2–5] have been reported to measure the depth of the surface-hardened layer or the thickness of the coating, owing to the different physical properties between the surface and base material.

Measurement of magnetic Barkhausen noise (MBN), which originates from the magnetic domain motion under external magnetic field, is an attractive option for layered material evaluation [6]. Numerous studies had shown that the features of the MBN are very sensitive to the change of case depth or the volume ratio of surface layer to the base material [7-9]. To clarify the correlation between the MBN and the thickness of the case depth or coating layer, experimental investigations are widely performed in current. Vashista et al. [10] experimentally observed the two-peak MBN profile obtained from isothermally annealed 18CrNiMo5 steel and 42CrMo4 steel, they found that the peak in lower field decreases with the decrease in volume fraction of ferrite and the peak in higher field increases with the increase in volume fraction of pearlite. Kleber et al. [11] assessed the change in volume fraction of individual phase in martensite-ferrite steels by means of MBN measurements. As the annealing temperature increases, the volume fraction of martensite phase decreases while the volume fraction of ferrite increases. The amplitudes of the two peaks of MBN profile were correlated with the annealing temperature as well as the volume fraction of individual phase. Lasaosa et al. [12] utilize the features of two-peak MBN profile for characterizing the case depth of induction hardened ball screw shafts. Their reported results show that the layer thickness between 150 and 2500 µm can be characterized by the MBN peak in lower field. To reveal the connections between the MBN profile and the compositions of multi-phase or multi-layer materials, theoretical or numerical models for predicting MBN profile in such materials are needed.

Ising model can be studied in microscopic level to perform MBN and hysteresis behavior simulations. Yamaguchi et al. [13] first combined Ising model with Monte Carlo algorithm to successfully do the MBN simulation. Thereafter, Ising model was incorporated with lattice defects, micromagnetic clusters and local dislocation [14–16] to predict the effect of microstructures on the MBN features. So far, the prediction of MBN in layered ferromagnetic materials is a challenge because the interactions between contiguous layers are quite complicated and are difficult to simulate using the classic Ising model.

Inside the investigated spin array of Ising model, exchange coupling efficiency (J) is used to characterize the interactions among the spins. The value of J is usually assigned as a constant being independent with the varying externally applied magnetic field. Such initialization of the exchange coupling efficiency limits the solution of the Ising model to predict the magnetic transitional behavior in layered ferromagnetic materials.

In this study, a double-Boltzmann function is first introduced to illustrate the dependency of the exchange coupling efficiency of Ising model on the applied magnetic field. Along with the ascending branch of the applied triangular magnetic field, the value of J continuously changes from J ₁ (hard component) to J ₂ (soft component) to simulate the magnetic transitional behavior between contiguous layers. Then the extended Ising model with double-Boltzmann function is successfully applied for MBN prediction in layered ferromagnetic materials. Experiments are performed on two-layer specimen composed of thin film of SAE1065 (hard component) carbon steel and base strip of SAE1045 (soft component) carbon steel. To draw the two-peak feature of MBN profile and extract the peak amplitudes, second-order Gaussian function is employed to fit the MBN profiles obtained from both experiments and simulations. The correlation between the two-peak feature of MBN profile and the volume fraction (p) of hard layer was investigated. Both the experimental and simulation results show that the ratio of peak amplitude demonstrates monotonously change trend as the value of p increases.

2. Experiments

To experimentally observe the MBN profiles of layered ferromagnetic materials, MBN measurements were performed in two-layer specimens of carbon steels. The top layer of the specimens is hard thin film of SAE1065 steel, and the bottom layer is soft strip of SAE1045 steel. The length and width of the SAE 1065 film were 125 mm and 25 mm, respectively. The dimension of bottom strip of SAE 1045 steel was 200 × 50 × 20 mm. A total of five specimens are prepared by alternatively replacing the top layer by SAE1065 film with different thickness ranging from 0.1 mm to 0.3 mm with an interval of 0.05 mm.

A self-developed sensor was used for MBN measurements. The schematic diagram of sensor and the tested specimen were shown in Fig. 1. An U-shape Fe-Si yoke wound by 350 turn excitation coils act as a magnetizer to produce magnetic field of 5Hz for specimen magnetization. A cylindrical inductive coil with a height of 9 mm and an outer diameter of 5.4 mm was placed in the middle of the yoke and near the specimen surface to measure the MBN signal. The inductive MBN coil is made by winding 200 turns of enameled copper wire with a diameter of 0.15 mm onto a cylindrical ferrite core with a diameter of 2 mm and a height of 9 mm. A SS39E Hall sensor was placed nearby the inductive MBN coil to measure the tangential magnetic field.

Fig. 1.

Schematic diagram of MBN sensor and test sample for (a) SAE 1065 carbon steel film, (b) SAE 1045 carbon steel strip and (c) layered SAE 1065 and 1045 steel.

Fig. 2.

Measured MBN signals of (a) SAE 1065 steel film, (b) SAE 1045 steel strip and (c) layered SAE 1065 and 1045 steel.

Fig. 3.

The transition behavior of exchange coupling efficiency under different cases of (a) standard deviation and (b) exchange field.

The raw signal output by the MBN coil was band-passed filtered (10--100 kHz) and amplified by 60 times in conditioning circuit before data acquisition. Finally, the acquired MBN signal was filtered by a digital four-order Butterworth band-passed filter (30 $\sim$ 70 kHz). Figure 2 show the MBN signals obtained from single layer of SAE 1065 steel, single layer of SAE 1045 steel strip and two-layer material. The position of the MBN envelop is related with the coercive force of the materials. As compared with the results of SAE1065 steel, the MBN peak position of SAE1045 steel is in a lower magnetic field, indicating that the SAE1045 steel is magnetic softer than SAE1065 steel. As shown in Fig. 2(c), the MBN signal obtained from two-layer material clearly demonstrates two peaks and the positions of the two peaks agree with the results of single hard and soft layer.

3. Theory

3.1. Ising model extended by double Boltzmann transition function

The spin Hamiltonian of Ising model can be written as [15,17]: $\begin{eqnarray}\displaystyle E=-\mathop{\sum }_{\langle i,j\rangle }JS_{i}S_{j}-K\mathop{\sum }_{i=1}^{n}S_{i}^{2}+\mathop{\sum }_{i>j}D\left({\displaystyle \frac{S_{i}S_{ji}}{r_{ij}^{3}}}-3{\displaystyle \frac{(S_{i}r_{ij})(S_{j}r_{ij})}{r_{ij}^{5}}}\right)+H\mathop{\sum }_{i=1}^{n}S_{i} & & \displaystyle\end{eqnarray}$ (1) where S _i, S _j denote the spin state of ith and jth cell, symbol 〈i, j〉 denotes two adjacent spins. The parameters of J, K, D and H stand for exchange coupling constant, magnetic anisotropic constant, magnetic dipole interaction constant, and applied magnetic field. n is the number of spin cells. Isotropy was supposed for the spin system. Magnetic dipole interaction was neglected for simplicity and reduction of calculations. In numerical simulation, S _i (or S _j) takes the value +1 for spins up and −1 for spins down. For simplification, spins only interact with neighboring spins through the exchange coupling effect. In addition, a periodic boundary was used to simulate an infinite spin array. In this study, the simulation was performed with a two-dimensional clusters, which was of 20 × 20 spins.

The exchange coupling efficiency J is the key parameter in Eq. (1) to determine the magnetic behaviors of the spin system. Simulated MBN as a function of exchange coupling constant J was illustrated in Ref. [18]. The value of parameter of J was usually selected as a constant and does not change with the applied magnetic field. For two-layer materials, each layer should be assigned with a specific value of parameter of J and transition characteristics between the two layers should be considered in the model to predict the magnetic behavior of the entire two-layer materials. Here, it is assumed that the magnetic behavior of two-layer material is determined by the transition from the properties of hard layer to the properties of soft layer as the applied magnetic field varies. To describe the transition characteristics between the two layers, double-Boltzmann function is introduced to modify the expression of the exchange coupling efficiency J, $\begin{eqnarray}\displaystyle J=J_{1}+(J_{2}-J_{1})\left[{\displaystyle \frac{p}{1+\text{e}^{\frac{H-H_{ex1}}{-k_{1}}}}}+{\displaystyle \frac{1-p}{1+\text{e}^{\frac{H-H_{ex2}}{-k_{2}}}}}\right] & & \displaystyle\end{eqnarray}$ (2) where J ₁ and J ₂ are the exchange coupling constant about the hard and soft component. p is the volume fraction of the hard component. k ₁ and k ₂ are the standard deviation of the field, which determine the transition behavior between the two layers. H _ex1 and H _ex2 are the exchange fields at which the transition between hard and soft layer occurs. The parameters of H _ex1 and H _ex2 have opposite sign but same absolute value, which is equal to the mean value of coercive forces of soft and hard layer materials. The setting principle for the parameters of H _ex1 and H _ex2 is the same as that has been reported in [19] and [20]. Figure 3 shows the dependency of exchange coupling efficiency (J), which is modified by double Boltzmann transition function, on the applied magnetic field under different cases. As sketched in Fig. 3a, selection of large values of k ₁ and k ₂ can achieve smooth transition between the hard and soft layer. Through increasing the absolute values of H _ex1 and H _ex2, the width of the platform for transition will be enlarged.

By substituting Eq. (2) into Eq. (1), an extended Ising model is derived. The extended model can be used to investigate the effect of volume fraction (p) on the MBN signal of two-layer ferromagnetic materials.

3.2. MBN simulation

Statistical Monte Carlo algorithm was employed to deal with the Hamiltonian equation of the proposed Ising model [21]. The general procedure of the Monte Carlo algorithm includes the following six steps

Step 1: To set the initial spin state x.

Step 2: To randomly flip a spin in the spin array to produce a new state x ^′.

Step 3: To calculate the change in the Hamiltonian value caused by the spin flip, δ(E) = E (x ^′) − E (x).

Step 4: If the Hamiltonian value of the system decreases, δ(E) < 0, the new state is accepted.

Step 5: If δ(E) ≥ 0, calculate the change with the probability $Z=\textit{e}^{-\frac{{\delta}(E)}{KT}}$ , where K is the Boltzmann constant and T is the Kelvin temperature. Then, a random number P between 0 and 1 is generated. If P < Z, the new state is accepted, else if P ≥ Z, the original state is retained.

Step 6: Return to Step 2 until the spin array gets the previously set Monte Carlo steps (MCS).

Each MCS indicates one cycle of the Monte Carlo algorithm for determining whether the spin flips or not. The Hamiltonian value of the system was mainly affected by the mechanism of spin flip. An example of spin state with increasing MCS was shown in Fig. 4. Under the condition of MCS = 10000, magnetic domain (white area) gradually formed and the Hamiltonian value of the system tends to be as low as possible.

Fig. 4.

Spin state under different MCS.

The arithmetic mean of the all the spins’ state is calculated as the spins magnetization M (t), t denote time. The profile of the Barkhausen noise is evaluated using the differential of the spins magnetization, V _BN = dM(t)∕dt. The extended Ising model is alternatively applied to simulate the MBN in hard layer, soft layer and two-layer materials. In the simulations, the parameters for hard and soft component are respectively selected as T = 1.25, J = 1.2 and T = 1.25, J = 0.8. Accordingly, the values of parameters for two-layer material are: T = 1.25, p = 0.2, J ₁= 1.2, J ₂= 0.8, k ₁= 0.01, k ₂= 0.01, H _ex1= $-$ 0.2 and H _ex2= 0.2. The simulated time-domain MBN signal are shown in Fig. 5. The MBN signal of two-layer material (Fig. 5c) can be regarded as the superposed results of the MBN of hard (Fig. 5a) and soft layer (Fig. 5b).

Fig. 5.

Simulation results of time-domain MBN signals obtained from (a) single hard layer and (b) single soft layer and (c) two-layer material.

Fig. 6.

MBN butterfly profiles of two-layer ferromagnetic material when the thickness of SAE 1065 steel film varies.

4. Results and discussion

To calculate the envelop of MBN signal, moving average operation was applied to the time-domain signals obtained from both experiments and simulations. For experimentally measured MBN signal, the length of sliding window was selected as 2000 data points. Figure 6 show the MBN butterfly profiles, which show the dependency of MBN envelop on the tangential magnetic field, of two-layer ferromagnetic materials when the thickness of SAE 1065 steel film changes. The experimentally measured MBN butterfly profile demonstrates two peaks. As compared with the results in Fig. 2, one can deduce that the peak in high magnetic field represents the SAE 1065 steel films and the peak in low magnetic field indicates the SAE 1045 steel strip. The available detection depth of MBN signal is limited to several hundreds of microns. When the thickness of the SAE 1065 steel film increases from 0.1 mm to 0.3 mm, the amplitude of the peak in high magnetic field increases and the peak in low magnetic field tends to be submerged in the profile.

It is assumed that the available detection depth of MBN signal is fixed as 1 mm. When the thickness of SAE 1065 steel film in the two-layer materials increases from 0.1 mm to 0.3 mm, the volume fraction ( $p$ ) of the hard layer also increases from 0.1 to 0.3. Therefore, in the simulation the volume fraction ( $p$ ) of the hard layer was alternatively selected in the range from 0.1 to 0.3 with an interval of 0.05. The values of the standard deviation k ₁ and k ₂ in Eq. (2) were defined as infinitesimal value to simulate the step changed transition behavior, the parameters of k ₁ and k ₂ were set as 0.01 in this simulation. The values of exchange field H _ex1 and H _ex2 were estimated using the experimentally measured MBN peak positions, which are related with the coercive forces of hard and soft layers.The interval of exchange coupling efficiency J was set in the range from 0.5 to 1.5 after a lot trials of MBN simulations. To satisfy this condition, the exchange coupling efficiency of hard component (J ₁) and soft component (J ₂) were assumed to be functions of volume fraction of hard component. The expressions of J ₁ and J ₂ are: $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}J_{1}=p+0.6\\ J_{2}=-p+1.4\end{array} & & \displaystyle\end{eqnarray}$ (3)For the simulation MBN signal, a sliding window with a length of 10 data points was used for calculating the moving root mean square (RMS) of the signal and such operation was repeated 30 times to obtain the smooth envelope of MBN signal. Figure 7 shows the simulation results of MBN butterfly profiles. With the increase of volume fraction $p$ of hard component, the peak amplitude of hard component gradually increases while downward trend can be observed for the peak amplitude of soft component. Figure 8 shows the hysteresis curves corresponding to MBN simulations. It can be seen from Fig. 8 that as the value of p increases the coercivity of the two-layer material increases while the distortion of hysteresis curve in high magnetic field weakens.

Fig. 7.

Simulated MBN butterfly profiles of two-layer material when the volume fraction of hard component varies.

Fig. 8.

Simulated hysteresis curves of two-layer material when the volume fraction of hard component varies.

Fig. 9.

Double-Gaussian function fitted results for measured MBN profile of two-layer material.

To draw the two-peak feature of MBN profile and extract the peak amplitude of hard and soft layer, second-order Gaussian function is employed to fit the MBN profiles obtained from both experiments and simulations. The second-order Gaussian function is: $\begin{eqnarray}\displaystyle y=y_{0}+A_{1}\text{e}^{-\frac{(x-x_{c1})^{2}}{2{\omega}_{1}^{2}}}+A_{2}\text{e}^{-\frac{(x-x_{c2})^{2}}{2{\omega}_{2}^{2}}} & & \displaystyle\end{eqnarray}$ (4)where y represent the fitted profile, x is the normalized magnetic field, y ₀ denotes the background noise. A ₁ and A ₂ represent the peak amplitude of hard and soft layer, x _c1 and x _c2 stand for the peak position of each peak. ω ₁ and ω ₂ is the peak width coefficient. The curve fitting operation was conducted in MATLAB software based on least squares algorithm. Figure 9 gives an example of the comparison between the profile of experimentally measured MBN signal and the fitted curve. Peak amplitude of hard layer and soft layer were extracted from the fitted curves. Table 1 gives the values of the extracted peak amplitudes for all the simulation and experimental cases.

The data of A ₁ and A ₂ as listed in Table 1 are respectively normalized by their maximum values.

Table 1

Peak amplitude of fitted experimental and simulation MBN envelope

Thickness or volume fraction of hard component (p)	Peak amplitude of experimental MBN		Peak amplitude of simulation MBN
	SAE 1065 film	SAE 1045 block	Hard component	Soft component
0.10	0.1113	0.0573	0.0324	0.0262
0.15	0.1254	0.0385	0.0364	0.0236
0.20	0.1262	0.0266	0.0503	0.0229
0.25	0.1368	0.0202	0.0604	0.0223
0.30	0.1450	0.0191	0.0664	0.0192

The dependency of normalized A ₁ or A ₂ on the thickness of SAE1065 film is shown in Fig. 10. With the increasing of hard layer thickness, upward tendency can be concluded for A ₁ and the value of A ₂ demonstrates exponentially decreasing trend. The result about the ratio of A ₁ to A ₂ is also plotted in Fig. 10. Linear correlation between the ratio of A ₁ to A ₂ and the thickness of SAE1065 film can be found. Therefore, the ratio of A ₁ to A ₂ can be used as good indicator for characterizing the thickness of surface hard layer or the volume fraction (p) of the hard layer in the two-layer material.

Fig. 10.

Normalized MBN peak amplitudes extracted from experimental results.

Fig. 11.

Normalized MBN peak amplitudes extracted from simulation results.

Figure 11 shows the normalized A ₁, A ₂ and A ₁/A ₂, which are extracted from the simulation MBN profiles. The correlation between the indicators (A ₁ or A ₂ or A ₁/A ₂) and the thickness of the surface hard layer was found to be very similar as that was concluded from experimental results. The agreement between the results in Fig. 10 and Fig. 11 verified that the proposed extended Ising model is feasible for MBN simulations in two-layer materials.

5. Conclusion

In this study, an extended Ising model is proposed to conduct MBN simulations in two-layer materials. A double-Boltzmann function is first introduced to modify the expression of the exchange coupling efficiency (J). Consequently, step-like transition of magnetic behavior from hard component to soft component are realized. Monte Carlo algorithm is employed to solve the extended model to derive the MBN signals and hysteresis curve of two-layer material.

MBN measurement experiments are performed on a two-layer materials made of SAE 1065 steel film and SAE 1045 steel strip. When the thickness of the surface SAE 1065 hard layer increases from 0.1 mm to 0.3 mm, the peak amplitudes of the MBN profiles are extracted to compare with the results obtained from simulation. Both the simulation and experimental works found similar conclusions that as the increasing of the surface hard layer, the MBN peak amplitude in low magnetic field decreases and the MBN peak amplitude in high magnetic field increases. The ratio of the MBN peak, A ₁/A ₂, demonstrate almost linear correlation with the thickness of surface hard layer or the volume fraction (p) of the hard layer in the two-layer material.

The extended Ising model was potential tool for predicting MBN in surface hardened layer material, in which smooth transition between two kinds of microstructures is involved. In future, various types of smooth transition function will be used to extend the Ising model for MBN simulations.

Footnotes

Acknowledgements

This study was supported by the National Key R&D Program of China (2018YFF01012300) and National Natural Science Foundation of China (Grant Nos. 11872081 and 11527801).

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