Optimization of Densification Behavior of a Soft Magnetic Powder by Discrete Element Method and Machine Learning

Jungjoon Kim; Dongchan Min; Suwon Park; Junhyub Jeon; Seok-Jae Lee; Youngkyun Kim; Hwi-Jun Kim; Youngjin Kim; Hyunjoo Choi

doi:10.2320/matertrans.MT-MB2022008

Abstract

Densification of amorphous powder is crucial for preventing magnetic dilution in energy-conversion parts owing to its low coercivity, high permeability, and low core loss. As it cannot be plastically deformed, its packing fraction is controlled by optimizing the particle size and morphology. This study proposes a method for enhancing the densification of an amorphous powder after compaction, achieved by mixing three types of powders of different sizes. Powder packing behavior for various powder mixing combinations is predicted by an analytical model (i.e., Desmond’s model) and a computational simulation based on the discrete element method (DEM). The DEM simulation predicts the powder packing behavior more accurately than the Desmond model because it incorporates the cohesive and van der Waals forces. Finally, a machine learning model is created based on the data collected from the DEM simulation, which achieves a packing fraction of 94.14% and an R-squared value for the fit of 0.96.

1. Introduction

Soft magnetic materials are used as major components in electronic devices owing to their advantageous properties in magnetization and demagnetization.¹^–⁴⁾ Recently, amorphous materials have attracted attention because of their excellent soft magnetic properties, which can eliminate factors that cause domain wall pinning effects such as the grain boundary.⁴^,⁵⁾ However, because an amorphous material must be manufactured before crystallization occurs, a fast cooling rate is required, limiting the manufacture of the bulk form. Therefore, amorphous material is manufactured and utilized in the form of a powder, which ensures uniform three-dimensional magnetic properties.⁶^–⁹⁾ In utilizing such amorphous powders, superior properties are derived through densification, insulation coating, and nanocrystallization processes. In particular, it is important to manufacture a compact material at a high density to increase the fraction of magnetic material to maintain a high saturation magnetization and magnetic permeability and prevent magnetic dilution by reducing non-magnetic pores.¹⁰^–¹²⁾ However, because of the negligible capability for plastic deformation and a low crystallization temperature of amorphous powder, the material cannot be densified with high pressure and temperature.⁸^,¹⁰^,¹³^,¹⁴⁾

Therefore, efforts have been made to increase the density of amorphous powders by controlling their morphology and size distribution. Various experimental and analytical approaches have been proposed to estimate and calculate the packing fraction of spherical powders.¹⁵^–²⁴⁾ In previous studies, the average particle size and particle size distribution of a powder were found to be the key factors that most dominantly influenced the packing fraction of the powder. However, these approaches only considered the geometrical factors, including powder shape, size, and size distribution, and did not consider factors such as the interaction of powders and consequent changes in flowability during the powder packing process.

A discrete element method (DEM) simulation, wherein the movement of each particle is numerically determined by a series of calculations that trace each individual particle within a population of independent particles,²⁵^,²⁶⁾ can be used to simulate powder packing behavior, which can accurately reproduce powder interactions and surface features, as well as geometrical properties such as the shape, size, and density of a powder.²⁷^–²⁹⁾ For particles in contact, the contact law is used to determine the contact force acting on the particles and, subsequently, the motion of the particles involved in the contact.³⁰⁾ The simulation is also appropriate for predicting the packing fraction by accurately reproducing the flow and behavior originating from the interaction of the powders. However, despite these advantages, it is still difficult to test all powder combinations within a simulation because there are thousands of combinations.

Artificial neural networks (ANN) have been applied to predict various phenomena because they have many advantages when it comes to resolving the complexity between dependent and independent parameters.³¹^,³²⁾ ANN techniques have been increasingly used in materials science to design alloys and predict their mechanical properties. They can also be used to design a mixing ratio for a novel compact of powder mixture without physical experimentation and high cost. Furthermore, particle swarm optimization can be used to determine specific ratios and target values for optimization using various combinations of input parameters.³³^,³⁴⁾ These optimization methods are approximate search methods for solving complex optimization problems.

In this study, a method for enhancing the densification of amorphous powders after compaction by mixing three types of powders with different sizes was proposed. The powder packing behavior for various powder mixing combinations was predicted by both an analytical model (i.e., Desmond’s model), and a computational simulation based on DEM. To predict the actual powder behavior, the particle-size distribution and angle of repose were implemented in the simulation. Finally, for a high-density packing fraction, the powder mixing ratio was optimized by performing machine learning based on an ANN.

2. Experimental Procedure

2.1 Process and materials

Commercial Fe–Cr–Si–B–C amorphous powder (KUAMET6B2, Epson Atmix, Japan) was used to validate the theoretical calculations, computer simulations, and machine learning results. To investigate the effects of the particle size distribution on the packing behavior, the amorphous powder was classified into three grades according to its size using a sieve shaker (Analysette 3 Pro, Fritsch, Germany) with mesh sizes of 45, 90, and 150 µm. Sieving was performed for 10 min at a vibration frequency of approximately 3600 times per minute and an amplitude of 1.0 mm. Three powder sets, designated as S, M, and L, were obtained from the sieving, which consisted of the following size ranges: less than 45 µm (S), 45–90 µm (M), and 90–150 µm (L). A laser particle-size analyzer (LPSA; Mastersizer 2000E, Malvern Instruments, UK) and field emission scanning electron microscope (FE-SEM; JSM 7410F, JEOL Co., Japan) were used to analyze the particle size distribution and the morphology of the powder, respectively. A metal funnel with a circular opening of 8 mm and a slope of 60° to the horizontal was used to measure the angle of repose. The circular opening was first blocked with a finger before 100 g of powder was poured into it. The removal of the finger permitted the powder to flow out of the funnel and onto a horizontal surface directly beneath it, thereby forming a stacked powder pile. Optical images were then taken to determine the angle of repose, which is defined as the angle that the powder surface makes with the horizontal plane on which it sits.³⁵^–³⁷⁾

A computational simulation was conducted using EDEM software (Dem Solution, UK) to predict how the powder packing density varies with powder mixing ratios. To investigate various combinations of powders, the combination ratio was designed by adjusting the weight ratio of the powders using integer values on a scale of 0 to 10. The Edinburgh elastoplastic adhesion model was used to comprehensively analyze the interactions among the S, M, and L powder. In general, this model employs stochastic calculations and predicts the behavior of powders using various variables, such as the constant pull-off force, contact plasticity ratio, and particle properties. Therefore, the measured properties of the powders were incorporated into the computational simulation. Furthermore, the bulk material properties of each powder were implemented into the simulation based on the bulk density and angle of repose. Additionally, the simulation was set to use the experimentally determined average particle size and particle-size distribution of the powders. The distribution of the particle sizes of the powder was found to be Gaussian using LPSA (Fig. 1(d)). To reproduce the particle size distribution of these powders in the simulation, the LPSA results of the powders were input at the particle setting steps for S, M, and L. Furthermore, the physical parameters (e.g., cohesive force, coefficient of restitution, and modulus of elasticity) controlling the interaction among the three powders were fitted based on the experimentally measured angle of repose (Fig. 2). Finally, particle calibration (e.g., shape, size, and angle of repose) was performed by adjusting the input parameters. The fitted powder was then set to be transferred through a funnel and into a cylinder within the simulation. The powder passing through the funnel was set to accumulate in the cylinder due to gravity (9.8 mm/s²), and no external force was applied except the interaction among the mixed powders. The mixed powder was passed into the cylinder until all powder was stationary. The powder was then compressed using a punch press of the same diameter as that of the cylinder. For the compressed body made in the simulation, the packing fraction was calculated by calculating the porosity using the DEM software.

Fig. 1

SEM images of classified powder by size (a) S, (b) M, and (c) L and (d) the corresponding particle-size distribution.

Fig. 2

Optical images of powders after the angle of repose test of (a) S, (b) M, and (c) L.

The change in the packing fractions with respect to the average particle size and particle-size distribution of the amorphous soft magnetic powders was calculated using the Desmond model. Recently, Desmond presented an equation to predict the packing fraction of a powder as a function of polydispersity (δ) and skewness (S) – two parameters that considerably influence the packing fraction¹⁶⁾ (eq. (1)). The effectiveness of the equation was validated using collectively jammed packing.

\begin{equation} \phi_{\textit{RCP}} = \phi_{\textit{RCP}}^{*} + c_{1}\delta + c_{2}S\delta^{2} \end{equation}

(1)

where ϕ_RCP and $\phi_{RCP}^{*}$ denote the packing fractions of the powder to be obtained and the powder with a single distribution, respectively. ϕ_RCP, c₁ and c₂ are experimentally measured constants, and their values were set to 0.57, 0.0658, and 0.0857, respectively. δ and S were expressed as follows:

\begin{equation} \delta = \sqrt{(\Delta R^{2})}/\langle R\rangle \end{equation}

(2)

\begin{equation} S = \langle \Delta R^{3}\rangle \langle \Delta \textit{R}^{2}\rangle^{3/2} \end{equation}

(3)

Specifically, δ was obtained by dividing the standard deviation of the powder radius (R) by the average powder radius. S represents the sum of the cubes of the deviations divided by the cube of the standard deviation.

Machine learning algorithms based on ANN were used to design the optimal powder mixing ratio for the maximum powder packing fraction. In general, an ANN learning process searches for a combination of hyperparameters corresponding to optimal performance. The hyperparameters include the number of hidden layers, neurons, and activation functions, among other factors. We used the rectified linear activation function (ReLU) for the hidden layers (two layers) and the linear transfer function for the output layer. Root mean squared propagation (RMSProp) was adopted the for optimizer. The learning rate of RMSProp was used 0.001. The dataset was divided into 64% of train dataset, 20% of test dataset, and 16% of validation dataset. The number of nodes was determined by changing from 1 to 15. As a result, the number of nodes was set to 8 and 7 for first layer and second layer respectively. The ratios of the S, M, and L powders were used as the input data, and the packing fraction was used as the target data. The trained ANN model was evaluated based on the coefficient of determination (R²).

Nevertheless, although the trained ANN model could be used to predict the packing fraction of the powders, determining the optimal mix ratio was challenging because of the diversity and complexity of the ratio combinations. To determine the optimal powder ratio with the best packing fraction, particle swarm optimization was employed, which can quickly determine the composition in the expansive space of mixtures of ratios.

3. Results and Discussion

Figure 1 shows SEM images of the S, M, and L powders and their particle size distributions. The atomized powder was spherical, and as the particle size increased, the degree of spheroidization decreased slightly, resulting in an oval shape. Furthermore, a slight roughness was observed on the surface of the L powder (Fig. 1(c)), which can be attributed to the partial crystallization on the powder surface. Generally, amorphous powders can be cooled quickly using water or gas during the atomization phase. For powders that have a relatively large particle size radius, the stored extra thermal energy takes longer to leave the powder, which causes the particle surfaces of these powders to exhibit a relatively lower cooling rate compared to their interiors, which in turn causes a crystallization on those surfaces.³⁸^,³⁹⁾ This partial crystallization may then lead to a rough surface of the large amorphous powder. The particle size distribution of the powder was entirely Gaussian, although there was a slight difference depending on the classification state (i.e., S, M, and L) (Fig. 1(d)). The volume median diameter (i.e., d(0.5)) of the S, M, and L powders was 28.6, 68.8, and 139.1 µm, respectively. The estimated powder size generally fitted the target size (i.e., the sieve size).

Figure 2 shows the optical images of the S, M, and L powders after the angle-of-repose tests. The angle of repose was similar regardless of the powder size; the measured angles were 35.9°, 32.8°, and 32.4° for the S, M, and L powders, respectively. In general, the smaller the powder size, the higher the surface energy; thus, the greater the cohesive force and angle of repose. However, in this study, owing to the crystallization observed in the morphology of the powder (Fig. 1(c)), the friction increased because of the roughness of the surface of the L powder. Therefore, the uneven surface of L powder impeded the flow, and as the powder fell through the funnel the interaction among the powders during the stacking increased the angle of repose.

The packing fractions calculated using Desmond’s model for various powder mixing ratios are shown in Fig. 3(a). Notably, as the powder is mixed, the packing fraction increases, and if the S powder accounts for more than 60%, a relatively high packing fraction is observed. In general, the mixed powder exhibited a higher packing behavior than the unmixed S powder, which is thought to be caused by the effect of polydispersity. Specifically, the polydispersity shows a higher value as the size distribution of the powder increases, which is driven by the mixing of the powder. Consequently, an increase in the packing fraction is observed. Desmond’s model, which can be applied to powders of various particle sizes and size distributions, shows the correlation between the powder packing fraction and parameters by introducing polydispersity and skewness as independent variables. Polydispersity is a value obtained by dividing the standard deviation of the powder radius by the average radius of the powder; furthermore, it indicates the degree of distribution of deviations based on the average value. The greater the polydispersity of the particle size distribution, the greater the variation in the powder, and thus, the wider the particle size distribution. Therefore, when polydispersity increases, particles pack to higher-volume fractions because the smaller particles pack more efficiently by either layering against the larger particles or by fitting into the voids created between neighboring large particles.¹⁶^,²¹^–²⁴⁾

Fig. 3

(a) Packing fractions calculated for various mixing ratio by (a) Desmond’s model and (b) computational simulation.

Skewness is the sum of the cubes of deviations divided by the cube of the standard deviation; it indicates the direction in which the distribution is skewed or the degree of asymmetry.¹⁶^,²³^,²⁴⁾ Skewness has a value of 0 when the left and right parts are symmetrical, and a positive value when the tail of the distribution is stretched to the right compared to the symmetrical distribution. Conversely, skewness has a negative value when the tail of the distribution is stretched to the left compared to the symmetrical distribution. Additionally, an increase in the absolute value of skewness indicates an increase in the asymmetry of the distribution.

Powder packing behavior after the compaction of the powder in computational simulation is shown in Fig. 3(b). Overall, the higher the mixing ratio of the S powder, the higher the density. When each powder was filled individually, the density increased as the size of the powder decreased. As the size of the powder increased, the size of the pores formed between the powders increased as well, while the density decreased. Similar to the theoretical calculation results discussed above (Fig. 3(a)), it was observed that the compact density of the mixed powder was higher than that of the individual powder. Furthermore, for the simulation, various parameters (e.g., particle size distribution of particles) were used to reproduce the experimental angle of repose using the DEM software, and the fitting process was performed to accurately implement the powder behavior by reflecting the actual data of the angle of repose. In our previous study,¹⁸⁾ it was confirmed that unlike analytical calculations using geometrical parameters (e.g., the average size of powder, powder size distribution, and deviation), simulation can make more accurate predictions because it considers the interaction among powder particles. Therefore, in the compression packing behavior of this study, determining whether the powders influence each other by frictional, cohesive, and repulsive forces was more important than the morphological analysis of the powder because this information can be used in the simulation to yield more accurate predictions.

Among the simulation results, the highest packing fraction obtained was 93.76%, which occurred when the S, M, and L powders were mixed at a 7:2:1 ratio. In this case, the simulation results show a dense arrangement throughout the compact mixture (Fig. 4(a)). Because the fraction of S powder was high, it was advantageous to position the S powder in the space formed around the M and L powders. However, in the 1:2:7 ratio of the powders (S, M, and L, respectively) the results of the packing analysis, which showed a relatively low packing fraction, exhibited a loosely packed arrangement of the powders that was characterized by a high porosity throughout the compact (Fig. 4(b)). A closer look at this structure (shown in Fig. 4(c)) reveals that the pores formed by the M and L particles are not sufficiently filled. In other words, at certain mixing ratios small powders lose the ability to effectively fill the pores formed by the relatively large powders, resulting in a lower packing fraction. In addition, as the fraction of large powders increases, the probability of adjoining of large powders also increases, which leads to the formation of isolated pores that prohibit the movement of the small powders.

Fig. 4

Simulation result images of (a) the highest packing fraction condition, (b) a relatively low packing fraction, and (c) a magnified image of the marked segment in (b).

Figure 5(a) compares the powder packing behavior predicted by the DEM simulation and machine learning based on the simulation data for various mixing ratios. An R-squared value of 0.96 indicates a very high reliability of the machine learning model. Figure 5(b) displays the packing fraction predicted for various powder mixing ratios based on machine learning. This is very similar to the relationship between the powder packing fraction and powder mixing ratio predicted by the DEM simulation (Fig. 3(b)). Based on the particle swarm optimization, the highest packing fraction was derived by inputting random variables into the mixing ratio. In this case the ratio of S, M, and L were 8.1:0.2:1.7, which resulted in the packing fraction of 93.86%. However, machine learning in the simulation achieved the highest packing fraction, which yielded 94.14% (for the same ratio). Notably, the fraction of M powder was relatively low under the optimal conditions. In the particle size analysis result shown in Fig. 1(d), there is a range in which the particle size distribution of M powder overlaps the particle size distribution of S and L powder. As the fraction of similarly sized powder increases, the pores formed increase, and the fraction of small powder that fills the pores decreases, which reduces the packing fraction.¹⁸⁾ However, the fraction of M powder was not derived as 0 in the machine learning results. A higher packing fraction was calculated at a mixing ratio containing an appropriate fraction of M powder. In addition, it can be seen that the fraction of the S powder is high enough to sufficiently fill the pores formed by the M powder, which is advantageous for high packing. Therefore, these results suggest that maintaining a tri-modal particle distribution in the mixing ratios should help increase the density of the compact.

Fig. 5

(a) Powder packing behavior predicted by DEM simulation, and by machine learning on the basis of DEM simulation data for various mixing ratios, and (b) packing fraction predicted for various powder mixing ratios on the basis of machine learning.

4. Conclusions

The compact density of amorphous soft magnetic powders was predicted through theoretical calculations based on Desmond’s model, and computational simulations based on DEM. The packing fraction prediction through theoretical calculations did not incorporate the cohesive and van der Waals forces of the actual powder, resulting in a notable deviation from the experimental results. However, in the DEM simulation, the consideration of the angle of repose of the powder helped to accurately reproduce the cohesive force and packing behavior of the actual powder. The machine learning model, which used the packing data collected from the compact density from the DEM simulation, yielded fairly accurate results. This model was used to determine the optimal mixing ratio of the S, M, and L powders as 8.1:0.2:1.7, which could not be obtained experimentally and corresponded to the best packing fraction of 94.14%. The ratio derived through machine learning was validated in the simulation, and an increase of approximately 7.06% was achieved compared to that obtained using monolithic L powder.

Acknowledgment

This research was financially supported by the Civil-Military Technology cooperation program (No. 18-CM-MA-15), the Ministry of Trade, Industry and Energy (MOTIE), and the Korea Institute for Advancement of Technology (KIAT) through the International Cooperative R&D program (P0006837).

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