A scaling up of flattening silver particles using dry ball milling by DEM simulation

Abstract

The rising demand for silver flakes, such as those used on solar cell electrodes, requires reliable scaling up of flattening silver particles. For this purpose, it was confirmed the applicability of the specific impact energy () calculated from the discrete element method (DEM) simulation to the scale-up of flattening silver particles. The normalized specific surface area (SSA) at arbitrary flattening times at large scales was predicted by the flattening rate constant () at small scales and . Silver particles were flattened in experiments using dry vibration mills of two different sizes under various conditions. was obtained through the increase in the normalized SSA during the flattening process. The simulations modeled the motion of the balls within the mills using DEM, and was calculated from the ball-ball and ball-wall collisions. To validate the scale-up, the predicted and experimental values of the normalized SSA at large scales were compared. A comparative analysis of prediction accuracy using various components of (normal, shear, and their total) was conducted. The results showed that the normal component of had the highest prediction accuracy. These findings suggest that the normal component of can be utilized to the scale-up of flattening processes.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-44107-1.

Introduction

Micron-sized metal particles, such as those made of silver, copper, and aluminum, are used in pastes and adhesives for electrodes in small electrical parts^1–5. Among the different shapes of metal particles, such as spheres, flakes, cubes, and dendrites, metal flakes are ideal for high conductivity and heat dissipation due to their large surface area and flat shape^5–9. Silver flakes are commonly used as conductive fillers in solar cell electrodes and die-attach for chip-mounting due to their high chemical stability and conductivity^6–9. Several studies have shown that the electrical and mechanical characteristics are enhanced by using silver flakes^8–10. Therefore, the demand for silver flakes has increased, making it essential to mass-produce them.

Various methods have been developed to produce silver flakes^11–16. Broadly categorized, these include build-up methods^11–13, such as chemical reduction and thermal decomposition of Ag⁺ ions, and ball-milling methods^14–16 of flattening silver particles. As ball mills, such as tumbling and vibration mills, are frequently used for grinding in mass-production^17,18, ball milling is also considered a suitable method for mass producing silver flakes. The large-scale ball mills are utilized for the mass-production because they have a high capacity. However, trial products of silver flakes are made by using the small-scale ball mills to minimize the loss of expensive silver particles. Therefore, scaling up of the flattening process with minimal tests is necessary to ensure that the industrial production of silver flakes while maintaining the same quality as the trial products.

Limited studies have reported on scaling up of the ball mills for flattening. The complexity of the parameters controlling these mills presents a challenging process. These parameters include the number of balls and input particles, ball size, rotation or vibration speed, and mill size. Several experimental studies on these parameters for flattening metal particles have been reported^14,19–23. Despite the numerous experiments conducted, the scale-up factor remains ambiguous.

The Discrete Element Method (DEM), proposed by Cundall and Strack²⁴, provides substantial information, such as the motion of balls during milling²⁵, the motion of balls and particles from a micro viewpoint²⁶, breakage and deformation behavior^27–31, the velocity of balls, and the specific impact energy³². Kano et al.^32–34 discovered the correlation between grinding rate constant, calculated from reduction in particle size, and the specific impact energy, calculated from DEM simulations^32–34. Additionally, Mio et al. proposed a scale-up method for planetary ball mills by confirming the correlation between the grinding rate constant and the specific impact energy at different scales³⁵. Therefore, specific impact energy is a key factor in predicting grinding phenomena³¹, predicting mechanochemical reactions³⁶, designing ball mills^{37, 38}, and scaling them up³⁵. The specific impact energy is a potential scale-up factor for flattening processes because flattening occurs when a particle is sandwiched between two balls or a ball and the mill wall.

Hence, the objective of this paper is to confirm the applicability of the specific impact energy calculated from the DEM simulation to the scale-up of flattening silver particles. This scale-up predicted the increase in the normalized specific surface area (SSA) at large scales by using the flattening rate constant obtained from flattening experiments at small scales and the specific impact energy calculated from the DEM simulation because the SSA increases as the metal particles become more flattened¹⁹. The scale-up method is organized as follows:

1. The flattening rate constant is obtained from flattening experiments using the small-scale vibration mill under various conditions. This constant is calculated from the increment of the normalized SSA over the flattening time.

2. The specific impact energy at small scales is obtained through simulation. The motion of the balls is modeled by the DEM simulation under the same conditions and scales as the experiments. The specific impact energy is then calculated from ball-ball and ball-wall collisions.

3. The prediction coefficient is calculated using the correlation between the flattening rate constant and the specific impact energy at small scales.

4. The simulations calculate the specific impact energy at large scales under several conditions.

5. The normalized SSA for the large-scale vibration mill is obtained from the prediction coefficient, the specific impact energy, and arbitrary flattening times.

To confirm the scale-up validity, flattening experiments were performed using the large-scale vibration mill, and mean absolute percentage error (MAPE) was calculated to compare the experimental and predicted values of the normalized SSA. Furthermore, this paper presents a comparative analysis of prediction accuracy using various components of the specific impact energy—including normal, shear, and total—to investigate whether one or both components can serve as the scale-up factor.

Methods

Experiments on silver particle flattening

To calculate the flattening rate constant and confirm the accuracy of the scale-up, flattening experiments were performed under various vibration speeds and at different scales. To calculate the flattening rate constant, the increment of the normalized SSA from the initial stage to certain flattening times was obtained experimentally. Additionally, to confirm that the increase in normalized SSA was caused by flattening, the particle shape was observed using a scanning electron microscope (SEM).

Equipment

Figure 1 shows a schematic diagram of vibration mills. Table 1 shows the geometry and material composition of the vibration mills used. B-1 made by Chuo-Kakohki-Shoji Ltd. was used as a small-scale vibration mill. FVR-20 made by Chuo-Kakohki Ltd. was used as a large-scale vibration mill. The small- and large-scale mills had volumes of 3.2 L and 70 L, respectively. Stainless-steel balls with a diameter of 1.6 mm and a density of 7.81 g cm⁻³ were used as flattening media in all experiments. The large-scale vibration mill had a separation screen, to separate balls and silver particles, and a separated space as shown in Fig. 1b. The silver particles were not flattened in the separated space because balls did not exist in the space.

Fig. 1.

Schematic diagram of a vibration mill.

Table 1.

Vibration mills for the experiments.

Scale	Material	Diameter (m)	Height (m)	Length (m)	Volume (L)	Area of side wall/volume ratio (m−1)
Small	Stainless Steel	0.1465	–	0.19	3.2	2.73
Large	Stainless Steel	0.452	0.412	0.452	70	0.91

Materials and experimental methods

Table 2 shows the material properties and experimental conditions. is the SSA at the initial stage. All input particles are irregular-shaped silver particles produced by DOWA Electronics Materials Co. Ltd. using a wet reduction process. They are covered by a fatty acid lubricant, and their median particle diameter D₅₀ is 4 to 6 μm, as measured by a laser diffraction/scattering particle size distribution analyzer (MT-3300 EXII, MicrotracBEL Corp.). The particle shape was observed using SEM (JSM-IT300LV, JEOL Ltd.). Cross-section SEM images were obtained from polished samples of particles and curing resin (SPECIFIX-20 Kit, Struers K.K.). An ion milling device (ArBlade5000, Hitachi High-Tech Corporation.) was used to polish the samples. The SSA of the silver particles was measured with a BET specific surface area meter (Macsorb HM-model 1210, MOUNTECH Co., Ltd.) via the single-point BET method. The amplitude and speed of the vibration mills were measured with a portable digital vibration meter (V-Checker, SINFONIA TECHNOLOGY CO., LTD.).

Table 2.

Material properties and experimental conditions.

Input particles SSA (m2 g−1)	Mill	Flattening condition
	Scale	Ball		Particles (kg)	Vibration speed (min−1)	Amplitude (m)
		Mass (kg)	Filling ratio (%)
0.22	Small	12.0	80	1.5	784	0.0072
0.23	Small	12.0	80	1.5	1200	0.0072
0.22	Large	256.0	80	32.0	784	0.0062
0.23	Large	256.0	80	32.0	910	0.0064
0.20	Large	256.0	80	32.0	1208	0.0066
0.41	Large	256.0	80	32.0	1465	0.0080

DEM simulation model for impact energy calculation

DEM simulation allows determination of the specific impact energy during the flattening process to be utilized in the scale-up. Figure 2 shows the flowchart of this simulation. This energy is calculated from individual collisions between two balls, or between a ball and the mill wall. The details are shown in the following paragraphs.

Fig. 2.

Flowchart of the DEM simulation.

Figure 3 shows the simulation model of the interaction force and the vector diagram between two balls. As shown in Fig. 3a, this simulation model consists of a spring, a dashpot, and a slider. The motion of the balls is calculated using the following governing equations:

Fig. 3.

a Model of interactive forces between two balls. b Vector diagram of the relative velocity.

1

2

where and represent the mass of the ball and the gravitational acceleration; respectively, represents the velocity vector of the ball; represents the number of participating balls; represents time; and represents the contact force. The indices and indicate the individual particles. The superscripts and refer to the normal and shear components, respectively. denotes the moment of inertia, denotes the angular velocity of the ball, denotes the radius of the ball, and denotes the rolling resistance.

The contact force is written as follows:

3

4

where, and are the spring and damping coefficients, respectively; is the relative displacement of the center of gravity between the two balls; is the relative angular displacement; and μ is the frictional coefficient. is the unit shear vector between the two balls. In the calculation of the shear contact force , the relative displacement considers the distance between the center of gravity of the two balls in the shear direction () and the distance derived from the rotation of balls (). The spring coefficient and the damping coefficient are calculated from the following equations^24,39:

5

6

7

8

9

where , , and represent Young’s modulus, Poisson’s ratio, and the reduced mass of the balls, respectively. Equations (5) and (7) are based on the Hertz’s theory of elasticity³⁹, and Lame’s constant were used to calculate in Eq. (6). As written in Eq. (8), the method proposed by Cundall et al.²⁴ was applied to the calculation of .

The rolling resistance is written as:

10

where is the rolling frictional coefficient and is the radius of the contact surface.

The impact energy is sum of the impact energy per collision obtained during simulation time as shown in Fig. 2. The collisions are detected per each time step . The impact energy is calculated by Eq. (11) as follows:

11

where and are the normal and shear relative velocity upon collision between two balls or between a ball and the mill wall. , , and are the normal, shear, and total impact energies, respectively. M is the reduced mass of the balls, as written in Eq. (9); is the simulation time; and is the number of collisions within . Figure 3b shows the vector diagram of the relative velocity upon collision between two balls. and resolved based on the method as shown in Fig. 3b are utilized to calculate and in Eq. (11). The specific impact energy is calculated by Eq. (12):

12

where is the mass of the silver particles charged in the mill. , , and are the normal, shear, and total specific impact energies, respectively.

Parameters and conditions in DEM simulation

DEM simulations were performed using in-house source code. The simulations are necessary to model the motion of the balls in the presence of the silver particles because the balls flow with the particles during the flattening process. Hence, the simulation parameters were determined based on the methods previously reported that simulated the motion of the balls in the presence of the particles²⁵.

Table 3 shows the simulation parameters used in this paper. As written in the Eqs. (1–10), the behavior of the balls is affected by DEM parameters such as Young’s modulus, Poisson’s ratio, the frictional coefficient, and the rolling frictional coefficient in the DEM simulation. Young’s modulus is commonly set to a lower value than the actual value to reduce the calculation time⁴⁰. According to the previous reports, a three-digit decrease in Young’s modulus had little contribution to the motion, velocity distribution, and energy consumption of the particles and balls^41–43. Thus, Young’s modulus was set to a value three orders of magnitude lower than the actual value. Regarding the frictional coefficient μ, Kano et al.²⁵ reported that the frictional coefficient of the balls correlated with the angle of repose of the input particles for grinding. Figure S1 shows the correlation between the frictional coefficient and the angle of repose²⁵. The angle of repose of silver particles was measured using the fixed funnel and free-standing cone methods, yielding a value of 50 degrees. Based on the correlation shown in Fig. S1 and this value, the frictional coefficient μ was estimated at 0.8. The balls used in the experiments are assumed to be spherical, and the rolling frictional coefficient is set to 1.885, which was used to represent spherical objects in the DEM simulation detailed in the previous report⁴⁴.

Table 3.

Simulation parameters.

Parameter	Unit	Value
Density of balls	(g cm−3)	7.81
Ball diameter DB	(m)	0.0016
Young’s modulus	(GPa)	0.193
Poisson’s ratio	(–)	0.3
Frictional coefficient	(–)	0.8
Rolling frictional coefficient	(–)	1.885
Frictional coefficient at the front and back wall (only at large scales)	(–)	0.3
Time step	(s)	2.4 × 10−6

Table 4 shows the simulation conditions. The simulations modeled the motion of the balls within the milling section, as shown in Fig. 1, across all scales because the balls could only exist inside of the milling section. Additionally, the simulations at large scales were performed under the assumption that the separation screen is equivalent to the side wall.

Table 4.

Simulation conditions of vibration mills.

Pot			Ball			Vibration speed (min− 1)	Amplitude (m)
Diameter (m)	Height (m)	Length (m)	Mass (kg)	Number NB (-)	Filling ratio (%)
0.1465	–	0.19	12.0	7.2 × 105	80	784	0.0072
0.1465	–	0.19	12.0	7.2 × 105	80	1200	0.0072
0.452	0.412	0.016	9.1	5.4 × 105	80	784	0.0062
0.452	0.412	0.016	9.1	5.4 × 105	80	910	0.0064
0.452	0.412	0.016	9.1	5.4 × 105	80	1208	0.0066
0.452	0.412	0.016	9.1	5.4 × 105	80	1465	0.0080

To reduce the calculation load at large scales, the length of the pot is set to 10 D_B, equivalent to 0.035 times the length used in the experiments shown in Table 1. This adjustment is based on the previous reports on this approach^45,46, which indicated that the pot length was set to 4 to 16 times the ball diameter. Since reducing the pot length increases the ratio of collisions between the balls and the front and back walls, these collisions are necessary to be excluded from the calculation of the specific impact energy. The ratio of collisions between the balls and the front and back walls in is small at small scales (see Table S1) and considered even smaller at large scales. However, the specific impact energy may be inaccurate when using this approach alone because these collisions may also affect the motion of the balls. It is considered that the specific impact energy can be accurately calculated if the motion of the balls in the simulation can represent that in the presence of the silver particles observed experimentally. Therefore, at large scales, the motion was fitted between the experiment and simulation. The angle of slope after vibration stopped was evaluated as the index of the motion. The frictional coefficient at front and back wall was changed to fit the angle between the experiment and simulation because the motion generally depends on the frictional coefficient^25,47. Figure 4 shows comparison of the angle of slope after vibration stopped between the experiment and simulation. Figure 4a suggested that the simulation with the shorter length had the potential to effectively simulate the motion at large scales because the angle experimentally remained constant along the z-axis direction. As shown in Fig. 4b and c, the angle was measured at 1s after vibration stopped in the simulation. Figure 4d indicated that the angles were comparable between the experiment and simulation when was 0.3. Hence, as written in Table 3, was set to 0.3 that could model the motion of the balls in the presence of the particles at large scales. Furthermore, when calculating the specific impact energy at large scales, the mass of the charged silver particles in the mill is reduced at a ratio consistent with the ratio of the pot length between simulations and experiments.

Fig. 4.

The angle of slope after vibration stopped. a The snapshot of balls and particles after vibration stopped observed in the experiment. Velocity distributions of ball motion b during vibration and c 1s after vibration stopped obtained from the simulation. d The relationship between the angle and frictional coefficient at the front and back wall .

Results and discussion

The flattening rate constant from the small-scale flattening experiments

Firstly, the changes of the particle shape during flattening experiments using the small-scale vibration mill are shown. Figure 5 shows the changes in particle shape during the flattening experiments using the small-scale vibration mill. SEM images show the particles under two vibration speeds during the flattening experiments. The number of flat particles and their size increase with flattening time. Cross-sectional SEM images indicate that particle thickness decreases as flattening time increases. The higher vibration speed of 1200 min^− 1 appears to produce more flat particles and reduce their thickness. Additionally, flattening is considered predominant because the particle population consists entirely of flat particles. The absence of severe agglomeration indicates that the SSA of the particles increases during flattening.

Fig. 5.

a-1, b, c-1, d, e-1 SEM images and a-2, c-2, e-2 cross-sectional SEM images of particles during flattening experiments using the small-scale vibration mill at vibrational speeds of b, d 784 min⁻¹ and c, e 1200 min⁻¹ and a flattening time of a 0 s, b, c 1800 s, d, e 3600 s.

According to the SEM observations of the particle shapes shown in Fig. 5, it is considered that the SSA can quantify the progressing degree of flattening. Accordingly, the variations of the SSA during flattening were evaluated. Figure 6 shows the variation of the normalized SSA , the quotient of the SSA at flattening time and the initial SSA, as a function of , using the small-scale vibration mill. The results indicate that increases proportionally to and that the increasing trends vary with vibration speeds. Therefore, the flattening rate constant, , an index for the quantitative evaluation of flattening, is obtained by the slope of and , as shown in Eq. (13):

13

Fig. 6.

Progression of normalized SSA during flattening using the small-scale vibration mill. The dashed lines are the results of fitting by Eq. (13).

Most of the experimental values in Fig. 6 align closely with the fitting lines of Eq. (13). These results indicate that Eq. (13) is an appropriate empirical equation and that is a proper representative value in flattening processes. Higher vibration speeds correspond to higher values of .

The specific impact energy at small scales from the DEM simulations

In the previous subsection, the changes in particle shape and during flattening were shown, and then was calculated. Here, the relationship between the specific impact energy and the flattening conditions based on the simulated motion of the balls and collision properties in the small-scale vibration mill is discussed.

Figure 7 shows the velocity distributions of ball motions in the small-scale vibration mill. Figure 7a and b illustrate the motion of the balls during one cycle at speeds of 784 min⁻¹ and 1200 min⁻¹, respectively. The balls repeatedly float and collide with the side wall. Figure 8 shows the impact energy distributions in the small-scale vibration mill. As the distribution of illustrated in Fig. 8a shows, the collisions between balls are the most frequent ones. A small peak in the higher range of (around 10⁻⁵ J) due to ball-wall collisions is also observed. The same peak is observed in the normal component distribution in Fig. 8b. This indicates that the higher range of is derived from the ball-wall collisions. In the vibration mill, the balls flow by propagation of collision from the side wall to the balls⁴⁸. The normal component of the ball-wall collisions is considered to show high energy because the side wall pushes the balls up and down as shown in Fig. 7.

Fig. 7.

Results from the small-scale vibration mill: velocity distributions of ball motion at vibration speeds of a 784 min⁻¹, b 1200 min⁻¹, respectively. The blue arrows indicate the vibrational direction of the mill.

Fig. 8.

Results from the small-scale vibration mill: a impact energy distributions at the collision points. b Impact energy distributions in the normal and shear components.

Next, the relationship between and flattening conditions is discussed. Figure 9 shows at two vibration speeds using the small-scale vibration mill. The normal component , the shear component , and their total specific impact energy increase with vibration speed. Figure 7a and b show that the velocity of the balls increases and the vibration period decreases with vibration speed. Thus, it is conceivable that both the collision frequency and relative velocity increase with higher vibration speed. Additionally, as shown in Figs. 6 and 9, both and showed that a tendency for the values to increase with vibration speed. Therefore, may correlate with .

Fig. 9.

Total specific impact energy and the proportions of its normal and shear components at small scales under two different vibration speeds.

Calculating the prediction coefficient

According to the previous report³⁵, it was indicated that the specific impact energy could be useful as the scale-up factor for the grinding process by confirming that the correlation between the grinding rate constant and the specific impact energy at different scales is expressed by a straight line. If the flattening process also shows the comparable correlation between and , the slope of the correlation can be utilized to predict at large scales. This subsection derives the scale-up prediction coefficient, , from the correlation between and using Eq. (14):

14

To determine which components of can serve as the scale-up factor, the normal prediction coefficient , the shear prediction coefficient , and the total prediction coefficient are calculated by substituting , , and for in Eq. (14). Figure 10 shows the relationship between and for the small-scale vibration mill. The three dashed lines represent the fitting results of the relationships substituted , , and for in Eq. (14), respectively. Strong correlations between and are observed on linear coordinates fitted by Eq. (14), regardless of the components of . This indicates that both and may both represent the same flattening phenomenon. The following subsections provide a comparative analysis of prediction accuracy using these coefficients.

Fig. 10.

Relationship between and the various components of the specific impact energy using the small-scale vibration mill. The dashed lines show the results of fitting Eq. (14). Error bars mean the fitting error of by Eq. (13).

The specific impact energy from the DEM simulations at large scales

In this subsection, is calculated under various conditions. Simulations at large scales were performed at four different vibration speeds. Figure 11 shows the velocity distributions of the balls and in the large-scale vibration mill. Figure 11a–d depict snapshots of the ball motion in the same vibration direction under different vibration speeds. These figures indicate that, in contrast to the motion of balls observed at small scales (Fig. 7a, b), the balls flow from the upper left to the middle right. These motions are considered general because they resemble the previously reported motions of steel balls in vibration mill experiments⁴⁸. Additionally, as in Fig. 7a and b, the velocity of the balls increases and the vibration period decreases with higher vibration speeds.

Fig. 11.

Velocity distributions of ball motion in the large-scale vibration mill. The vibration speeds are a 784 min⁻¹, b 910 min⁻¹, c 1208 min⁻¹, and d 1465 min⁻¹, respectively. The blue arrows indicate the vibrational direction of the mill.

Figure 12 shows at four different vibration speeds using a large-scale mill. , as well as its components and , increase with higher vibration speeds. According to Figs. 9 and 12, the proportion of in is lower at large scales than at small scales. This behavior arises because the ratio of the side wall area to the volume in the mills decreases as the pot diameter increases (Table 1), thereby reducing ball-wall collisions. Thus, the normal component due to ball-wall collisions decreases at large scales.

Fig. 12.

Total specific impact energy and the proportions of its normal and shear components at large scales under four different vibration speeds.

Prediction of the normalized SSA and validation of the scale-up method

The normalized SSA () at arbitrary flattening times is predicted using the large-scale vibration mill and compared with experimental values () to validate the accuracy of the scale-up. is calculated from and using Eq. (15), derived by substituting in Eq. (13) with Eq. (14):

15

and are substituted by the corresponding components of the specific impact energy obtained from DEM simulations. Figure 13 shows the progression of during flattening using the large-scale vibration mill. In Fig. 13, the dots represent the experimental values of and the dashed lines represent the predicted values of . As in Fig. 6, increases with higher vibration speeds.

Fig. 13.

Progression of normalized SSA () during flattening using the large-scale vibration mill. The dashed lines show the predicted results using Eq. (15) with a and , b and and c and .

To quantitatively evaluate the prediction accuracy of the scale-up, the mean absolute percentage error (MAPE) was calculated using Eq. (16) and is given in Fig. 13:

16

where is the number of data points in Fig. 13. The MAPE predicted from the normal components, and , is lower than that predicted from the other components. Therefore, at large scales can be accurately predicted using the normal component of the specific impact energy obtained in the simulations, as well as the flattening rate constant , obtained in the small-scale experiments. Accordingly, it is reasonable to conclude that can be utilized for the scale-up of the flattening process.

The findings are consistent with previous experimental studies in that normal collisions affect flattening behavior. Fujita and Motozuka et al. reported that the plastic deformation of iron particles during ball milling was induced by uniaxial compression, as revealed by analyzing changes in the crystallographic texture of the particles^49,50. Hence, it is also considered that the findings can relate the mechanisms of the flattening metal particles with the motion of the balls and the collisional properties in the ball mills.

This scale-up approach might have a wider range of applicability than the existing scale-up approaches, such as experimentally power-based scaling⁵¹. The collisional properties could be changed in scales, conditions, and mill types³⁶, and it is challenging to analyze the collisional properties such as the collisional direction from only power consumption. In contrast, this approach might be also applied to the scale-up using other types of mills and design mills because it can analyze the collisional properties. Further investigation is necessary to validate the applicability of this approach to the scale-up using other mill types.

Conclusion

To confirm the applicability of the specific impact energy calculated from the DEM simulation to the scale-up of flattening silver particles, the normalized SSA during flattening at large scales was predicted from the specific impact energy and the prediction coefficient. The prediction coefficient was obtained from the correlation between the flattening rate constant and the specific impact energy. These were derived from experiments and simulations at small scales. To confirm the validity of the scale-up, MAPE values were calculated from the predicted and experimental normalized SSA at large scales. Furthermore, prediction accuracies using different components of the specific impact energy (normal, shear, and total) were compared to investigate whether one or both components could serve as the scale-up factor.

In the experiments, silver particles were subjected to flattening using dry vibration mills at different vibration speeds, and the SSA of the silver particles was measured. The flattening rate constant was calculated from the increment of the normalized SSA and the flattening time in the small-scale experiment. In the simulations, the motions of the balls within the vibration mill were modeled by DEM simulations under the same conditions as the experiments. The specific impact energy was calculated from the ball-ball and ball-wall collisions.

At small scales, the flattening rate constant and the specific impact energy increased with vibration speed. Strong correlations were found between them on linear coordinates, so the prediction coefficient could be determined. Although the specific impact energy increased with vibration speed at large scales, the proportion of the normal component was higher at small scales. Validation showed the lowest MAPE when using the normal component of specific impact energy for predictions.

Therefore, it can be concluded that the normal specific impact energy calculated by the DEM simulation can be utilized to the scale-up of the flattening process. This finding could be valuable for designing equipment to flatten metal particles.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

T. K. – Conceptualization, Data curation, Investigation, Methodology, Software, Visualization, Writing – original draft.K. K. – Software, Supervision, Writing – review & editing.D. O. – Data curation, Software.Y. H. – Supervision, Funding acquisition.J. K. – Supervision, Writing – review & editing.All authors reviewed the manuscript.

Funding

Kizuku Kushimoto and Junya Kano disclose support for the research of this work from DOWA Holdings Co. Ltd. [the DOWA TECHNO FUND]. The funder provided financial support for research and did not have any additional role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

Takuya Kojima, Daisuke Oka, and Yutaka Hisaeda are employees of DOWA Electronics Materials Co. Ltd. Two related technologies are applied for patents in Japan. (Japanese Patent Application No. 2024-57995(“The method and equipment and program of prediction of flattening”), Japanese Patent Application No. 2024-58001 (“The equipment and program of simulation”)). Kizuku Kushimoto and Junya Kano declare no competing interests.