TECHNICAL PAPER

Effects of the friction coefficient of a barrel on the grindingperformance of a planetary ball mill

Bumcho Kim1,2• Gunhee Jang1

Received: 31 October 2017 / Accepted: 24 May 2018 / Published online: 31 May 2018� Springer-Verlag GmbH Germany, part of Springer Nature 2018

AbstractWe investigated the effect of the friction coefficients of a barrel on the grinding performance of multi-layer ceramic

capacitors (MLCCs) in a planetary ball mill. We formulated the motion equation of the MLCCs, including gravitational

force, interactive force due to collisions and the drag force in a planetary ball mill, using the three-dimensional discrete

element method. The contact model in a collision between two MLCCs is represented by the Voigt model. The linear

velocity, impact and shear energy according to the friction coefficients of a barrel were calculated. The trajectory of a

single MLCC and the linear velocity distributions of many MLCCs were analyzed. It was found that an increase of the

friction coefficient of a barrel not only improves the shear energy, but also reduces the impact. Finally, grinding exper-

iments using four barrels with different friction coefficients were conducted to verify the simulated result. The measured

wear height accurately matches the height expected from the simulated shear energy.

1 Introduction

A multi-layer ceramic capacitor (MLCC) is a general-type

capacitor that temporarily stores electric charge and

removes noise in electronic circuits. The main body is

composed of dielectric layers with interlayered internal

electrode layers, as shown in Fig. 1. An external electrode

connects an MLCC to a printed circuit board or a hybrid IC

module in a physical and electrical manner. A planetary

ball mill which is used to grind the sharp corner of MLCCs,

consists of a barrel on a disc. The rotation of the barrel and

the revolution of the disc randomly mix tens of thousands

of the MLCCs and grinding media in the cylindrical barrel.

If the rotational speed increases for better productivity,

micro cracks can occur due to strong impacts. If the rota-

tional speed decreases to reduce the number and size of

micro cracks, the productivity decreases. Therefore, it is

important not only to minimize the damage caused by

impacts but also to increase the mill grinding efficiency.

This optimization improves both the quality and efficiency

of the MLCC manufacturing process.

Numerical tools that accurately predict the motion of the

MLCCs are required for better understanding of grinding

operations and improving grinding performance as well as

grinding processes. The discrete element method (DEM),

also called a distinct element method, is a family of

numerical methods for computing the motion and effect of

a large number of small particles such as MLCCs. Though

DEM is very closely related to molecular dynamics, this

method is effectively utilized by the inclusion of rotational

degrees-of-freedom as well as complicated geometries. The

DEM was first proposed by Cundall and Strack (1979) to

model the behavior of soil particles under dynamic loading

conditions. This method was applied to grinding mills for

the first time by Mishra and Rajamani (1992). Numerical

investigation and experimental verification of mixing and

segregation were carried out by Cleary et al. (2003),

Kwapinska et al. (2006) and Liu et al. (2008). Some

researchers have studied the motion of balls and mixing

behavior in a planetary ball mill using DEM. Mori et al.

(2004) investigated a method of simulating the motion of

balls in tumbling mills operated under wet conditions at

different solid concentrations. This simulation method is

based on the three-dimensional DEM and takes into

account the effects of suspension, drag force, and

& Gunhee Jang

ghjang@hanyang.ac.kr

1 PREM, Department of Mechanical Engineering, Hanyang

University, 17 Haengdang-Dong, Seongdong-Gu,

Seoul 133-791, Republic of Korea

2 Samsung Electro-Mechanics Co. Ltd., 314, Maetan 3-Dong,

Yeongtong-Gu, Suwon-Si, Gyeonggi-Do 443-743, Republic

of Korea

123

Microsystem Technologies (2018) 24:4465–4470https://doi.org/10.1007/s00542-018-3974-3(0123456789().,-volV)(0123456789().,-volV)

buoyancy. Xu et al. (2010) focused on the effects of the

physical properties of particles on their mixing behavior in

a qualitative way through the use of two-dimensional DEM

simulation and quasi two-dimensional experiments. They

demonstrated that the rotational speed of the barrel and the

density and size of the particles are the main factors

affecting mixing behavior. Mio et al. (2002) investigated

the effect of the direction of the barrel rotation and the ratio

of the barrel rotation to the disk revolution on grinding

performance and investigated the specific impact of balls

during milling. However, none of these works studied the

relation between grinding performance of the balls and

friction coefficient of the barrel. Also, no previous

researchers have studied the trajectory of the balls

according to the friction coefficient of the barrel.

In this paper, we formulated the motion equation of the

MLCCs including gravitational force, normal force, and

tangential force due to collisions and drag force using the

three-dimensional DEM. Further, we investigated the

effect of the friction coefficient of the barrel on the

grinding performance in a planetary ball mill. The linear

velocity, impact, and shear energy according to the friction

coefficients of the barrel were calculated and the trajectory

of a single MLCC was analyzed. Finally, grinding tests

using four barrels with different friction coefficients were

conducted to verify the simulated results.

2 Simulation

2.1 Equation of motion of balls

The equation of motion of balls (MLCCs and ceramic

media) in a planetary ball mill can be written as follows:

F!¼ F

!G þ F

!I þ F

!D ð1Þ

where F!

G, F!

I and F!

D are gravitational force, interactive

force due to collision and drag force. All of these forces are

summed to find the resultant force acting on each ball. An

integration method is employed to compute the change in

position and velocity of each ball during a certain time step

using Newton’s second law, expressed by Eq. (2):

a!¼ F!

mð2Þ

where a!, m, and F!

are acceleration of a ball, mass of a

ball, and resultant force, respectively. The new positions

are used to compute the forces during the next step, and

this loop is repeated until the simulation ends. The contact

in collision between two balls is represented by the Voigt

model, as shown in Fig. 2, which describes a spring–

dashpot to represent normal motion and additional slider

for the friction resulting from the tangential motion. The

interactive force,F!

I , acting in the collision can be written

as follows:

F!

I ¼ K u!þ C _u~ ð3Þ

where K, C, u~, and _u~ are the spring and damping coeffi-

cients, the relative displacement, and relative velocity

between two balls, respectively. The impact (I) and shear

energy (J), which is the energy index in the tangential

direction, can be written as follows:

I ¼Z

F!

I

������Dt ð4Þ

J ¼Z

F!

TI � v!T

������Dt ð5Þ

where F!

I , F!

TI , v~T , and Dt are interactive force, interactiveforce in the tangential direction, relative velocity in the

tangential direction, and time step, respectively. The

magnitude of the tangential force which is kinetic friction

between two balls is proportional to the magnitude of the

normal force and can be written as follows:

Fig. 1 Sectional view of an MLCC

Fig. 2 Voigt model a normal force, b tangential force

4466 Microsystem Technologies (2018) 24:4465–4470

123

F!

TI

������ ¼ ld F

!NI

������ ð6Þ

where F!

TI , F!

NI , and ld are tangential force, normal force

and coefficient of kinetic friction, respectively. The drag

force can be written as follows:

F!

D ¼ q2CDA u!

�� �� u! ð7Þ

where q, CD, A, and u! are density of air, the drag coef-

ficient expressed as a function of the Reynolds number

(Re) given by Eqs. (8) and (9), the projection area of a ball,

and the relative velocity between two balls, respectively:

CD ¼ 24

Reþ 6

1þffiffiffiffiffiffiRe

p þ 0:4 ð8Þ

Re ¼dB u!�� ��qa

ð9Þ

where dB, u!, q, and a are the diameter of a ball, the

relative velocity, and the density and viscosity of air,

respectively.

2.2 Simulation condition

Figure 3 shows a schematic diagram of the planetary ball

mill. The Young’s modulus and Poisson’s ratio of the

barrel are 1000 MPa and 0.30, respectively. Other detailed

grinding conditions and material properties are listed in

Table 1. It is assumed that the time step is sufficiently

small such that no new contacts are generated in the course

of the motion of the balls in that time step. In calculating

the contact forces, the balls are allowed to overlap.

Therefore, every such overlapping contact is modeled by a

pair of spring-dashpots in both the normal and shear

directions. All of the balls have the same size, which is

invariant under collisions. A computer program called

SAMADII, developed by Metariver Technology (2016),

was used to calculate the equations of motion of the balls in

the planetary ball mill. The time step for the simulation was

0.02 s, and 15 total seconds were simulated. It took

approximately 85 h to calculate the motion of MLCCs in

the barrel using a single GPU. In order to analyze the

dependency of the grinding performance on the height of

the barrel where the balls rotate, the barrel was divided into

three regions, as shown in Fig. 4. The bottom, middle, and

top regions were defined from 0 to 5, 30 to 35, and 60 to

65 mm from the bottom of a barrel, respectively.

3 Results and discussion

3.1 Simulated motion of MLCCs in the barrelaccording to barrel revolution

Figure 5 shows the simulated motion of the MLCCs in the

barrel for a single barrel revolution in a planetary ball mill.

It shows that the MLCCs within the barrel are displaced in

the direction of the revolution (counterclockwise), which is

the most common feature of the charge dynamics in the

mill (Mishra 1995).

Fig. 3 Schematic diagram of the planetary ball mill

Table 1 Grinding conditions and material properties

Design variable Value Unit

Barrel diameter 150 mm

Barrel depth 65 mm

Revolution radius 230 mm

Revolution speed 170 rpm

Rotational speed 68 rpm

MLCC diameter 1.44 mm

Density of MLCC 9392 kg/m3

Number of MLCCs 32,861 EA

Media diameter 3 mm

Density of media 3954 kg/m3

Number of media 4830 EA

Fig. 4 Top, middle, and bottom regions in a barrel

Microsystem Technologies (2018) 24:4465–4470 4467

123

3.2 Simulated grinding performance accordingto friction coefficients of the barrel

Figure 6 shows the average linear velocity, impact, and

shear energy according to the coefficients of friction of the

barrel. While the shear energy increases with an increasing

coefficient of friction, both the linear velocity of the

MLCCs and the average impact decrease. To analyze the

cause of these results, both a path-line of a single MLCC

and the linear velocity distribution of the MLCCs were

calculated and compared when the friction coefficients of

the barrel were 0.2 and 0.8, respectively. Figure 7 repre-

sents the simulated path-line of a single MLCC in a barrel

with a friction coefficient of 0.2 and 0.8, respectively. A

single MLCC in the barrel with a low friction coefficient

moves only in the revolution direction of the barrel

(counterclockwise) because there is little friction between

an MLCC and the barrel wall. However, an MLCC in the

barrel with a high friction coefficient moves not only in the

revolution direction of the barrel (counterclockwise), but

also in the rotation direction of the barrel (clockwise). In

particular, movement of the MLCC in the direction of

rotation occurs due to the frictional force of the barrel when

the MLCCs are near the wall of the barrel. As the friction

coefficient of the barrel increases, the linear velocities of

the MLCCs and their distribution are affected by the

rotation of the barrel. Figure 8 presents the simulated linear

velocity distribution results of the MLCCs in barrels with

friction coefficients of 0.2 and 0.8, respectively. When the

friction coefficient of the barrel was 0.8, the average linear

velocity of the MLCCs was 4.93 m/s, which was 1.6%

lower than that found with a coefficient of 0.2, while the

standard deviation of the linear velocity was increased by

44% to 0.36 m/s. These results mean that the decrease in

the linear velocity of the MLCCs decreases the number and

energy of impacts and increases the shear energy due to the

increase of the relative velocity between the two balls.

Therefore, increasing the friction coefficient of the barrel

increases the shear energy and reduces the damage caused

by impacts.

3.3 Experimental verification

The wear height is proportional to the applied force, the

velocity in the tangential direction, and the time step

(Powell et al. 2011). Therefore, the wear height is

expressed as a function of shear energy. We verified the

simulated result by measuring the wear height of MLCCs

and compared it with the simulated shear energy. Grinding

experiments were carried under identical conditions except

for the friction coefficient of the barrel. Four barrels were

made of polycarbonate using three-dimensional printing

technology, each printed to have a different surface

roughness. The grinding experiment was performed over a

period of 20 min. When the experiment time was short, the

wear height of MLCCs was too small to measure. On the

contrary, when it was long, the grinding efficiency

decreased due to loss by frictional heat. The wear height

was defined as the difference in the thickness of an MLCC

before and after grinding experiments, as shown in Fig. 9.

The wear height of 32 MLCCs were measured before and

Fig. 5 Motion of MLCCs in the barrel according to barrel revolution

Fig. 6 a Average velocity and b impact and c shear energy of MLCC

according to friction coefficient of the barrel

4468 Microsystem Technologies (2018) 24:4465–4470

123

after the grinding experiments and was compared with the

calculated shear energy, as shown in Fig. 10. The simu-

lated shear energy results show a similar trend to the

measured wear height results according to the friction

coefficient of the barrel.

4 Conclusions

We investigated the effect of the friction coefficients of a

barrel on the grinding performance of MLCCs in a plane-

tary ball mill using DEM, with the Voigt model repre-

senting collisions between two MLCCs. The linear

velocity, impact, and shear energy according to the friction

coefficients of the barrel were calculated, and the trajectory

of a single MLCC in the barrel was analyzed. An increase

of the friction coefficient of the barrel not only increases

the shear energy, but also decreases the impact. Finally,

grinding tests using four barrels with different friction

coefficients show that the wear height of MLCCs, which is

dependent on the shear energy, increases with an increase

of the coefficient of friction of the barrel. This research will

contribute to increases in the grinding efficiency in future

manufactured planetary ball mills.

Fig. 7 Path-line of single

MLCC in a barrel with friction

coefficient of a 0.2 and b 0.8

Fig. 8 Velocity distributions of

MLCCs in a barrel with a

friction coefficient of a 0.2 and

b 0.8

Fig. 9 Defined of the wear

height of an MLCC

Fig. 10 Simulated shear energy and measured wear height according

to friction coefficient of the barrel

Microsystem Technologies (2018) 24:4465–4470 4469

123

References

Cleary PW, Morrisson R, Morrell S (2003) Comparison of DEM and

experiment for a scale model SAG mill. Int J Miner Process

68(1–4):129–165

Cundall PA, Strack ODL (1979) A discrete numerical model for

granular assemblies. Geotechnique 29:47–65

Kwapinska M, Saage G, Tsotsas E (2006) Mixing of particles in

rotary drums: a comparison of discrete element simulations with

experimental results and penetration models for thermal pro-

cesses. Powder Technol 161(1):69–78

Liu X, Ge W, Xiao Y, Li J (2008) Granular flow in a rotating drum

with gaps in the side wall. Powder Technol 182:241–249

Metariver Technology (2016) Samadii user manual. 4–5

Mio H, Kano J, Saito F, Kaneko K (2002) Effects of rotational

direction and rotation-to-revolution speed ratio in planetary ball

milling. Mater Sci Eng 332:75–80

Mishra BK (1995) Charge dynamics in planetary mill. KONA

13:151–158

Mishra BK, Rajamani RK (1992) The discrete element method for the

simulation of ball mills. Appl Math Model 16:598–604

Mori H, Mio H, Kano J, Saito F (2004) Ball mill simulation in wet

grinding using a tumbling mill and its correlation to grinding

rate. Powder Technol 143–144:230–239

Powell MS, Weerasekara NS, Cole S, LaRoche RD, Favier J (2011)

DEM modeling of liner evolution and its influence on grinding

rate in ball mills. Miner Eng 24:341–351

Xu Y, Xu C, Zhou Z, Du J, Hu D (2010) 2D DEM simulation of

particle mixing in rotating drum: a parametric study. Particuol-

ogy 8:141–149

Publisher’s Note Springer Nature remains neutral with regard to

jurisdictional claims in published maps and institutional affiliations.

4470 Microsystem Technologies (2018) 24:4465–4470

123