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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
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
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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
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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
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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
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