1. Transactions of NAMRI/SME, Vol. 36, 2008, pp. 57-64
TRANSACTIONS OF NAMRI/SME Discrete Element Modeling of Micro-
Feature Hot Compaction Process
authors
P. CHEN
J. NI
University of Michigan
Ann Arbor, MI, USA
abstract
In the forming of porous microfeatures using a hot compaction process, it is costly
and time consuming to determine a proper experiment setting (force, temperature,
and time) by trial and error. Product qualities, such as mechanical strength and
porosity, are significantly affected by the setting of those process variables. To
analytically study the effect of force and temperature on particle bonding strength
and porosity, a discrete element model for pressure-assisted sintering was developed
for the forming of porous microfeatures. The model was first validated with
experimental results for a unit problem (two particles). It was then expanded for a
10-particle channel hot pressing problem. With this model, it was feasible to
conveniently assess the effects of force and temperature on the particle bonding
strength and shrinkage, which then gave insight on deciding a proper process setting
before actual operations.
terms
Porous Microfeatures
Pressure-Assisted Sintering
Hot Compaction
Discrete Element Modeling
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3. DISCRETE ELEMENT MODELING OF MICRO-FEATURE
HOT COMPACTION PROCESS
Peng Chen and Jun Ni
Department of Mechanical Engineering
University of Michigan
Ann Arbor, Michigan
KEYWORDS INTRODUCTION
Pressure Assisted Sintering, Network Model, Porous micro-features with high aspect ratio
Hot Compaction, Porous Micro-Features are becoming more and more important in the
modern industry, especially for high efficiency
heat transfer applications (Liter and Kaviany
ABSTRACT 2001). As discussed in our previous studies
(Chen et al. 2007), hot compaction process is
In the forming of porous micro-features using one of the most promising ways to produce such
hot compaction process, it is costly and time- features, and its capabilities have already been
consuming to determine a proper experiment experimentally demonstrated. However, it is very
setting (force, temperature and time) by trial and costly and time-consuming to determine a
error. Product qualities, such as mechanical proper experiment setting (force, temperature
strength and porosity, are significantly affected and time) by trial and error. As investigated by
by the setting of those process variables. Chen et al., product qualities (such as
mechanical strength and porosity) are
In order to analytically study the effect of the significantly affected by the setting of the
force and temperature on the particle bonding process variables (Chen et al. 2007). Therefore,
strength and porosity, a discrete element model in order to reduce the time and efforts spent on
for pressure assisted sintering was developed trial and error in physical experiments, this study
for the forming of porous micro-features. The aims to develop a computational model to
model was first validated with experimental analytically study the effect of process variables
results for a unit problem (two particles). And on the particle bonding strength and porosity.
then it was expanded for a 10-particle channel
hot pressing problem. With this model, we could Hot compaction processes combine the
conveniently assess the effects of force and simultaneous application of pressure and
temperature on the particle bonding strength temperature, which is also termed as pressure
and shrinkage, which then give us insight on assisted sintering. During sintering, particles are
deciding a proper process setting before the bonded together by atomic transport events. The
actual operations. driving force for sintering is a reduction in the
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4. system free energy, which is achieved by sintering (Hwang and German 1984; Parhami
reduction of surface curvatures and elimination and McMeeking 1998), which is dedicated to
of surface areas (German 1994). Initially, a grain simulate the diffusion and mass transport
boundary is formed at the contact between mechanisms near the particle surface without
neighboring particles. Atoms travel along this grain growth; (2) models for late intermediate
boundary and along the particle free surface to and final stage sintering, which is focused on the
the neck regions. modeling of grain growth and pore shrinkage
(Hassold et al. 1990; Tomandl and Varkoly
Starting from late 1950s, numerous 2001). The model for final stage sintering is
researchers have studied the computer especially important for ceramic sintering since
simulation of sintering processes (German large shrinkage is often encountered in this
2002). More than 1,000 publications can be case. Since this study is only concerned with the
found on this topic. According to the different initial and early intermediate stages of sintering,
scales of constitutive modeling, the existing only the first model will be discussed in detail in
computer models for sintering could be divided this work.
into three classes: (1) continuum model
(Olevsky 1998; Delo et al. 1999; Sanchez et al. During the initial and early intermediate stages
2002); (2) discrete model (German and Lathrop of sintering, necks between neighboring
1978; Parhami and McMeeking 1994); (3) particles grow up; and no densification occurs.
molecular dynamics model (Zavaliangos 2002; Therefore, a mathematical expression of neck
Raut et al. 1998). However, most research growth as a function of temperature and time will
efforts were on the modeling of free sintering be sufficient to model the free sintering process
process, where no external mechanical loading (no external load) (Hwang and German 1984). In
was considered. Continuum models are most the case of hot pressing (pressure assisted
suitable for free sintering process, and they also sintering), to accurately simulate the particle
require accurate material testing in high behavior under the influence of both elevated
temperature condition, which is difficult to temperature and external pressure, sintering
perform. In addition, no microstructure stress induced diffusion and external pressure
information could be obtained from continuum induced diffusion should be integrated together.
simulation. Molecular dynamics method is highly An efficient way to achieve this goal is to
accurate but is difficult to implement for our combine the existing neck growth model with
problem due to time and length scale limitations. Discrete Element Model, which is called network
In addition, a real particle usually has a model by some researchers (Parhami and
polycrystal structure, which imposes another McMeeking 1994; Parhami and McMeeking
difficulty in the MD modeling, that is, how to 1998), or truss model (Jagota and Dawson
effectively define the grain boundary in a single 1988). In this model, every particle center is
particle. Relatively speaking, discrete models represented by a node and every contact
stand out to be a sound candidate for the between neighboring particles by an element.
simulation of hot compaction of powders into Figure 1 is a two dimensional representation of a
micro-features. pair of particles bonded together at a neck. A
relative axial velocity of the particles centers is
In this study, a discrete element model for the consequence of atomic flux from the
pressure assisted sintering was developed for interparticle grain boundary to the free surface.
the forming of porous micro-features. The model This process, coupled to mass transport on the
was first validated with experimental results for a free surface, leads to the development of grain
unit problem (two particles). And then it was boundary area at the contact and the generation
expanded for a 10-particle channel hot pressing of thermodynamically induced normal stresses
problem. on the grain boundary.
DISCRETE ELEMENT MODELING OF FORMULATION OF THE NUMERICAL MODEL
SINTERING PROCESS FOR HOT COMPACTION (NETWORK
MODEL)
Generally speaking, there are two categories
of particle-level models for sintering: (1) models Based on the network model mentioned
developed for initial and early intermediate stage above, a numerical model was developed which
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5. could predict the pressure assisted sintering (hot −
Qg
compaction) behavior of a particle system as a 8δ g D g 0 e Rs T
Ω ⎧σ γ ψ ψ ⎫
vn = ⎨ 2 − 4 [4 R(1 − cos ) + r sin ]⎬
function of temperature, external force and time. kT ⎩ r r 2 2 ⎭
(3)
x The second terms on the right hand side of
Eq. (1) and Eq. (3) drive free sintering.
According to Swinkels and Ashby (1981), the
Fn1 ,Vn1 Fn2 ,Vn2 values of the above coefficients for copper are
2r
shown in Table 1.
R
ψ TABLE 1. MATERIAL PROPERTY OF COPPER
(SWINKELS AND ASHBY 1981).
FIGURE 1. 2D REPRESENTATION OF A TWO-
PARTICLE NECK GROWTH MODEL. Material constant Copper
3
δ g Dg 0 (m /s) 5.12 × 10
-15
γ (J/m ) 2
1.72
Only the initial stage sintering was considered
in our study, in which case the dominant mass Q g (J/mole) 105000
transport mechanisms are surface diffusion and R s (J/mole) 8.31
grain boundary diffusion. As shown in Eq. (1), Ω (m3) 1.18 × 10
-29
the neck growth rate equation was derived k (J/Kelvin) 1.38 × 10
-23
based on neck growth rate equation proposed ψ
by Parhami and McMeeking (1998) and the 146°
diffusion coefficient equation used by Exner
(1979).
Before pressure After pressure
−
Qg assisted sintering assisted sintering
8 Rδ g D g 0 e RsT
Ω⎧ σ γ ψ ψ ⎫
r= ⎨− 3 + 5 [ 4 R(1 − cos ) + r sin ]⎬
kT ⎩ r r 2 2 ⎭
(1)
1
where δ g is the effective grain boundary Fn , v1 , x1
n n
thickness, Dg 0 is the maximum grain boundary
diffusion coefficient (at infinite temperature), γ is
the surface energy per unit area, Qg is the Neck
2
activation energy of grain boundary diffusion, Fn , v , x 2 2
n n
Rs is the gas constant, Ω is the atomic volume
and k is Boltzmann’s constant. T denotes
absolute temperature (Kelvin). As shown in
FIGURE 2. ILLUSTRATION OF THE TWO-
Figure 1, r is the neck radius, ψ is the dihedral
PARTICLE PRESSURE ASSISTED SINTERING
angle at the neck and R is particle radius. σ is MODEL.
the normal stress on the contact.
Fn1 F2 Assuming that the neck growth rate and axial
σ= = − n2
πr 2 πr (2) velocity remains the same in a very small time
step, the axial displacement of the particle and
Similarly, the axial velocity of the particle was neck radius are updated using central finite
derived based on the equation used by Parhami difference method (Cundall and Strack 1979).
and McMeeking (1998):
x N +1 = x N + (v n ) 1 Δt
N+
2 (4)
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6. where Δt is the critical time step and is found to
-7
be around 10 s for our case (Martin et al.
2002).
Modeling of Unit Problem
For a simplified two-particle model as shown
on the left of Figure 2, tangential force and
moment are ignored. A numerical model for
pressure assisted sintering was developed using
MATLAB based on Eqs. (1)–(4) and Table 1.
The right diagram in Figure 2 is an illustration of
calculation result for hot pressing in the format of
neck growth.
Modeling of Multi-Particle Problem with
Boundary Conditions
The discrete element model for cold FIGURE 3. ILLUSTRATION OF THE 10-PARTICLE
MODEL.
compaction developed by Cundall and Strack
(1979) is based on the original particle
A code was developed for this multi-particle
dynamics, where contacts between particles are
pressure assisted sintering problem using
not sustained. It is not well-suited, however, for
MATLAB. The step-by-step computing structure
application where the contacts undergo large
of the code is shown in Figure 4.
deformations and, once made, rarely break. In
our case (hot compaction after pre-press), the
particle assembly may be assumed to be in
equilibrium at all stages of the process (Jagota
and Dawson 1988), permitting solution for
velocities implicitly, as discussed below. Based
on the study of Fleck (1995) and Heyliger and
McMeeking (2001), shearing tractions between
particles was neglected, which was found to play
a minor role in the particle assembly, especially
after pre-press.
Particle packings were treated as frameworks
of links that connect the centers of particles
through inter-particle contacts. The behavior of
each link in the framework was based on unit
problems for the interaction between individual
spheres as described in the previous section. As
shown in Figure 3, a network model for the
pressure assisted sintering of 10 particles in a V-
shape channel was developed. The angle
between two V-channel walls was 60°. The FIGURE 4. CALCULATION SCHEME FOR MULTI-
particle diameter was 200 µm. Each particle was PARTICLE PRESSURE ASSISTED SINTERING
assigned a number as shown in Figure 3. PROBLEM.
Identical force was applied on particles 7, 8, 9,
and 10 to account for the compression load. The In the initialization step, constants such as
interaction force between each particle pair was material properties and temperature are defined.
obtained via frame analysis. Geometry and dimension of the channel and
particles are defined in the assembly step. The
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7. coordinates of the particles are first defined in RESULTS AND DISCUSSION
global coordinates, and then transformed into
local coordinates via rotation matrix for the ease Unit Problem and Validation
of computational operation. Before the hot
compaction, the pre-pressed particles will have To validate the above numerical model, the
an initial neck radius due to elastic or plastic simulation results (r/R and shrinkage) were
deformation, which was solved using the original compared with the experiment results provided
discrete element model proposed by Cundall by Exner (1979), as shown in Figure 5. In
and Strack (1979). The subsequent five steps Exner’s experiments, 20 large copper spheres
compose an iteration loop, which solves the were sintered at 1027°C without any external
pressure assisted sintering process continuously force loading. In Figure 5, the relationship
until a pre-defined sintering time is reached. The between neck radius / particle radius ratio (r/R)
approaching velocity between every two and the relative center approach ([X0 – XN]/R)
contacting particles was calculated using Eq. (which is the ratio between the approaching of
(3), which is stored in an approaching velocity two particle centers and their original distance
matrix as shown in Eq. (5). and is an indication of the shrinkage of the
particle system) were presented. Simulation
⎡Vn11 Vn12 .... Vn1n ⎤
results agreed well with the experiment results,
⎢Vn ⎥ (5) and the predicted trend of the evolution of
Vn = ⎢ 21 ⎥ shrinkage as a function of r/R matched well with
⎢ ⎥
⎢ ⎥ the experimental observations.
⎣Vnn1 Vnnn ⎦
where Vnij denotes approaching velocity on
particle i caused by particle j. The matrix was
constructed this way such that the absolute
velocity of the particle could be assembled
conveniently in the velocity summation step with
only on matrix operation as shown in Eq. (6).
′
⎡Vn11 Vn12 .... Vn1n ⎤ ⎡θ11 θ12 .... θ1n ⎤
⎢Vn ⎥ ⎢ ⎥
V = ⎢ 21 ⎥ * cos ⎢θ 21 ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎣Vn n1 Vnnn ⎦ ⎣θ n1 θ nn ⎦
(6)
FIGURE 5. COMPARISON BETWEEN SIMULATION
where θ ij denotes the angle between local y axis AND EXPERIMENT RESULTS.
(orthogonal to the axial direction) and the vector
direction on particle i caused by particle j.
The displacement of each particle is updated
using central finite difference method [Eq. (4)]
with forced boundary conditions imposed by the
V-channel. At the end of each iteration, the neck
radius is updated using Eq. (1). Post-processing
step store and plot out data.
Simulations were run for the above problem
with a force of 10 N for eight minutes of pressure
assisted sintering at different temperatures.
Each simulation took about five hours of FIGURE 6. EFFECTS OF TEMPERATURE AND
computational time on a Sun Ultra 20 (1.8 GHz) FORCE ON r/R.
workstation.
After validation, a further study of the pressure
assisted sintering process was performed using
the numerical model. Figure 6 shows the effects
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8. of temperature and compaction force on the contributing factor; the interaction force caused
pressure assisted sintering process. The by surrounding particles also affected the neck
following conclusions could be drawn from this size. For example, the axial force between
figure: (1) the rate of neck growth was very low particle 6 and particle 8 was the highest, but
at a low temperature (25-150°C), in which case their neck was not the largest. The largest neck
neck did not grow much even if a compaction occurred at the particle 2 and 5 interface, which
force was applied; (2) an external compression was more than twice the size of other necks. But
force significantly increased the neck growth its growth rate after the first 10 seconds was
rate at a higher temperature range (300- also the lowest comparing to other necks. A
1000°C), which was due to the fact that the review of Eq. (1) reveals that the neck growth
3
material was softened in this temperature range. rate is proportional to 1/r , which results in a
Especially in the cases of 1 N at 700°C and 0.1 lower growth rate at a larger neck size.
N at 850°C, there was a dramatic increase in the
neck growth rate. Figure 8 shows the relative center approach of
the particles during pressure assisted sintering
at 350°C. While most particles were
Multi-Particle Problem with Boundary approaching each other, some particles were
Conditions departing from others. However, the general
trend was that all the particles were shrinking
As shown in Figure 7, at an isothermal into the center of the particle packing. In this
temperature setting (350°C), the neck growth of case, particle 5 became the center of
different particle pairs were different. The growth approaching. Similarly to the neck growth, the
of the neck was very rapid in the first 30 lower the interaction force, the slower the
seconds, after which the growth slowed down approaching.
dramatically and appeared as seemingly linear
increase over the time.
FIGURE 8. RELATIVE CENTER APPROACHING
DURING PRESSURE ASSISTED SINTERING
FIGURE 7. NECK RADIUS DURING PRESSURE (350°C, 10 N).
ASSISTED SINTERING (350°C, 10 N).
Figures 9 and 10 show the neck growth and
Depending on the axial interaction force relative center approaching of the network
between two particles, the size of the formed model at 422°C. As the temperature increased,
neck was different. Generally speaking, the the neck size and approaching speed increased
higher the axial force, the larger the neck is. For as well. But the general growth trend remained
example, the neck between particle 7 and the same.
particle 8 was the smallest, since the axial force
between them is the lowest. However, the axial
force between each given pair was not the only
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9. The numerical model developed in this study
effectively captures the atomic diffusions caused
by both pressure and heat, and provides means
to extend this model for more particles with the
consideration of boundary conditions, which
could be a convenient tool for engineers and
scientists to study the effects of force,
temperature and time on the quality of the
formed micro-features.
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FIGURE 9. NECK RADIUS DURING PRESSURE
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