# DEM simulation of the powder application in powder bed fusion

Vasileios Angelidakis  Michael Blank  Eric J. R. Parteli  Sudeshna Roy  Daniel
Schiochet Nasato  Hongyi Xiao  and Thorsten Pöschel

##  1 Introduction

The packing behavior of powders is significantly influenced by various types
of inter-particle attractive forces, including adhesion and non-bonded van der
Waals forces yu1997modelling (1, 2, 3, 4, 5, 6). Alongside particle size and
shape distributions, the inter-particle interactions, in particular frictional
and adhesive forces, play a crucial role in determining the flow behavior and
consequently the packing density of the powder layer. The impact of various
types of attractive forces on the packing density of powders with different
materials and particle size distributions remains largely unexplored and
requires further investigation. Accurately comprehending these effects through
experiments while considering specific particle size distributions and
material properties poses significant challenges. To address these challenges,
we employ Discrete Element Method (DEM) simulations to characterize the
packing behavior of fine powders. We can demonstrate quantitative agreement
with experimental results by incorporating the appropriate particle size
distribution and using an adequate model of attractive particle interactions.
Furthermore, our findings indicate that both adhesion, which is modeled using
the Johnson–Kendall–Roberts (JKR) model johnson1971surface (7), and van der
Waals interactions are crucial factors that must be taken into account in DEM
simulations.

Characterization of the packing structure is essential to achieve simulation-
driven optimization of the layer quality. While local packing density is
commonly employed to assess granular packing, relying solely on this parameter
is inadequate for identifying structural defects in disordered packing
richard2020predicting (8). This is due to the fact that numerous packing
arrangements can exhibit the same local density. Conversely, local structural
anisotropy is a fundamental characteristic of non-crystalline packing and
plays a significant role in crucial mechanical properties within disordered
packings, such as jamming rieser2016divergence (9), plasticity
richard2020predicting (8), and shear band formation xiao2020strain (10, 11,
12). We use local structural anisotropy as a measure to characterize the
structures of the deposited particles. This approach is threshold-free and
provides a meaningful distribution that accurately reflects the variations in
the packing structure within a deposited powder layer. It holds particular
relevance in discerning structural differences between homogeneous packing
observed in non-cohesive powders and the heterogeneous packing tendencies
prevalent in highly cohesive powders.

The heat transfer within the powder bed is significantly impacted by the
quality of the final powder layer, including aspects such as the packing
density and surface profile zhao2023multiscale (13). It is worth noting that
many DEM simulations commonly disregard the influence of temperature on inter-
particle interaction forces. Nevertheless, it is widely recognized that
temperature can have a substantial impact on the mechanical behavior of
particle interactions. The temperature dependence of viscosity is one of the
most important variables for polymers, which is given by an Arrhenius type
equation glasstone1941theory (14, 15). As the temperature approaches higher
values, nearing the melting point, the particles within a stressed state
experience a reduction in stiffness. Consequently, this leads to increased
deformation and larger overlaps between particles due to the compressive
forces acting upon them. Luding et al. luding2005discrete (16) introduced a
new discrete model for the sintering of particulate materials with a
temperature-dependent elastic modulus.

The role of particle shape coupled with its thermal properties is hardly ever
explored in the literature and thus is a subject of great potential. We
propose an extension of the Discrete Element Method (DEM) that incorporates
heat transfer capabilities by combining a multisphere algorithm with a thermal
discrete particle model, enabling the simulation of non-spherical particles.
To showcase the applicability of this model, we conduct simulations of a
powder spreading process utilizing irregularly shaped Polyamide 12 (PA12)
powder particles. This demonstration illustrates the effectiveness of the
proposed model in capturing the behavior of non-spherical particles in real-
world scenarios.

Improving the quality of the layer during powder spreading can be achieved by
various means. In addition to exploring the thermal, cohesive, and overall
mechanical properties of the powder material itself, an optimization of the
steps of the powder application process can result in improved layer
characteristics. To this end, the efficiency of a vibrating recoating
mechanism has been explored with respect to the layer density and surface
roughness.

##  2 Particle-based tool for powder spreading

The first particle-based numerical tool was developed by parteli2013simulation
(17) parteli2013simulation (17) based on an existing DEM solver
kloss2012models (18) to model the dynamics of geometrically complex particles
subjected to dynamic boundary conditions. Snapshots of the simulation setup
with complex shaped particles and dynamic boundary conditions are shown in
Figure 1.

Figure 1: Snapshots of a simulation of particles with complex geometric
shapes. Particles constructed with the multisphere method are inserted into
the system with dynamic boundary conditions that mimic the device used in
additive manufacturing.

This setup mimics the device used for powder spreading in additive
manufacturing. In this numerical setup, the moving boundaries (walls) are
modeled by triangular meshes, which can be imported into the solver. The
device consists of a rake for powder application (which moves from left to
right in Figure 1) and a building tank (central area) placed on top of a
vertically adjustable platform. The simulation starts by releasing about
$6500$ particles from a short distance above the platform, as shown in Figure
1(a). After falling under gravity, the settled particles are transported on
the platform as the rake moves from left to right, as shown in Figure 1(b-d).
This small setup helps us to address several questions relevant to the real
powder spreading process, such as the role of particle shape and size
distribution on the flowability of the powder within the device. This, in
turn, allows us to understand the surface profile and packing density of the
powder bed.

In 19, parteli2016particle (19) parteli2016particle (19) established a real
scale setup for the first time to simulate powder spreading using a roller as
the spreading device, as shown in Figure 2(a). The numerical tool based on the
Discrete Element Method (DEM) accounts for a realistic description of inter-
particle forces, particle size distributions, and complex geometric shapes of
the powder particles, which play a major role in the static and dynamic
characteristics of granular systems. With this numerical setup, the authors
predicted the powder packing and surface roughness profile dependent on the
recoating velocity and the powder particle size distribution.

##  3 Geometrically complex particles

The first step to achieve reliable numerical simulations of the powder
spreading application in additive manufacturing is the accurate representation
of complex geometric shapes of the particles. While preliminary studies are
done by parteli2013using (20) parteli2013using (20, 17), this representation
is first accomplished by parteli2016particle (19) parteli2016particle (19) in
the context of powder spreading for powder bed fusion using the multisphere
method, which consists of building clumps of spherical particles to model the
complex shape of the target. Within the Discrete Element Method (DEM), there
are various simulation techniques available for modeling non-spherical
particles. One of the oldest and most versatile techniques is the multi-sphere
approach, which involves rigidly connecting a group of spheres to create
irregular particles abou2004three (21, 22, 23). In the multisphere approach,
each individual subsphere serves the purpose of contact detection, enabling an
accurate interaction between particles. However, when it comes to calculating
inertial characteristics and integrating particle motion, the entire
collection of spheres is considered as a single particle. This simplification
reduces computational complexity and facilitates efficient simulations. The
fidelity of the multisphere particle’s morphology, or how closely it
approximates the shape of the target particle, is determined by the number and
size of the spheres used in its construction. A higher number of spheres or
smaller sphere sizes yield greater morphological fidelity, providing a better
representation of the target shape. However, this also increases the
computational cost associated with interactions between neighboring particles,
as more contact points need to be considered. Therefore, the choice of the
number and size of spheres in a multisphere particle involves a trade-off
between morphological accuracy and computational efficiency. Researchers must
carefully consider these factors based on the specific requirements of their
simulation to strike an appropriate balance. Overall, the multisphere approach
offers a flexible and widely used method for modeling non-spherical particles
within DEM simulations, providing a compromise between accuracy and
computational efficiency.

(a)

(b)

Figure 2: (a) (Top) Light microscope images of commercially available PA12
powder particles (first row) and corresponding particle models using the
multisphere method (second row) for implementation in the DEM. (Bottom)
Cumulative distribution and volume density distribution as a function of the
particle diameter. (b) Snapshot of the simulation indicating the main elements
of the powder application process.

Figure 2(a) shows some examples in which target shapes were obtained from SEM
images of commercially available PA12 powder as described in
parteli2016particle (19). The translational and rotational motion of the
multisphere particles is driven by the forces acting on the constituent
spheres, which include gravity, inter-particle forces, and the forces due to
particle and wall interactions. The small size particles ($1-100\,\upmu{}$m)
involved in the process require attractive interaction forces to be modeled
and incorporated into the DEM. We developed a particle-based model which takes
into account contact forces, as well as inter-particle attractive
interactions. The contact forces are modeled using the Cundall and Strack
model cundall1979discrete (24) considering elastic and dissipative forces in
the normal collision direction and in the tangential direction. This model was
extended by Parteli et al. parteli2014attractive (25) to incorporate
attractive particle interaction forces. This improved DEM model takes into
account bonded adhesion contacts as well as non-bonded van der Waals forces.
The simulation predictions of the solid fraction of powders covering a wide
range of size distributions were quantitatively in agreement with the
experimental results parteli2014attractive (25).

Figure 2(b) shows the simulation snapshot indicating the main components of
the powder spreading setup implemented for numerical simulation using the DEM
model. The roller moves in the positive $x$ direction with a translational
velocity because it rotates in a counterclockwise direction with a constant
rotational velocity.

###  Influence of particle size

The packing behavior of the deposited powder layer is strongly dependent on
the transport dynamics of the powder over the substrate. In ideal powder
spreading conditions, the goal is to enhance the production speed while
maintaining flat, densely packed powder layers. To investigate this behavior,
we analyzed the roughness of the powder bed as a function of the roller
velocity $V_{R}$ as shown in Figure 3. We observe a clear increase in the bed
roughness with the increase in the roller’s translational velocity as
represented by the blue squares in Figure 3.

Figure 3: Roughness of the deposited powder layer as a function of the
translational velocity of the roller $V_{R}$ parteli2016particle (19, 26).

To further analyze how the particle size distribution affects the roughness of
the bed, we modified the size distribution shown in Figure 2(b) by removing
all particles of diameter smaller than $60\,\upmu$m. The surface roughness
resulting from particles of size $d\geq 60\,\upmu$m is shown as a function of
the roller’s translational velocity, represented by the circles in Figure 3.
Overall, we achieve a smaller roughness with the larger particle size
distribution, and the roughness increases in the same fashion with the
roller’s velocity as it does for the original size distribution.

Note that the attractive inter-particle forces (adhesion and non-bonded van
der Waals forces) lead to the formation of large agglomerates in the powders
associated with the blue squares in Figure 3. For very small particles
$<30\,\upmu$m, the gravitational forces are dominated by the cohesive forces,
and thus, the particles are transported as irregular clumps and deposited on
the surface. This results in a higher roughness than the original size
distribution of the powder. Thus, a way to decrease the surface roughness of
the deposited powder layers in additive manufacturing is to modify the
particle size distribution of the powder by filtering out the smaller
particles from the system.

###  Influence of particle shape

The role of real particle shape on the powder layer quality has been explored
in detail by Nasato et al. nasato2020influence (27). This is done by
multisphere reconstruction of 3D images of the powder particles obtained using
the CT-REX tomograph as shown in Figure 4. Samples from commercial polyamide
11 (PA11), modified PA11, polyamide PA12, and reconstructed images from PA12,
named S1, S2, S3, and S4, respectively, are considered for this study.

Figure 4: Multisphere reconstruction of particles used in additive
manufacturing. From left to right: original template (SEM image), all spheres
before optimization, and optimized multisphere representation. (a) S1 sample
(PA11), (b) S2 sample (PA11 powder rounded through precipitation), (c) S3
sample (PA12), and (d) S4 sample (PA12 reconstructed from SEM images)
nasato2020influence (27, 28).

The surface roughness is measured for all four samples for different recoating
velocities of the powder spreading tool. Interesting to note from Figure 5(a)
that for every recoating velocity, the surface roughness is the highest for
either the spheres or the sample S2 (rounded PA11). Samples S1, S3, and S4
have similar results of roughness within the standard deviation. Note that
samples S3 and S4 have a smaller roughness than sample S1, particularly for
the higher recoating velocity of 250 mm/s. Samples S3 and S4 have the lowest
value of aspect ratio and elongation ratio that is measured. Thus, it is
speculated that the smaller surface roughness of S3 and S4 samples is related
to the alignment of the particles during the powder spreading process, leading
to a smoother surface. An analysis of the packing density results for the
different powder samples is shown in Figure 5(b). For up to recoating velocity
of 200 mm/s, samples S1 to S4 achieve higher packing densities than the sphere
sample. The sample using spheres reveals a higher packing density only for the
high recoating velocity of $250$ mm/s. This is explained by the better
flowability of the spheres at high velocities compared to the non-spherical
particles. Thus, a competitive mechanism influences the quality of the powder
layer in the recoating process. Spherical particles generally have better
flowability and are thus preferred for additive manufacturing applications.
However, this holds true only for high recoating velocities. For lower
recoating velocities, the elongated shape of the particles plays a dominant
role in maintaining higher packing densities.

(a)

(b)

Figure 5: (a) Roughness and (b) packing density of the deposited powder layer
as a function of the translational velocity of the roller $V_{R}$.

##  4 Attractive interactions between particles

It is crucial to consider not only repulsive contact forces but also
attractive forces arising from van der Waals interactions when modeling the
dynamics of powder particles accurately. Van der Waals forces are a type of
intermolecular force that results from fluctuations in electron distributions,
creating temporary dipoles and inducing attractive forces between particles.
In this section, our focus is on investigating the effects of quantum
mechanically based van der Waals forces and tensile forces arising from
surface adhesion on the macroscopic behavior of powder particles on the
packing structure of powder. While van der Waals forces are generally
associated with intermolecular interactions at the atomic or molecular level,
their influence can extend to macroscopic particles of small size range
parteli2014attractive (25).

###  Significance of attractive forces

In cooperation with experimental validation, it must always be verified that
the models used for DEM simulations lead to results that are consistent with
the physical reality schmidt2020packings (29). Therefore, the numerical tool,
which is extended to include a more detailed description of particle geometry
and the electrostatic inter-particle interactions, is first tested for
simulations of a granular system in a simple geometry by Parteli et al.
parteli2014attractive (25). They verified the bulk packing density of fine
glass powders in the diameter range $(4-52)\,\upmu$m by comparing the results
with experiments and DEM simulations. The numerical setup is demonstrated in
Figure 6(a), where silica glass beads are deposited in a rectangular box. We
adopt periodic boundary conditions in the $x$ and $y$ directions and a
frictional wall at $z=0$. The height of the box is considered large enough to
produce packings of depth larger than $30\langle d\rangle$. To validate our
models, we obtained qualitative agreement between the experimental and
numerical results, taking into account attractive forces for particle
interaction, which include adhesion force and non-bonded van der Waals forces.

(a)

(b)

Figure 6: (a) Numerical simulation of the powder packing of size distribution
corresponding to sample $i$ with radius varying from $10-55\,\upmu$m and $R$
representing the radius of the particles and (b) packing density as a function
of the average particle diameter.

We performed different sets of simulations assuming different particle
interaction force models, namely (i) viscoelastic interaction (no cohesion in
Figure 6(b)), (ii) adhesive and viscoelastic interaction (adhesion in Figure
6(b)), and (iii) full model, including viscoelastic, adhesive, and van der
Waals interaction (adhesion and non-bonded in Figure 6(b)). The results are
shown in Figure 6(b). The packing density obtained from the experimental
results was compared to the given sets of simulations. For the pure
viscoelastic interaction model, the obtained packing density is almost
independent of the mean particle size, $\langle d\rangle$, with a slight
tendency of increasing packing density for smaller particles. This may be
explained by the geometric effects of smaller particles filling the pore
space. The simulation results with the viscoelastic interaction models agree
with experimental results for large particle size, $\langle d\rangle\gtrsim
40\,\upmu$m but disagree for smaller particle size. Simulations incorporating
JKR-type adhesive forces reveal a decay in $\phi$ with decreasing $\langle
d\rangle$. A good agreement is obtained with the experiment for $\langle
d\rangle\gtrsim 20\,\upmu$m, but the data diverge for smaller particles.
Simulations with the full model, including viscoelastic, JKR-adhesive, and van
der Waals interaction, allowed us to reproduce the experimentally measured
packing fraction for the entire particle size interval
$4\,\upmu\mathrm{m}\lesssim\langle d\rangle\lesssim 52\,\upmu$m. Thus, we
conclude from this that, for a predictive simulation of fine powder behavior,
both adhesive and van der Waals forces are essential and should, thus, be
considered in DEM simulations. Neglecting any of these contributions in
simulations of fine powders may lead to unreliable results.

###  Effect of inter-particle cohesion on packing structure

Inter-particle cohesion plays a key role on the structure and density of the
deposited powder layer, and thus, in the overall quality of the end product
he2020linking (30, 31). Recently, we explored the structural anisotropy,
packing density, and surface profile roughness of the deposited powder layer
in spreading processes with varying cohesion using DEM simulations
roy2022local (32). The local density and surface roughness of the produced
layers were also quantified to identify the influence of cohesion on the
formation of surface texture. Remarkably, based on the statistics of the local
anisotropy and the surface roughness, we observe a synchronized transformation
of the structural anisotropy and the surface roughness profile with increasing
cohesion. In particular, the structure changes from homogeneous to
heterogeneous, the layer surface roughness increases, and its peaks change
shape, around $Bo\approx 10$, where $Bo$ is a measure of inter-particle
cohesion strength in dimensionless form. In the following subsections, we
highlight our main observations based on the aforementioned characteristics of
powder layer packings.

Figure 7(a) illustrates an exemplary deposited layer of highly cohesive
particles, characterized by the Bond number $(Bo\approx 30)$. On examination,
we observe a heterogeneous structure comprising both densely packed and
loosely packed regions. This variation in packing density highlights the
complexity and non-uniformity of particle arrangements within the deposited
layer. The presence of regions with varying packing densities within the
deposited layer of highly cohesive particles renders the overall packing
density inadequate as a sole characterization metric. In order to address the
challenge of characterizing the structural inhomogeneity in the deposited
layer, we propose a novel method that utilizes a threshold-free measure based
on the local structural anisotropy. The method of characterizing the
structural inhomogeneity using a threshold-free measure based on the local
structural anisotropy, measured by $Q_{k}$, and calculated from Voronoi
tessellation, is explained in roy2022local (32).

Figure 7: Characterizing the structural anisotropy in the deposited layer. (a)
Granular packing of the powder layer for $Bo\approx 30$. The points represent
the projections of the center points of the particles in the plane. The
corresponding 2D Voronoi tessellation is also shown. Also shown in (a) is a
schematic representation of the particle packing with the superimposed Voronoi
tessellation (blue) and Delaunay triangles (green). The vectors $C_{p}$ (red)
point from the center of the particles to the centroids of the Voronoi cells.
The calculated $Q_{k}$ for (b) $Bo=0$ and (c) $Bo\approx 30$, where the
Delaunay triangles are colored by the corresponding values $Q_{k}$.

As stated by Rieser et al. (2016) rieser2016divergence (9), in regions of
dense and homogeneous packing, the values of $Q_{k}$ tend to exhibit random
fluctuations, resulting in a Gaussian-shaped distribution. However, in regions
characterized by inhomogeneity and high anisotropy, the distribution of
$Q_{k}$ deviates from the Gaussian shape. In such cases, both highly positive
and highly negative values of $Q_{k}$ appear at the tails of the distribution.

Figure 8(a)

(a) (b)

Figure 8: (a) Probability density of normalized divergence of center-to-
centroid vectors for the quasi-2D packing of powder deposited for different
$Bo$. $Q_{k}>0$ regions are more densely packed than their surroundings;
hence, we call these regions overpacked. $Q_{k}<0$ regions are more loosely
packed than their surroundings and are therefore labeled underpacked. (b)
Standard deviations (red circles) and skewness (blue squares) vs. $Bo$. We
also compared the standard deviations and skewness of the distributions with
the divergence calculated from the condition without 2D projections (hollow
points) and the condition with 2D projections (solid points), as explained in
roy2023structural (33). Error bars reflect standard errors in three simulation
repetitions. The dashed lines indicate the standard deviation and the skewness
of the distribution $Q_{k}$ for $Bo=0$.

shows the distribution of $Q_{k}$ for different Bond numbers ($Bo$), providing
visual evidence of this behavior. The non-Gaussian shape observed in the
distribution indicates the presence of inhomogeneous regions with pronounced
anisotropy, where the particle arrangements deviate significantly from random
fluctuations. This analysis enables the characterization and differentiation
of regions with varying degrees of structural complexity, providing valuable
insights into the packing behavior as influenced by the Bond number. Referring
to Figure 8(b), we observe interesting transition behavior of the standard
deviation and skewness of the $Q_{k}$ distribution for different Bond numbers
($Bo$). Initially, up to $Bo\approx 10$, the standard deviation remains
relatively constant, indicating a consistent level of structural anisotropy.
Similarly, the skewness also exhibits minimal variation within this range.
However, beyond $Bo\approx 10$, both the standard deviation and skewness start
to exhibit a monotonic increase and decrease, respectively. This suggests a
significant shift in structural anisotropy for cohesive materials, with
$Bo\approx 10$ serving as a transition point. For higher cohesion values
$(Bo>10)$, particles have the ability to maintain larger voids during the
spreading process, resulting in a more pronounced anisotropic structure. The
observed fluctuations in the trends for higher $Bo$ values, even within
repeated simulations, can be attributed to the inherent high anisotropy of the
structures. These fluctuations highlight the complexity and variability of
particle arrangements in cohesive systems with enhanced anisotropy. By
analyzing the standard deviation and skewness of the $Q_{k}$ distribution, we
gain valuable insights into the transition in structural anisotropy for
cohesive materials and the impact of different Bond numbers on the packing
behavior.

##  5 Thermal and mechanical properties

This section aims to expand our understanding of the various aspects involved
in powder application and gain a more comprehensive insight into the process
dynamics by combining the thermal and mechanical effects of the process and
material parameters on the powder application process. To incorporate heat
transfer capabilities into our Discrete Element Method (DEM) model for
simulating non-spherical particles, we have extended the existing framework by
merging a multisphere algorithm with a thermal discrete particle model
ostanin2023rigid (34). The heat transfer for multi-sphere particles includes
conduction, convection, and radiation. Conduction occurs when a clump overlaps
with another clump or a boundary, while convection and radiation occur for the
exposed portion of a clump’s surface with the background. The total thermal
transfer rate is expressed as the sum of the heat flux from conduction,
convection, and radiation. To demonstrate the effectiveness of our extended
DEM model with heat transfer capabilities, we conducted simulations of a
powder spreading process using Polyamide 12 (PA12) powder particles of
irregular shapes.

###  Multi-sphere particle generation

In this study, multi-sphere particles were generated to approximate the shape
of Polyamide 12 (PA12) powder particles. To achieve this, an image-informed
particle generation procedure was implemented, utilizing scanning electron
microscopy (SEM) images of PA12 particles as a reference nasato2021influence
(35). The procedure involved extruding the SEM images into three-dimensional
(3D) representations, which served as the target geometry for generating
realistic multi-sphere particles. The generation of multi-sphere particles was
performed using a recently developed method based on the Euclidean transform
of 3D images. The open-source software CLUMP angelidakis2021clump (36) was
employed, which offers various techniques for the multisphere generation. For
adequately approximating the target particle shapes, a total of 10 sub-spheres
were used per particle. This choice of the sub-sphere count was determined to
strike a balance between accurately representing the target particle
morphology and maintaining computational efficiency. By employing the image-
informed particle generation procedure and the multi-sphere approach with 10
sub-spheres per particle, the study aimed to create realistic representations
of PA12 powder particles for subsequent simulations and analysis. The
different irregular particle shapes that are included in our studies are from
light microscope images of commercially available PA12 as illustrated in
Figure 2(a).

###  Simulation of powder spreading of multisphere particles with thermal
properties

A small part of the powder bed (width $0.75$ mm) is simulated using periodic
boundary conditions in the $y$-direction. The contact model used for this
simulation is Hertz-Mindlin model combined with viscous damping
muller2011collision (37). In this numerical setup, the substrate is assumed to
be smooth. We insert multispheres of 10 different irregular shapes generated
from images of commercially available PA12 particles as shown in Figure 9. The
multispheres are inserted in front of the spreader tool until the total bulk
particle volume equals $0.7$ mm3, which is sufficient material to create a
powder layer of $5$ mm length, $0.75$ mm width and $100\,\upmu$m height. We
allow the particles to settle and relax under the effect of gravity before we
start the spreading process. These simulations begin with particle
temperatures set at $T_{\mathrm{part}}=393$ K, and the powder is spread over a
pre-heated plate held at $T_{\mathrm{plate}}=430$ K, as previously documented
in li2021experimental (38, 39, 40). After the particles have settled down and
the system is relaxed, the spreading process starts by moving the tool at a
constant speed $v$ mm/s. Particles reaching the end of the powder bed (at
$x=5$ mm) are not used for the analysis. Finally, the simulation is stopped
when the system is static, i.e., the kinetic energy of the system is
sufficiently low. Thus, the maximum simulation time is chosen as
$t_{\mathrm{max}}=1$ s, taking into account the time for initial settling and
relaxation of particles, the spreading time, and the time for the system to
reach a static state. A detailed description of the setup can be found in
roy2023thermal (41).

Figure 9: (a) Powder spreading setup with a blade speed $v=10$ mm/s, showing
(b) front view after $0.08$ s and (c) top view after $1.0$ s of the spread
powder layer.

###  Packing density analysis with thermal influence

The elastic modulus is dependent on the temperature as luding2005discrete (16)

$\displaystyle
E^{*}(T_{ij})=\frac{E_{0}^{*}}{\alpha}\bigg{[}1+(\alpha-1)\tanh\bigg{(}\frac{T_{m}-T_{ij}}{\Delta
T}\bigg{)}\bigg{]}$ (1)

where $T_{m}=451$ K is the melting temperature, $\Delta T=20$ K is the
temperature interval over which phase change occurs, and $\alpha$ is the
parameter defining the dependency of the elastic modulus on the temperature.
The $E^{*}(T_{ij})$ model proposed by Luding et al. luding2005discrete (16)
conforms to $\alpha=2$, which needs further experimental validation. Note that
according to Equation 1, an extreme softening of the material of up to $80\%$
occurs at a large value of $\alpha$ and at an extreme temperature of the
particles, close to their melting point. However, in our powder spreading
setup, such a large temperature is not achieved by particles without
sintering.

We characterized the packing density of the powder layer subjected to varying
blade velocity and the dependence of the particle stiffness on temperature. In
Figure 10, we observe that the packing density is significantly influenced by
the blade velocity. The packing density corresponding to a lower blade speed
of 50 mm/s in Figure 10(a) is lower than that in Figure 10(b) for blade speed
of $250$ mm/s. Further, the packing density is more sensitive to the values of
$\alpha$ at a lower blade speed of $v=50$ mm/s. Note that the packing density
is more sensitive to a smaller value of $\alpha$, i.e., there is a significant
difference in packing fraction between $\alpha=2$ and $20$ than between the
values for $\alpha>20$. A lower blade speed allows more time for inter-
particle heat exchange, and thus the value $\alpha$ plays a significant role
in softening the material. This leads to a change in the packing density of
the powder layer.

(a)

(b)

Figure 10: Packing density as a function of substrate length $X$ for different
coefficient $\alpha$ for (a) $v=50$ mm/s and (b) $v=250$ mm/s for initial
elastic modulus $E_{0}^{*}=10^{4}$ Pa.

In Figure 11(a)

(a)

(b)

Figure 11: Packing fraction for different blade speed $v$ for different
initial elastic modulus (a) $E_{0}^{*}=10^{4}$ Pa and (b) $E_{0}^{*}=10^{6}$
Pa for $\alpha=2$.

and (b), we compare the packing density for different blade speeds. The real
modulus of PA12 powder is on the order of $10^{8}$ Pa. However, for the sake
of computational efficiency, we use a lower elastic modulus in our
simulations. To further test the effect of the elastic modulus on the packing
density of the powder layer, we simulate the powder spreading process using
two different values of the initial elastic modulus, $E_{0}^{*}$: $10^{4}$ Pa
and $10^{6}$ Pa, respectively, while keeping $\alpha=2$ as constant. We
observe that the packing density is less sensitive to blade velocity for
material with higher elastic modulus. This is because stiff particles allow
less interparticle compression, and thus the packing density is less sensitive
to the blade speed. For soft particles, the packing density is influenced by
the flow dynamics of the material, which depends on the blade speed.

##  6 Vibrating recoating mechanism

Improving the layer quality can be achieved via modification of the powder
application process marchais20213d (42, 43). Nasato et al. (2021)
nasato2021influence (35, 44) explored the applicability of vibrating recoating
mechanisms as a technique that can be tailored according to material
characteristics, such as particle size, shape, and cohesion, to produce layers
of improved characteristics. They performed DEM simulations using cohesive,
multisphere particles of realistic shapes of PA12 particles to explore powder
spreading systems using two spreading tools, namely a blade and a roller.
Figure 12 shows the simulation system.

(a)

(b)

Figure 12: Powder spreading simulation setup using (a) a blade and (b) a
roller recoating tool. Red boxes indicate the powder bed region where porosity
is evaluated after the layer is recoated. Porosity is additionally evaluated
on the top of the part (source nasato2021influence (35)).

Vibration applied on the PA12 material via the recoating tool was used as a
means of compacting the powder particles during their application to form the
layered bed. The influence of the vibration characteristics on the layer
density and surface roughness was investigated. The vibration frequency was
varied to values of 100, 250, 500, and 1000 Hz, while the amplitude was varied
to 2, 5, 10, and 20 µm, for recoating velocities of 100, 150, 200, and 250
mm/s.

Figure 13: Porosity of granular layer obtained for different amplitudes
(x-axis) and frequencies. The powder spreading tool is a blade, operating with
a translational velocity from 100 mm/s (top left) to 250 mm/s (bottom right).
The dashed line indicates the reference case where no vibration was applied.
The porosity was measured on the powder recoated on the top of the granular
bed, see also nasato2021influence (35)).

Figure 13 demonstrates that for recoating velocity 100 mm/s, considering large
frequency (i.e., 1000 Hz) combined with intermediate amplitude (10 µm) leads
to a major improvement of the powder bed density, reducing the porosity by
13.2%. However, for larger amplitudes, no significant change in the porosity
is observed. This indicates an optimum between 10 and 20 µm for the amplitude.
In all cases, increased vibration energy seems to disturb the powder layer,
and leads to a loosening of the granular bed, as seen e.g., for 1000 Hz and 20
µm. This effect can be observed also for recoating velocity 150 mm/s, where
the porosity of the granular bed increases for large frequencies and
amplitudes by 20.9% for 1000 Hz and 20 µm.

Figure 14: Formation of bumps on the surface of the powder bed as a result of
the vibrating mechanism. (a) Snapshots taken at different stages of the
recoating process show the formation of the bumps for (b) recoating velocity
250 mm/s, frequency 100 Hz and amplitude 20 µm (c) recoating velocity 100
mm/s, frequency 100 Hz and amplitude 20 µm. $D_{b}$ is 2.5 mm and 1.0 mm for
cases (b) and (c), respectively. The particles marked in dark blue indicate a
region between the periodic bump, and correspond to $D_{b}$. (d) No bump could
be identified for recoating velocity 100 mm/s, frequency 250 Hz and amplitude
20 µm. (source nasato2021influence (35)).

The influence of vibrations on the recoating mechanism using a roller was
investigated by nasato2021influence (35). They found that vibrations increased
the porosity of the recoated layer in all cases for recoating velocities above
150 mm/s, which they attributed to being an artifact of the roller geometry,
largely influencing the powder bed. The roller introduces an artificial
bumpiness on the layer surface, as shown in Figure 14, which is, however,
minimized for small recoating velocities, such as 100 mm/s.

##  7 Conclusions

We developed a numerical tool specifically designed for particle-based DEM
simulations in the context of powder application within additive manufacturing
devices. Our tool takes into account the intricate geometric shapes of the
particles involved. To assess the capabilities of our model, we utilized it to
investigate the transport of powder particles using a roller as coating
system. Through our simulations, we were able to discern that the process
velocity has a significant influence on the packing characteristics of the
applied powder. The findings of our study highlight the importance of process
speed in determining how the powder particles pack together during the coating
process. This understanding can be valuable in optimizing the additive
manufacturing process, allowing for improved control over the final properties
and quality of the manufactured objects. By incorporating complex particle
geometric shapes into our simulations, our numerical tool provides a more
realistic representation of the powder application process. This enables us to
gain insights into the underlying mechanisms governing particle transport and
packing, aiding in the development of more efficient and effective additive
manufacturing techniques.

Subsequently, we expand upon our existing Discrete Element Method (DEM) model
by introducing inter-particle attractive forces. We emphasized the importance
of considering two essential contributions to the particle interaction model:
adhesion and van der Waals forces. Adhesion is incorporated into our model
using the Johnson-Kendall-Roberts (JKR) theory, while van der Waals
interactions are also taken into account. We assert that neglecting either of
these contributions leads to significant deviations in the simulation results,
particularly for fine powders. Such deviations render the DEM method
unreliable in accurately capturing the packing behavior.

Furthermore, our investigation considered the influence of powder cohesion on
the structural anisotropy of the deposited layers. Structural anisotropy
refers to the variation in the spatial distribution of particles within the
layers. It provides insights into the overall organization and packing
behavior of the powder particles during the spreading process. Understanding
the effect of powder cohesion on structural anisotropy is important for
optimizing processes where particle arrangement plays a significant role, such
as in additive manufacturing, powder compaction, or coating applications.

Finally, the focus of the research was to further explore the powder
application process through process simulations. In particular, the thermal
and mechanical effects of the process and material parameters on the powder
application process are the primary research objectives. The integration of
the multisphere algorithm with the thermal discrete particle model in the DEM
framework done during this phase of research represents a significant
advancement in simulating non-spherical particles with heat transfer
capabilities. This extension opens up new possibilities for studying a wide
range of systems, such as granular materials, powders, and particulate flows,
where particle shape and thermal effects are of critical importance.

The addition of vibrations during powder application was investigated via DEM
simulations with non-spherical, cohesive particles as a new application
system. The effect of vibration and process characteristics was explored.
Small frequency and amplitude, combined with small recoating velocity, lead to
denser granular beds of reduced porosity. Large frequency and amplitude,
however, lead to a vibro-fluidized state of the particles and granular beds
with loose configuration characterized by an increased porosity. For practical
applications, the choice of frequency and amplitude should be considered in
combination with a specific translational velocity of the recoating process.

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## Index