# Combined thermal and particle shape effects on powder spreading in additive
manufacturing via discrete element simulations

Sudeshna Roy\dagger\dagger$\dagger$Equal contribution.1∗ Hongyi
Xiao11footnotemark: 11,2 Vasileios Angelidakis11footnotemark: 11,3 Thorsten
Pöschel1 1\. Institute for Multiscale Simulation, Friedrich-Alexander-
Universität Erlangen-Nürnberg, Erlangen, Germany
2\. Department of Mechanical Engineering, University of Michigan, Ann Arbor,
USA
3\. School of Natural and Built Environment, Queen’s University Belfast,
Belfast, UK

###### Abstract

The thermal and mechanical behaviors of powders are important for various
additive manufacturing technologies. For powder bed fusion, capturing the
temperature profile and the packing structure of the powders prior to melting
is challenging due to both the various pathways of heat transfer and the
complicated properties of powder system. Furthermore, these two effects can be
coupled due to the temperature dependence of particle properties. This study
addresses this challenge using a discrete element model that simulates non-
spherical particles with thermal properties in powder spreading. Thermal
conduction and radiation are introduced to a multi-sphere particle formulation
for capturing the heat transfer among irregular-shaped powders, which have
temperature-dependent elastic properties. The model is utilized to simulate
the spreading of pre-heated PA12 powder through a hot substrate representing
the part under manufacturing. Differences in the temperature profiles were
found in the spreading cases with different particle shapes, spreading speed,
and temperature dependence of the elastic moduli. The temperature of particles
below the spreading blade is found to be dependent on the kinematics of the
heap of particles in front, which eventually is influenced by the temperature-
dependent properties of the particles.

###### keywords:

Discrete Element Method, powder spreading, thermal, particle shapes

††journal: Additive Manufacturing

##  1 Introduction

Powder-based additive manufacturing is an effective and sustainable
manufacturing technique that has the potential to manufacture objects of
arbitrary and complex shapes [1, 2, 3]. One specific method with promising
development is powder bed fusion, where a laser or an electron beam
selectively melts sequentially deposited layers of powders to form intricate
parts upon solidification, while the un-melted powders act as support.
Although this method presents many merits, such as rapid prototyping, low
energy consumption and reduced material waste, compared to traditional
manufacturing techniques, the manufacturing quality of the built part is far
from ideal with the preparation of the powder bed being one of the major
sources of problems.

The powder packing before melting is crucial for the part quality [4], and it
is affected by the recoating process [5, 6, 7], which typically involves a
blade or a roller spreading a layer of powder over a loose powder bed or
previously sintered powders. Powders are out-of-equilibrium complex materials
often made of cohesive particles with irregular shapes that typically have
poor flowability. The packing structure achieved in spreading is highly
dependent on the material properties and process parameters [8, 9], which are
difficult to optimise, as powder flow is far more complicated than that of
Newtonian liquids. As a result, problems exist such as voids sustained in the
powder bed due to friction, interlocking due to irregular particle shape, or
cohesion [10, 11]. Although high-speed powder spreading can efficiently
enhance the productivity of laser powder bed fusion, it also reduces the
packing density of powder layer and dimensional accuracy in the building
direction, which is unfavorable for the part quality in additive
manufacturing. Consequently, a higher bed temperature is achieved before the
next layer melting, reducing fusion defects and minimizing temperature
gradients, which is an advantage of high-speed spreading [12]. For polymer
powders that are amorphous or semi-crystalline, their mechanical properties
can have rather strong temperature dependence. This is highly relevant as
their temperatures are elevated significantly during spreading due to combined
effects of pre-heating, heat lamps, and spreading on printed parts. This
effect can further compromise the quality of the powder layer, making
obtaining a dense powder bed with smooth surface even more challenging.

Understanding the influence of temperature-dependent properties for
irregularly-shaped particles can be challenging at the particle-level, as it
is hard to probe in-situ powder particles experimentally, due to their small
sizes. Multiphysics simulations that simultaneously model the mechanics and
heat transfer can be utilised to study the spreading process, which often
treat the powder as a continuum phase with bulk properties. However, it is can
be difficult for continuum-based computational models to include these
effects, as the relation between microscopic details and continuum-level
properties are missing, such as how the bulk stiffness, density, and heat
transfer parameters can be deduced from material properties for a packing of
complex-shaped particles. To resolve particle-level physics in power
spreading, simulations using the Discrete Element Method (DEM) that directly
calculate trajectories and interactions of individual particles have proved to
be effective. The heat conduction between spherical particles has also been
simulated with discrete element methods [13], which considers the conduction
(e.g. [14, 15, 16, 17, 18]), convection (e.g. [16, 19]), and radiation (e.g.
[19, 16, 20, 21]) between the particles and the ambient environment.

Real powder particles feature a variety of irregular shapes. Particle shape
affects the packing density and packing structure of granular assemblies, as
irregular particles tend to form complex structures compared to spherical
particles. Particle shape exhibits various aspects based on the considered
level of observation, which typically entail form, roundness and surface
texture [22, 23]. Particle form plays a crucial role in the way irregular
particles pack into many-particle assemblies, and is typically characterised
in terms of elongation, flatness and compactness, which are used to classify
particles as flat, elongated, bladed (i.e. both flat and elongated) and
compact [24]. Different particle shapes demonstrate different packing
characteristics; for instance, flat particles tend to align their largest
projected area to a surface, when deposited under gravity, elongated particles
tend to align their longest dimension to the surface, while compact particles
have no clear preference based on their form characteristics alone. In the
context of powder spreading processes, particle shape influences the surface
roughness of the produced powder layer, which in turn affects the part quality
of the object manufactured via the powder fusion process [25]. While
irregularly shaped particles offer a larger exposed surface area, leading to
improved conductivity within the powder bed, they come with inherent
disadvantages, most notably reduced flowability. This limitation poses a
challenge to the efficiency of additive manufacturing processes, particularly
in applications like laser beam melting, where factors such as processing
speed and material selection are crucial [26]. To overcome the issue of
reduced flowability, a strategic shift to the utilization of spherical
particles with a narrow size distribution becomes advantageous [27, 28, 29,
30, 31, 32]. These spherical particles, often derived through innovative
production methods such as melt emulsification [33] and wet grinding with a
rounding process [34], are preferred in additive manufacturing processes due
to their ability to enhance flowability and mitigate the drawbacks associated
with irregular shapes. Specifically, spherical particles of polyamides also
find widespread use, contributing to the overall efficiency of the additive
manufacturing process.

This work addresses the combined effects of heat transfer and particle shape
within a powder-layer-spreading framework. To this end, a multi-sphere
algorithm [35] is coupled with a thermal particle formulation in the
MercuryDPM [36] open-source software package. First, imaging data of polyamide
12 (PA12) particles are used to generate three-dimensional multi-sphere
particles of realistic shape, using a recently proposed open-source platform
for particle generation. Then, mixtures of these realistic multi-spheres with
thermal properties are used to used to simulate a powder spreading process to
investigate the effects of varying blade speed and particle stiffness on the
powder bed temperature profile, both during the powder spreading phase and in
the time immediately after spreading. To explore the effect of particle shape
in the thermal properties of the system, powder bed temperature profiles are
compared for different particle shapes, namely for the realistic irregular
shapes versus spherical particles of same size. Conclusions are drawn on the
effect the varied parameters have on the powder flowability and, consequently,
on the temperature profile of the produced powder layers.

##  2 Methods

###  2.1 Multi-sphere particle generation

Several simulation techniques exist to model non-spherical particles in the
Discrete Element Method (DEM). The multi-sphere approach is one of the oldest
and more versatile techniques, where a collection of spheres are connected
rigidly to form irregular particles. In this approach, individual subspheres
are only used to facilitate contact detection, while the collection of spheres
is considered as one particle when it comes to calculating inertial
characteristics and integrating the motion of the particle. The number and
size of the spheres used to create a multi-sphere particle defines its
morphological fidelity, i.e. the degree in which it approximates the
morphology of a target particle shape, as well as the computational cost
associated to its interactions with neighbouring particles.

In this work, multi-sphere particles of PA12 are generated to approximate the
shape of real powder particles. To this end, an image-informed particle
generation procedure is put in place, where scanning electron microscopy (SEM)
images of PA12 particles [7] are extruded into three-dimensional ones, which
are then used as a target geometry for the generation of realistic multi-
sphere particles. Multi-sphere generation is carried out utilising a recently
developed particle generation method based on the Euclidean transform of
three-dimensional images. In particular, the open-source software CLUMP is
employed [37], which provides various techniques for multi-sphere generation.
Figure 1 shows ten multi-spheres generated using the SEM imaging data of PA12
powders. The number of subspheres, here $N_{\mathrm{sub}}=10$, is chosen so
that the multi-spheres approximate adequately the particle morphology compared
to the imaging data, while its effect on the thermal particle properties is
explored in this paper.

Figure 1: Non-spherical PA12 powder particles. Multi-sphere particles
generated based on binarised SEM images of the material.

###  2.2 Thermal contact model

Table 1: Material and model parameters. Properties | Value | Reference
---|---|---
Particle diameter $D_{50}$ |  $55~{}\mu$m | [38, 7]
Density $\rho$ |  $1000$ kg/m-3 | [7]
Young’s modulus $E$ |  $2.3\times{10}^{6}$ Pa | [39, 38]
Poisson ratio $\nu$ | $0.40$ | [7]
Dissipation coefficient $A_{n}$ |  $1.85\times{10}^{-7}$ s | [40]
Coulomb’s friction coefficient $\mu$ | $0.50$ | [7]
Latent heat of fusion $L_{f}$ |  $101.66\times{10}^{3}$ J/kg | [41]
Solid heat capacity $c_{p}$ |  $1200$ J/kgK | [42]
Thermal conductivity coefficient $k_{s}$ |  $0.12$ W/mK | [42, 43, 44]
Thermal convection coefficient $h_{\mathrm{conv}}$ |  $15$ W/m2K | [45, 46, 47]
Emissivity $\epsilon$ | $0.80$ | [47, 48, 46]
Melting temperature $T_{m}$ |  $451.15$ K | [42]
Temperature interval $\Delta{T}$ |  $20$ K | [49, 50]
Ambient temperature |  $393$ K | [42]
Particle temperature |  $393$ K | [46, 51, 52]

The contact behaviour between the powder particles is described using the
simplified, viscoelastic Hertz-Mindlin contact law (no-slip solution), which
is enhanced with a temperature dependence. The normal contact force is given
by

$\vec{F_{n}}=\min\left(0,-\rho\xi^{3/2}-\frac{3}{2}A_{n}\rho\sqrt{\xi}\dot{\xi}\right)\vec{e_{n}}$
(1)

with

$\rho=4/3\,E^{*}_{T}\,\sqrt{R^{*}}$ (2)

where $E^{*}_{T}$ is the temperature dependent effective Young’s modulus and
$R^{*}=R_{1}R_{2}/\left(R_{1}+R_{2}\right)$ is the effective radius. For
contact between two identical materials, Equation 2 simplifies to
$\rho=\frac{2E\sqrt{R^{*}}}{3\left(1-\nu^{2}\right)}$, where $E$ and $\nu$ the
Young’s modulus and Poisson ratio of the material, respectively. The Young’s
modulus considered here is two scale of magnitude smaller than the real
Young’s modulus of PA12 to decrease computational time, as in Parteli and
Pöschel [38].

The employed viscous damping model is used to simulate non-constant normal
restitution for different impact velocities between particles, in alignment
with experimental evidence [40]. The dissipative constant A is calculated for
a given coefficient of restitution at a given impact velocity, based on the
particle elastic properties and size, as in [40]. Here, a coefficient of
restitution of 0.7 is considered at a characteristic impact velocity set to 50
mm/sec, which is the blade speed of the powder spreading process.

A temperature-dependent effective Young’s modulus $E^{*}$ is considered, which
varies during the simulation, based on the current temperature of the
particles in contact, as in [53, 36].

$E^{*}=\frac{E^{*}_{0}}{\alpha}\left[(1+(\alpha-1)\tanh{\left(\frac{T_{m}-T}{\Delta{}T}\right)}\right]$
(3)

where $\alpha$ is the degree of Young’s modulus change near melting
temperature,
$E^{*}_{0}=\left(\frac{1-\nu_{i}^{2}}{E_{i}}+\frac{1-\nu_{j}^{2}}{E_{j}}\right)^{-1}$
is the effective Young’s modulus at room temperature, $T_{m}$ is the melting
temperature, $T=\frac{T_{i}+T_{j}}{2}$ is the average temperature between the
two particles in contact and $\Delta{}T$ is the temperature interval around
the melting temperature, which is used to calculate the energy required for
phase change, as in [36].

In the original implementation of this thermal model by [36], a temperature-
dependent Hertzian model was combined with a linear tangential force
formulation of constant stiffness, to facilitate the calculation of particle-
particle and wall-particle contact forces. Here, we extend their formulation,
to implement a fully-temperature-dependent Hertz-Mindlin contact law. The
contact force in the tangential direction is given by

$\vec{F_{t}}=-\min\left[\mu|\vec{F_{n}}|,\int_{path}8G^{*}_{T}\sqrt{R^{*}\xi}\,ds+A_{t}\sqrt{R^{*}\xi}v_{t}\right]\vec{e_{t}}$
(4)

where the the no-slip solution of Mindlin [54] is used to calculate the
elastic tangential contact forces, and the approach of Parteli and Pöschel
[38] is used to calculate the tangential dissipative constant $A_{t}\approx
2A_{n}E^{*}$. Regarding the elastic material parameters, $G^{*}_{T}$ is the
temperature-dependent effective shear modulus. For non-thermal particles, the
effective shear modulus is calculated as
$G^{*}=\left(\frac{2-\nu_{i}}{G_{i}}+\frac{2-\nu_{j}}{G_{j}}\right)^{-1}$
which for two identical materials simplifies to $G^{*}=\frac{4G}{2-\nu}$. To
reflect the temperature dependence of the shear modulus, the following
assumption was made in this study: According to the Hertz-Mindlin contact
model (no-slip solution), the shear to normal stiffness ratio is defined as
$\kappa=k_{t}/k_{n}$, where the tangential elastic stiffness for non-sliding
contacts is $k_{t}={dF_{t}}/{d_{s}}=8G^{*}\sqrt{R^{*}\xi}$ and the normal
elastic stiffness is $k_{n}={dF_{n}}/{d\xi}=2E^{*}\sqrt{R^{*}\xi}$, thus
leading to $\kappa=k_{t}/k_{n}=4G^{*}/E^{*}$. Considering the stiffness ratio
constant for the PA12 material, the evolving, temperature-dependent shear
modulus is calculated as $G^{*}=\kappa E^{*}/4$, which changes throughout the
simulation, according to the change of the Young’s modulus. For two identical
materials in contact, the stiffness ratio can be written as
$\kappa=\frac{2\left(1-\nu\right)}{\left(2-\nu\right)}=$ which highlights that
the current modelling approach assumes no change of the Poisson ratio during
heating of the PA12 particles. A future extension of the model would be to
update the stiffness ratio with a potential thermal-dependent evolution of the
Poisson ratio, throughout the simulation, if such an effect is established in
the literature.

Cohesion between multi-sphere particles are modelled via adding the Johnson-
Kendall-Roberts (JKR) model [55] as an attractive force when two slave
particles come into contact:

$\vec{F}_{\mathrm{JKR}}=4\sqrt{\pi a^{3}\gamma E^{*}}\;\vec{e_{n}}$ (5)

where $\gamma$ is the surface energy density and $a$ is the contact radius
related to deformation, calculated using
$\xi=\frac{a^{2}}{R^{*}}-\sqrt{\frac{4\pi a\gamma}{E^{*}}}$. The maximum
interaction distance at which the contact breaks under tension is given by
$\xi_{t}=\frac{1}{2}\frac{1}{6^{1/3}}\frac{a^{2}}{R^{*}}$. For the JKR model,
Hærvig et al. [56] proposed a scaling law where the surface energy density is
scaled with the modified effective Young’s modulus as
$\gamma=\gamma_{r}{(E/E_{r})}^{2/5}$, where $\gamma_{r}=32$ mJ/m2 is the real
surface energy density of PA12 [57, 58] and $E_{r}$ is the real Young’s
modulus of PA12.

###  2.3 Heat transfer for multi-sphere particles

The heat transfer for multi-sphere particles can include 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 other particles and the ambient.
Here, we consider the spreading of pre-heated powder over the surface of the
printed part that has a temperature at the melting point. In this case, the
temperature of the powder and the ambient is assumed to be the same and major
heat exchange should occur between the particles and the solidified part. We
do not consider thermal convection here as we do not expect strong upward
convection current from spreading. For each clump $i$, a single temperature,
$T_{i}$, is assigned, and is updated by

$\frac{\partial T_{i}}{\partial t}=\frac{Q_{i}}{m_{i}c_{p}},$ (6)

where $c_{p}$ is the heat capacity and $Q_{i}$ is the total thermal transfer
rate. which can be expressed as

$Q_{i}=\sum_{j=1}^{N}Q_{\mathrm{cond},ij}+Q_{\mathrm{rad},i},$ (7)

where $Q_{\mathrm{cond},ij}$, and $Q_{\mathrm{rad},i}$ are the respective rate
for conduction and radiation, which are demonstrated in Figure 2. The
conduction rate for clump $i$ needs to be summed over the $N$ neighbouring
clump $j$ that $i$ is in contact with. Note that a pair of clumps can have
multiple contacts which are essentially overlaps of their slave spheres. Thus,
$Q_{\mathrm{cond},ij}$ is calculated as the summation of the conduction rate
of the slave particles for clump $i$, which is

$Q_{\mathrm{cond},ij}=\sum_{s=1}^{N_{s}}\sum_{n=1}^{N_{\mathrm{neb},s}}k\frac{s_{sn}}{r}(T_{j}-T_{i}),$
(8)

where $N_{s}$ is the number of slaves of clump $i$, $N_{\mathrm{neb},s}$ is
the number of contacts a slave sphere $s$ has with the slave spheres of $j$,
$k$ is the thermal conductivity, and $s_{sn}$ is the overlap area of a pair of
slave spheres in contact. The temperature gradient is calculated at the clump
level, with $T_{j}$ being the temperature of clump $j$ and $r$ being the
centroid-to-centroid distance between clumps $i$ and $j$. As the heat
conduction flux is proportional to the contact area of two contacting slave
particles, it is strongly affected by the fact that a smaller Young’s modulus
is used here, and so the value of the conduction coefficient has to be scaled.
To this end, we correct the conduction coefficient as
$k=k_{r}(E/E_{r})^{{2}/{3}}$, where $k_{r}$ and $E_{r}$ are the realistic
conductivity and modulus, and the ${2}/{3}$ exponent comes from the relation
between contact area and the Young’s modulus.

To derive this scaling, we assume two spherical PA12 particles of equal size
in equilibrium, where one particle is resting on top of the other under its
self weight. For the real Young’s modulus, we can calculate the deformation
and contact area for these particles in contact via the Hertzian contact
theory. For the (softer) system with scaled Young’s modulus, the contact area
between the particles will be larger. Hence, since the thermal conduction is
proportional to the contact area, we scale the thermal conduction coefficient
of the system with softer Young’s modulus, so that the thermal conduction of
the scaled and the real system are the same, leading to the scaling law
proposed above.

Figure 2: Sketch for heat transfer mechanism of multi-spheres for conduction
and radiation.

For radiation, we consider the heat exchange between particles and the hot
substrate (solidified part from previous spreading cycle), and the the
respective heat transfer rate can be approximated as

$\displaystyle
Q_{\mathrm{rad},i}=\epsilon\sigma_{SB}A_{ix}(T_{i}^{4}-T_{\mathrm{substrate}}^{4}),$

where $\epsilon$ is the material emissivity, $\sigma_{SB}=5.67\times 10^{-8}$
W/(m${}^{2}/K^{4}$) is the Stefan-Boltzmann constant [59], and
$T_{\mathrm{substrate}}=443$ K represent the temperature of the substrate [52,
51, 46]. The exposed area of each clump is approximated as
$A_{ix}=G(z)A_{i}/2$, where $A_{i}$ is the surface area of clump $i$, and
$G(z)$ is a height dependent shielding factor accounting for the blockage of
radiation from the substrate to particles in the upper levels. The dividing
factor of two comes from an approximation that assumes that on average the
lower half of a clump is exposed. The functional form for $G(z)$ is given by
$G(z)=0.5\bigg{(}1-\mathrm{erf}\frac{z-D_{50}}{D_{50}/16}\bigg{)}$, where
$D_{50}=55\,\mu$m is the mean diameter of the non-spherical particles and
$D_{50}/16$ is the width of the error function which is considered here based
on the requirement of a smooth transition of the radiation heat transfer
across the bed in the $z$ direction. Inter-particle radiation is neglected on
the assumption that the temperature gradient between particles is small.
Admittedly, the real situation is more complicated, and the additional
convection and radiation can be modelled by adapting existing algorithms for
spherical particles, via switching in the surface area of the clumps [19, 20,
21].

##  3 Effect of number of subspheres on heat conduction

Representing an irregularly-shaped particle with different number of
subspheres ($N_{s}$) can have an influence on its behavior, especially when it
comes to the number of contacts with a neighboring particle or boundary and
the effective stiffness of the multi-sphere, both of which influence heat
conduction. Per our multi-sphere construction, as $N_{s}$ increases, the
average size of subspheres decreases, as every additional sphere is of smaller
or equal size to the previous one, which in turn decreases the prefactor
$\rho$ in Equation 2 for the normal contact force defined in Equation 1. For a
larger subsphere, a larger overlap area can be expected under the same load.
On the other hand, having more subspheres potentially leads to a higher number
of contacts, reducing the load on each subsphere. It is hard to derive the
dependence of heat conduction (the total contact area) on $N_{s}$ through
these two effects, as other factors also come to play, including the specific
multi-sphere generation method as well as the shape and orientation of the
particles at contact, or how densely they are packed.

(a)

(b)

Figure 3: Snapshot after $0.50$ s of simulation of a sample layer of multi-
spheres with (a) $5$ subspheres and (b) $25$ subspheres. The particles are
coloured according to their temperature $T$. The simulations are conducted for
$\alpha=2$.

Alternatively, we perform auxiliary simulations of a sample layer of
multipshere particles in a sample box with a hot substrate and periodic
boundary conditions in the lateral directions. Radiation is not included in
this test. For each test, we generate loose packings of particles with a
certain $N_{s}$, settle under gravity on the hot substrate and monitor the
temperature rise due to heat conduction with the hot substrate. The initial
particle temperature is $T_{\mathrm{in}}=393$ K and the wall temperature is
fixed to $443$ K. Two examples using $N_{s}=$5 and $N_{s}=$25 are shown in
Figure 3. The particles are colored by their temperature, which shows that the
bottom layer of particles are heated up by the hot boundary. The temperature
difference between the bottom particle layer and the upper layers is still
much smaller than the difference with the hot wall, which explains the lack of
temperature rise in the upper layers. In both cases, some variations in
temperature can be seen in the bottom layer, with the more horizontally
orientated particles being heated up more significantly due to the increased
number of contacts with the bottom in comparison to the more vertically
orientated particles. In addition, packing of particles with more subspheres
is less dense as reflected by the increased layer height, which could be a
result of the change of effective particle surface roughness and cohesion,
which is also a function of subsphere size.

(a)

(b)

Figure 4: The rise of average temperature of a sample particle layer with
conduction only for particles made of different numbers of subspheres. The
dashed line indicates a linear regression fitting line
$(T_{\mathrm{avg}}-T_{\mathrm{in}})=0.0012\times{N_{s}}+0.0197$ as a potential
correction method.

To quantify the influence of $N_{s}$, we plot the average particle temperature
in the layer, $T_{ave}$, vs. time in Figure 4(a), with the curves colored by
the number of subspheres. Heat conduction begins at about $t=0.05$ s after the
particles settle down, and we focus on the initial period up to $t=0.50$ s,
during which the temperature rise is significantly less than the $50$ K
temperature difference between the particles and the hot substrate, and the
temperature rise is therefore linear in time. The result clearly indicate that
heat conduction is more significant for larger $N_{s}$, and we quantify the
temperature rise, $T_{\mathrm{ave}}-T_{\mathrm{in}}$, at $t=0.50$ s for
different as shown in Figure 4(b). The temperature rise, which is roughly
proportional to the total contact area between the particles and the
substrate, doubles as $N_{s}$ increases from 3 to 30. This means that heat
conduction is artificially enhanced by using a larger $N_{s}$ as a result of
subsphere softening and increased number of contact, despite having looser
packing. As the dependence of the temperature rise is rather linear with
$N_{s}$, it is possible to empirically scale the conductivity to remove the
dependence of $N_{s}$ according to the fitting in Figure 4(b). However, more
rigorous solutions potentially exist with controlling the subsphere size
during multi-sphere generation, which is beyond the scope of current study. In
the rest of the paper, we use results of $N_{s}=10$. The thermal radiation, on
the other hand, is much less sensitive to $N_{s}$ as it is not a function of
contact area.

##  4 Simulation of powder spreading

###  4.1 Numerical setup

(a)

(b)

(c)

Figure 5: Powder spreading setup showing (a) side view during initial particle
deposition, (b) side view during spreading and (c) deposited powder layer
structure after $0.05$ s of powder spreading process with a blade velocity
$v=50$ mm/s.

A thin layer of the powder bed of width $0.75$ mm is simulated using periodic
boundary conditions in the $y$-direction. In this numerical set-up, the
substrate is assumed to be flat and frictional. We inserted ca. $20000$ multi-
sphere particles with a mean diameter $D_{50}=55~{}\mathrm{\mu}$m as shown in
Figure 5(a) in front of the spreader tool, at $(x,y,z)\in[0.5,1.5]$ mm
$\times[0,0.75]$ mm $\times[0,h]$ mm until the total bulk particle volume
equals $0.7$ mm3, which is sufficient to create a powder layer of $5$ mm
length, $0.75$ mm width and $0.1$ mm height. In the beginning of the
simulations, the particles are allowed to deposit and relax under the effect
of gravity. Then, the spreading process begins with the tool starting to move
the at a constant speed $v$. The particles are spread in a layer where the
tool gap is set to $H=100\,\mu$m, which corresponds to about $1.8D_{50}$ in
$z-$direction as shown in Figure 5(b). The structure of the deposited powder
layer is shown in Figure 5(c). Particles reaching the end of the powder bed
(at $x=5$ mm) are not used for the analysis. The powder layer, spread on the
hot substrate, undergoes heating through both conduction and radiation.
Finally, the simulation is stopped when the temperature of the spread powder
layer stabilizes, indicating minimal temperature changes. 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 temperature of the powder layer to achieve a stabilized
state.

###  4.2 Parameters for powder spreading simulations

The packing density of the powder layers decreases with increasing spreading
velocity, as reported by Nasato et al. [25]. Consequently, the effective
thermal conductivity of the powder bed at the bulk scale should vary with the
spreading velocity. Therefore, we analyze the effect of spreading blade
velocity on the temperature evolution of the deposited powder layer,
considering two blade velocities: $v=50$ and $250$ mm/s.

Above the glass transition temperature of $322$ K (i.e. $48.8^{\circ}$C), the
Young’s modulus of PA12 undergoes a sharp decline as a function of temperature
[60]. This temperature-induced decrease in the Young’s modulus is modelled as
a function of temperature by Equation 3, where $\alpha=2$ is a commonly used
model as proposed by Luding et al. [53], however little experimental evidence
exists to verify this value. To investigate the effect of the Young’s modulus
decrease on the powder layer’s local structure, we conducted simulations for
various $\alpha$ values: $2,20,$ and $200$.

Particle shapes play a crucial role in the packing density as well as in the
contact area between the particles and between the particles and the
substrate. Non-spherical particles of PA12 have a higher contact area with the
substrate as well as inter-particle contact area, leading to a higher
conductivity of the powder layer compared to spherical particles. Spherical
particles of PA12, on the other hand, are often used in the powder spreading
process due to their better flowability. Therefore, we analyze the effect of
particle shapes on the temperature evolution of the deposited powder layer,
considering real PA12 particle shapes and spherical particles of the same
volumes of PA12.

###  4.3 Temporal evolution of powder bed temperature

In this sub-section, we examine the temporal evolution of average temperature
$T_{\mathrm{avg}}$ within a specific section of the powder layer ($1$ to $2$
mm in length) under various parametric conditions during powder spreading.
From the temporal evolution trend, we assess the temperature attained after
$1.0$ s of powder spreading simulation in different parametric scenarios.

####  4.3.1 Effects of blade velocity

In Figure 6(a), the packing structure of a deposited powder layer is
visualized at two distinct blade velocities with uniform resolution. Notably,
the powder layer at a blade velocity of $250$ mm/s appears thinner compared to
the layer at $50$ mm/s. Simultaneously, in Figure 6(b), the mean temperature
evolution within a section of the powder layer is shown over a time period of
$1.0$ s. The powder layer subjected to a blade velocity of $250$ mm/s exhibits
a notably higher mean temperature compared to the layer at $50$ mm/s. The
faster heating rate and the higher temperature for higher blade velocity are
primarily due to two reasons. First, as powders are being spread faster, more
particles are brought to contact with the substrate given the same amount of
time, thus $T_{avg}$ rises faster. Second, the thinner layer with the faster
spreading means that heat can propagate through the height of the layer
faster. The thinner layer also results in a higher mean temperature $T_{avg}$
of the powder bed.

(a)

(b)

Figure 6: (a) Packing of powder layers for different blade velocities $v=50$
mm/s (top) and $v=250$ mm/s (bottom) and (b) average temperature of the powder
bed $T_{\mathrm{avg}}$ as a function of time for simulations with the two
different blade velocities.

####  4.3.2 Effects of degree of Young’s modulus change

In Figure 7(a), the packing structure of the deposited powder layer is
depicted for two degree of Young’s modulus change parameter, denoted by
$\alpha$. Interestingly, the powder layer structures remain notably similar
for $\alpha=2$ and $\alpha=200$. This visual similarity is further confirmed
by the analysis of the mean temperature evolution illustrated in Figure 7(b)
over a time frame of $1.0$ s. Here, we observe the evolving temperature is
slightly lower for $\alpha=2$, but no substantial differences in the
temperature profiles are observed for $\alpha=20$ and $200$ values. The higher
temperatures observed for $\alpha=20$ and $200$ are primarily attributed to
increased particle softening with rising temperatures. This phenomenon results
in a greater overlap of particles, leading to enhanced heat conduction between
the substrate and the powder bed.

(a)

(b)

Figure 7: Packing of powder layers for different degree of Young’s modulus
change parameter $\alpha=2$ (top) and $\alpha=200$ (bottom) and (b) average
temperature of the powder bed $T_{\mathrm{avg}}$ as a function of time for
simulations with different $\alpha$.

####  4.3.3 Effects of particle shapes

In Figure 8(a), the packing structure of a deposited powder layer is
presented, showcasing two distinct particle shapes: irregular shaped particles
and regular spherical particles. Notably, the packing structure involving
irregular particles displays a significantly larger contact area, both among
the particles themselves and with the heated substrate. This structural
difference is further explored in Figure 8(b), which illustrates the evolution
of the mean temperature within the identical section of the powder layer over
a duration of $1.0$ s. As expected, owing to the larger contact area, the
powder layer comprising irregular particles exhibits a notably higher mean
temperature in contrast to the layer composed of spherical particles.

(a)

(b)

Figure 8: Packing of powder layers for different particle shapes: non-
spherical particles (top) and spherical particles (bottom) and (b) average
temperature of the powder bed $T_{\mathrm{avg}}$ as a function of time for
simulations with the two different particle shapes with $v=50$ mm/s and
$\alpha=2$.

###  4.4 Spatial evolution of powder bed temperature

In this sub-section, we explore the in-situ average temperature of particles
beneath the blade $T_{\mathrm{blade}}$ during the powder spreading process.

(a)

(b)

Figure 9: (a) Temperature of powder layer under the moving blade as a function
of the spreading length for two different blade velocities $50$ mm/s and $250$
mm/s. (b) Powder spreading after $t=$ s and $t=$ s for blade velocity $250$
mm/s.

Figure 9(a) illustrates the comparison of the average particle temperature
beneath the moving blade in relation to the spreading length for two distinct
blade velocities. At a blade velocity of $250$ mm/s, the faster motion
restricts the duration for particles to absorb heat, resulting in a notably
lower temperature profile in contrast to the $50$ mm/s blade velocity. The
temperature of the particles demonstrates a consistent increase concerning the
spreading length $x$. Conversely, the temperature profile exhibits an initial
steady rise for the blade velocity of $50$ mm/s, followed by a sharp decline
in temperature for $x>4$ mm. This decline can be attributed to the movement of
cooler particles within the heap in front of the blade. In cases of a
substantial heap, cooler particles from the heap’s surface cascade downward,
settling well ahead of the blade. This provides ample time for heat
absorption. As the powder spreading process nears completion and the heap
diminishes, cooler particles settle closer to the blade, reducing the time
available for heat absorption and resulting in a corresponding temperature
decrease. Higher blade velocity leads to a faster spreading process, offering
minimal time for heat absorption. Figure 9(b) illustrates the flow profile of
particles corresponding to the higher blade velocity.

Figure 10: Temperature of powder layer under the moving blade as a function of
the spreading length for different $\alpha$ with spreading velocity $50$ mm/s
for non-spherical particles.

In Figure 10, it can be observed that the temperature profile exhibits an
initial steady rise for $\alpha=2$, followed by a sharp decline in temperature
for $x>4$ mm as also observed in Figure 9(a). In contrast, the temperature
profile increases with $x$ for $\alpha=20$ and $200$ and there are no
significant drop in temperature for $x>4$ mm for the two cases. This suggests
that the temperature sensitivity of Young’s modulus play a critical role in
affecting heat transfer during the in-situ powder spreading process. Further,
we visualise the local temperature of particles during the spreading process
as shown in Figure 11(a) and (b) for the two extreme cases of $\alpha=2$ and
$\alpha=200$. For $\alpha=2$, the hot particles at the tip of the heap get
overrun by the intermittent avalanche of cold particles from the top of the
heap. In cases of a substantial heap, cooler particles from the heap’s surface
cascade downward, settling well ahead of the blade. This provides ample time
for heat absorption. As the powder spreading process nears completion and the
heap diminishes, cooler particles settle closer to the blade, reducing the
time available for heat absorption and resulting in a corresponding
temperature decrease. Therefore, the temperature of the particles at the tip
of the heap is relatively lower as observed in Figure 11(a). For the higher
alpha case, the flow is more cohesive and the heap of particles in front of
the blade remains quite rigid. Therefore, the temperature of the particles at
the tip of the heap is consistently higher as observed in Figure 11(b).
Therefore, the temperature profile of the particles during spreading
consistently rises with spreading distance $x$.

(a)

(b)

Figure 11: Powder spreading after $t=0.04$ s and $t=0.07$ s for (a) $\alpha=2$
and (b) $\alpha=200$.

Next, we show the average particle temperature beneath the moving blade
$T_{\mathrm{blade}}$ compared for non-spherical and spherical particles for a
blade velocity $50$ mm/s and $\alpha=2$ as shown in Figure 12(a).
Interestingly, although for both cases $\alpha=2$, the temperature profile of
the spherical particles do not show in drop for $x>4$ mm. This is because,
spherical particles have higher flowability and hence cold particles from the
top of the heap get deposited much farther ahead of the tip of the heap. As a
result, the tip of the heap consistently have particles at higher temperature,
resulting in a consistent increase in $T_{\mathrm{blade}}$ as a function of
$x$. Figure 12(b) illustrates the flow profile of spherical particles, which
can be compared to that of irregularly shaped particles in Figure 11(a).
Clearly, the cold spherical particles from the top of the heap get deposited
much further ahead of the tip of the heap compared to the deposition of
irregularly shaped particles.

(a)

(b)

Figure 12: (a) Temperature of powder layer under the moving blade as a
function of the spreading length for particles with spherical and irregular
shapes. (b) Powder spreading after $t=0.04$ s and $t=0.07$ s for spherical
particles.

##  5 Conclusions

In this study, we enhanced the multi-sphere algorithm by integrating it with a
thermal particle formulation within the open-source code MercuryDPM. This
modification allowed us to comprehensively explore the interplay between heat
transfer and properties of irregularly-shaped powders in the spreading
process. Utilizing the refined algorithm, we systematically examined the
impact of blade speed, particle stiffness, and particle shapes on the
temperature profiles of the powder bed.

For the temporal evolution of the average temperature, $T_{\mathrm{avg}}$,
within a specific section of the spread powder layer, higher blade velocity
leads to faster heating rates during powder spreading and higher temperatures
of the powder layer due to increased particle-substrate contact and faster
heat propagation through the thinner powder layer. As for the temperature
dependence of the material modulus, a marginal decrease in temperature is
noted for $\alpha=2$, while no significant variations in temperature profiles
are observed for $\alpha=20$ and $200$. Higher temperatures for the latter
cases are attributed to increased particle softening, fostering greater
particle overlap and improved heat conduction between the substrate and the
powder bed. For the particle shape, the layer with irregular particles
exhibits a significantly higher evolving temperature compared to the layer
with spherical particles, attributed to the larger contact area of irregular
particles.

We explored the in-situ temperature of particles beneath the blade,
$T_{\mathrm{blade}}$, in relation to the spreading distance $x$ during the
powder spreading process. For faster blade velocity, the temperature increases
with spreading length as it leads to a wider spreading. For slower blade
velocity ($\alpha=2$, irregular shape), the temperature initially rises with
spreading length $x$ and then sharply declines beyond $4$ mm due to cooler
particles from the diminishing heap. For different $\alpha$, e.g. $\alpha=20$
and $200$, the temperature again consistently increases with spreading length,
indicating that the sensitivity of Young’s modulus significantly affects heat
transfer during spreading. The local temperature dynamics depict distinct
behaviors, with $\alpha=2$ showing intermittent avalanches of cold particles
impacting heat absorption, and $\alpha=200$ displaying a more cohesive flow,
resulting in consistently higher temperatures at the tip of the heap. For the
comparison of different particle shape, we observe a lack of temperature drop
for spherical particles beyond $4$ mm. This can be attributed to their higher
flowability that leads to the deposition of colder particles further ahead of
the tip of the heap to be heated up, ensuring a consistent increase in
$T_{\mathrm{blade}}$ with spreading distance $x$.

## Acknowledgement

We would like to express our gratitude to the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) for their generous funding of the
Collaborative Research Center 814 (CRC 814), Project Number 61375930-SFB 814
‘Additive Manufacturing’, sub-project B1. Additionally, we extend our thanks
to the Humboldt Research Foundation for awarding the ‘Humboldt Research
Fellowship’. This work received support from various institutions, including
the Interdisciplinary Center for Nanostructured Films (IZNF), the Competence
Unit for Scientific Computing (CSC), and the Interdisciplinary Center for
Functional Particle Systems (FPS) at Friedrich-Alexander-Universität Erlangen-
Nürnberg.

## Conflicts of interests

There are no conflicts to declare.

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