# Numerical simulation of thermal conductivity of stainless steel and Al-12Si
powders for additive manufacturing

Vladimir Ankudinov Corresponding author  Udmurt State University
Institute of Mathematics, Information Technologies
and Physics
Izhevsk, 426034
Russia

Vereshchagin Institute for High Pressure Physics
Russian Academy of Sciences
Moscow (Troitsk), 108840
Russia
Email<EMAIL_ADDRESS>Georgii A. Gordeev  Udmurt State University
Institute of Mathematics, Information Technologies
and Physics
Izhevsk, 426034
Russia
Email<EMAIL_ADDRESS>Evgeniy V. Kharanzhevskiy  Udmurt State
University
Institute of Mathematics, Information Technologies
and Physics
Izhevsk, 426034
Russia
Email<EMAIL_ADDRESS>Mikhail D. Krivilyov  Udmurt State University
Institute of Mathematics, Information Technologies
and Physics
Izhevsk, 426034
Russia

Udmurt Federal Research Center
Ural Branch of the Russian Academy of Science
Izhevsk, 426067
Russia Email<EMAIL_ADDRESS>
###### Abstract

A three-dimensional model of a partially melted powder bed with particles
stochastically distributed in size and space coordinates has been developed.
Numerical simulation of temperature distributions in stainless steel AISI 316L
and Al-12Si powders in vacuum, air and argon has been performed to analyze
unsteady heat transfer in a porous medium. The numerical model demonstrates a
large effect of heat transfer through the gas phase in case of powders with
low thermal conductivities like stainless steels. At the porosity level of 65%
and above, the mechanism of heat transfer drastically changes and a linear
dependence of thermal conductivity on porosity frequently used in literature
becomes incorrect. The effects of the consolidation coefficient and size
distribution on effective heat transfer in powders are discussed. The obtained
dependencies of the effective thermal conductivity on porosity and the
consolidation coefficient could be used in additive manufacturing
applications.

Keywords: selective laser melting; numerical simulation, random particle
distribution, unsteady heat transfer, Al-12Si, stainless steel AISI 316L

##  1 Introduction

Transport of heat through a porous medium is of great interest in relation
with its morphological properties since in many technological processes
effective thermal properties control the dynamics of heat conduction, melting
and solidification [1, 2]. The important technical applications where the
metallic powders are used include powder metallurgy, selective laser sintering
(SLS) and selective laser melting (SLM) [3]. Usually, the structure of a
porous media is complex owing to a wide distribution of particles in size and
irregular morphology of pores in a metal matrix [4, 5, 6]. Due to the fact
that the commonly utilized in SLM powders are spheroidized (see scanning
electron microscopy (SEM) image of raw commercial Al-based alloy powder in
Fig. 1(a)), the simplest particle’s approximation by spheres can be used [3,
5]. This approximation substantially simplifies the results of thermal
analysis for real industrial powders [5]. Thus, the detailed prediction of the
effective thermal conductivity of powder media requires consideration of the
shape and size distribution, conductivity of each particle and gas atmosphere,
and heat exchange between the metal and gas phases [7]. In the most cases, for
an anisotropic porous medium saturated by gas or liquid the thermal
conductivity has significantly lower values than for the same non-porous
material [8, 6]. The thermal conductivity of dry powders could differ
significantly for Van-Der-Waals force bonded, ionic-bonded or covalent-bonded
materials [5, 9, 10]. This difference comes from the contact resistance effect
between particles which plays a significant role and is ruled mainly by three
factors: (i) elastic deformation of the particles caused by the external
compressive force, (ii) chemical composition of the particle’s surface, and
(iii) Kapitza resistance. The thermal contact resistance is successfully
studied for the case of particles with low-area contacts. However, in the case
of SLM processing the formation of improved thermal contacts between
particle’s surfaces due to solid-phase, liquid-phase sintering and pre-melting
leads to the reduction of these effects.

Figure 1:  (a) SEM image of raw commercial Al12SiMg SLM powder, grade CL30AL
[11] studied in the present work; (b) the same powder exposed for 1 hour heat
treatment at $T=623$ K. The necks between powder particles due to solid-state
consolidation are formed and coagulation of the particle’s conglomerates
started. The SEM voltage is 10kV at magnification of 5000X.

Due to surface pre-melting of particles during laser treatment, the
neighboring particles start to form necks between them which significantly
improves heat exchange in a porous medium [12, 3], see Fig. 1(b). The improved
thermal contact between particles or the effect of dynamical consolidation
under conditions of laser treatment may significantly affect the effective
heat transport in the processed powder bed. To precisely control processing
parameters, one can account for the above-mentioned effect by introduction of
a consolidation coefficient which influences the effective thermal
conductivity $k_{eff}$ [13]. Then simulation of SLM processes may be carried
out in the approximation of a continuous media [14, 15, 16, 17, 18, 19, 20]
where the effective heat conductivity coefficient is treated as a state
functions.

The existing approaches in modelling of heat transfer in powder beds can be
reduced to two major cases:
(i) discrete particles modelling where the powder bed is described as a domain
filled with the separate particles [21, 5, 22, 23, 24, 25, 26];
(ii) elementary cell approach where heat transfer in one particle and its
elementary volume is considered as periodically repeated in the whole volume
[5, 27, 28, 8, 29, 30, 31, 32, 4, 33, 34].
To establish a given porous medium’s properties using the discrete particles
approach (i) one have to measure averaged sample’s properties. If the sample
is too small, parameters such as heat conduction coefficients and other mass-
related parameters tend to oscillate. With the increasing of the sample size,
the oscillations begin to dampen out. The sample size related to the
consistent averaged properties and designated as representative volume element
(RVE) when its elastic constants, porosity and thermal characteristics are the
same as for the composite or powder medium [35, 5, 3]. Calculation of the
effective heat conductivity in the powder with spherical particles can be done
by combination of the classic packed bed approach [4, 7] and the stochastic
random filling models (see [5] and references therein). This approach and its
comparison to the existed analytical models is presented in this article.

The objective of this paper is to study the effect of powder bed morphology,
porosity and particle consolidation on the effective thermal conductivity of
commercial Al-12Si and stainless steel SLM powders during the initial and
intermediate stages of heating or melting/sintering. The numerical study of
heat transfer is carried out for randomly generated powder beds filled with
spherical particles of different sizes at different consolidation levels in
vacuum, air and argon. We shall consider this process as solid-state
consolidation, though in general the dynamics of powder compaction involves
also solid-liquid phase transitions and motion dynamics of phase interfaces
[36, 1, 2, 20].

Table 1: Technical specification of Al-12Si and stainless steel 316L powders
and thermophysical parameters used in the present paper.

Parameter | Value
---|---
| Al-12Si |  Stainless steel
Commercial name | CL30AL |  CL20ES, AISI 316L
Material | Al-based, 10.5..13.5 wt.% Si |  Fe, $0.03$ wt.% C, $16..18$ wt.% Cr,
| 0.05 wt.% Mn, 0.25 wt.% Fe [11] |  $10..14$ wt.% Ni, $2..3$ wt.% Mo [11]
Shape of particles | Quasi-spherical or spherical
Mean / min / max diameter, $\langle d\rangle$, $\upmu$m | 4.9 / 1.3 / 18.6a |  37.5 / 15.0 / 50.0 [37, 38, 39]
Bulk thermal conductivity, $k^{\textrm{metal}}$, W/(m$\cdot$K) | 155 [11, 40] |  15
Bulk heat capacity, $C^{\textrm{metal}}_{p}$, J/(K$\cdot$kg) | 947 [41, 42] |  500 [11]
Bulk density, $\rho^{\textrm{metal}}$, kg/m3 | 2640 [43, 41] |  8000 [11]
Melting temperature, $T_{m}$, K | 854 [41, 42] |  1713 [11, 41]
| Air |  Argon
Gas thermal conductivity, $k^{\textrm{gas}}$, W/(m$\cdot$K) | 2.42$\times 10^{-2}$ [41, 44] |  1.78$\times 10^{-2}$ [41, 44]
Gas heat capacity, $C^{\textrm{gas}}_{p}$, J/(K$\cdot$kg) | 1005 [41, 44] |  519 [41, 44]
Gas density, $\rho^{\textrm{gas}}$, kg/m3 | 1.29 [41, 44] |  1.63 [41, 44]

  * a

Obtained in the present work

Figure 2:  Particle size distribution of raw commercial Al-12Si SLM powder
CL30AL [11] evaluated on the basis of SEM images Fig. 1(a). The distribution
is approximated by the lognormal probability density function (solid line)
with the parameters of $\sigma=0.25$, $\mu=0$, shifted and normalized.

##  2 Model of effective heat transfer in randomly allocated powder beds

In this section, a mathematical model of heat transfer in the partially-melted
(consolidated) metallic powder beds is formulated. The powder bed is generated
using random stochastic deposition of particles in the representative volume
element (RVE) with up to $10^{3}$ metallic particles. Using direct simulations
of heat transfer in such layer, one can find the dynamics of heat propagation
in a metallic powder processed in gas atmosphere or in vacuum for the given
porosity $\varepsilon$, particle size distribution and particle contact spot
diameter $d_{C}$. The model allows one to choose the consolidation coefficient
of particles and the bed porosity as input parameters during generation of
powder filling.

###  2.1 Analysis of local thermal equilibrium

For macroscopic simulations of transport phenomena in discontinuous or two-
phase systems the microscopic governing equations are averaged using the
coarse-graining procedure [45]. Additionally, the effective transport
characteristics of powder media are described in the approximation of
continuous media [5, 6]. These characteristics like the effective thermal
conductivity depend on temperature, porosity and other powder parameters and
may be implemented as state functions during modeling of powder shrinkage and
melting [18, 14, 15, 16, 17, 19, 20]. To that end, three important aspects are
to be considered as follows for proper space and time discretization in
numerical simulation of heat transfer in porous media.

(i) Local thermal equilibrium. The classic approach [5, 46] for modeling is
based on justification of the condition of local thermal equilibrium between
gas and powder particles. The general condition for local thermal equilibrium
in a discrete volume element can be formulated as [5]:

$\frac{1}{V^{\textrm{gas}}}\int_{V^{\textrm{gas}}}T^{\textrm{gas}}dV=\frac{1}{V^{\textrm{metal}}}\int_{V^{\textrm{metal}}}T^{\textrm{metal}}dV,$
(1)

where $V=V^{\textrm{gas}}+V^{\textrm{metal}}$ is the total volume of a
discrete element, the superscripts “gas” and “metal” belong to the designate
phases, and $T$ is local temperature.

(ii) Representativity of discrete volumes. First, the effective thermal
conductivity of a discrete volume should be independent on the size of this
volume. This criterion allows one to find a size of the representative volume
element (RVE) and the upper limit of the mesh size for discretization of the
powder bed. Typically, RVE contains no less than $5^{3}..7^{3}$ particles [5].
Figure 1 provides the SEM images of Al-12Si powder at initial state and after
solid-phase sintering while the size distribution and mean diameter are
provided in Fig. 2. The specifications of two SLM powders are given in Table
1. The particle size has an average of $\langle d\rangle=4.9$ $\upmu$m.
Therefore, one can estimate that a linear size of RVE is about
$L_{\textrm{RVE}}=25..35$ $\upmu$m. Second, representativity means continuity
of the thermal field. Continuity is satisfied in the finite element method,
since regularity of the temperature field in each finite element is
automatically provided owing to linearization of the non-stationary second-
order conduction equation in the case of zero heat source [47].

(iii) Consistency of discrete volumes with macroscopic characteristic lengths.
Let one consider the local temperature difference between gaseous and metallic
phases to be $\Delta T^{\textrm{gas/metal}}$. The maximum attainable
temperature difference in the RVE is $\Delta T_{\textrm{RVE}}$. RVE fulfills
the upper limit for discretization on the macroscopic scale $L$ if the general
temperature difference $\Delta T_{L}$ is much higher than one in the RVE

$\Delta T^{\textrm{gas/metal}}<\Delta T_{\textrm{RVE}}\ll\Delta T_{L}.$ (2)

The typical temperature difference attained during SLM processing is as
follows [14, 20]:

$\displaystyle\Delta T^{\textrm{gas/metal}}$ $\displaystyle\sim 20..50\,K,$
(3) $\displaystyle\Delta T_{\textrm{RVE}}$ $\displaystyle\sim 150\,K,$ (4)
$\displaystyle\Delta T_{L}$ $\displaystyle\sim 500..600\,K.$ (5)

Therefore, to consider media in the RVE in the continuous approximation the
temperature criterion Eq. (2) satisfies if $L>200$ $\upmu$m.

(iv) Time discretization. For time discretization, the condition Eq. (1) of
local thermal equilibrium [48, 5] has to be applied to a single discrete
element. This condition formulated for heat propagation in connection to a
time step $\delta t$ in the discrete element with a size of $\ell$ may be
formulated as

$\frac{\varepsilon(C_{p}\rho)^{\textrm{gas}}\,\ell^{2}}{\delta
t}\left(\frac{1}{k^{\textrm{gas}}}+\frac{1}{k^{\textrm{metal}}}\right)\ll 1,$
(6) $\frac{(1-\varepsilon)(C_{p}\rho)^{\textrm{metal}}\,\ell^{2}}{\delta
t}\left(\frac{1}{k^{\textrm{gas}}}+\frac{1}{k^{\textrm{metal}}}\right)\ll 1.$
(7)

This time and length scales must be satisfied for the system at local thermal
equilibrium, requiring the heat fluxes to be small enough to calculate the
thermal conduction equation. If the size of discrete elements is
$\ell=1~{}\upmu$m, then the time step $\delta t$ in calculations should be of
the order of $\delta t\sim 10^{-8}$ s or lower to meet the conditions Eqs.
(6)–(7) for Al-12Si SLM powder. Finally, the particles of $\langle d\rangle=5$
$\upmu$m must be discretized by at least 5 finite elements. In the present
work, we chose the size of the adaptive mesh elements to be $\ell\sim 0.05..1$
$\upmu$m and the time step of about $\delta t\sim 10^{-8}$ s to fulfill Eqs.
(6)–(7).

###  2.2 Powder bed generation

Figure 3:  Schematic view of the contact between two consolidated particles in
partially-consolidated powders. The diameters of two particles are $d_{i}$ and
$d_{j}$ correspondingly, $d_{C}$ is the contact spot diameter, $\langle
d\rangle$ is the mean diameter of two particles, $l_{ij}$ is the distance
between the sphere’s centers and $\xi$ is the overlapping coefficient
according to Eq. (8).

The powder bed was generated by placing randomly the spherical particles in
nodes of a hexagonal closed-pack (HCP) lattice. We assumed the realistic log-
normal size distribution [1, 33] with the parameters of $\mu=0$,
$\sigma=0.25$, see Fig. 2. This size distribution parameters correspond to the
morphology of commercial SLM CL30AL powder. The diameter $d_{C}$ of the
contact spot and thus dimensionless overlapping coefficient $\xi$ between
particles was controlled by proper displacement of each particle

$\displaystyle\xi=\frac{\langle d\rangle-l_{ij}}{\langle
d\rangle},\quad\quad\langle d\rangle=\frac{d_{i}+d_{j}}{2},$ (8)

where $d_{i},d_{j}$ are the sizes of the particles $i$ and $j$, $\langle
d\rangle$ is the mean diameter of two particles, $l_{ij}$ is the distance
between their centers, see Fig. 3. Then the contact spot diameter $d_{C}$
could be found from

$\displaystyle
16d_{C}^{2}l_{ij}^{2}=\left((d_{i}+d_{j})^{2}-4l_{ij}^{2}\right)\times\left(-(d_{i}-d_{j})^{2}+4l_{ij}^{2}\right).$
(9)

The specified porosity was achieved by removal of random particles from the
bed till the target average porosity $\varepsilon=V^{\textrm{gas}}/V$ is
attained. Here $V$ is the domain volume and $V^{\textrm{gas}}$ is the volume
of the gaseous phase only. Due to the HCP packing limit, generation of powder
beds at low porosities $\varepsilon<0.26$ was performed by random allocation
of the gas voids between particles which reproduces a powder briquette or a
partially consolidated high-density powder bed. The porosity obtained for dry
powder beds experimentally varies between $\varepsilon=0.41$ and $0.46$ for
loose-packed stainless steel [39, 49]. Once the particles are deposited, their
positions and radii are used as input parameters for generation of the finite
element mesh and implementation of the heat transfer model. A heat source was
allocated at the top boundary of the computational domain with dissected
particles (see Fig. 4).

Figure 4:  (a) Generated particle bed with the log-normal distribution of
particle sizes (see Fig. 2) at $\langle d\rangle=4.9$ $\upmu$m,
$\varepsilon=0.72$. (b) Projection of the instantaneous thermal field $T$ in
the Al-12Si powder bed calculated for SLM in air with a fixed particle size of
$\langle d\rangle=4.9$ $\upmu$m, size deviation of $\sigma=0$, contact
diameter of $d_{C}=3$ $\upmu$m, and porosity of $\varepsilon=0.65$. The
physical time is $t=3.05\times 10^{-5}$ s. The evaluated effective heat
conductivity is obtained by Eq. (14) and it equals to $k_{eff}=16.2$
$\textrm{W}/(\textrm{m}\cdot\textrm{K})$.

###  2.3 Governing equations and computational domain

The transient problem of heat conduction in a two-phase “gas-metal” domain was
formulated. The effect of heat transfer by radiation between particles was
neglected due to its small contribution to the overall energy balance. The
convective mechanism of heat exchange through the gaseous phase was also
eliminated for the specific case at the considered time scales
$\tau=10^{-8}-10^{-5}$ s because viscosity of the gas phase significantly
reduces convection in small pores [5, 6]. More details are provided below in
Sec. 3.1. Accounting for nice contact between the pre-melted particles with
formed necks we neglect the surface effects and contact thermal resistance
since its influence is drastically reduced with compaction evolvement in SLM-
processing. Heat conduction in the gaseous and metallic phases correspondingly
is described by the following equations

$\displaystyle C_{p}^{\textrm{gas}}\rho^{\textrm{gas}}\frac{\partial
T}{\partial t}$ $\displaystyle=k^{\textrm{gas}}\nabla^{2}T,$ (10)
$\displaystyle C_{p}^{\textrm{metal}}\rho^{\textrm{metal}}\frac{\partial
T}{\partial t}$ $\displaystyle=k^{\textrm{metal}}\nabla^{2}T,$ (11)

where $C_{p}$ is the specific heat capacity, $\rho$ is the density, $k$ is the
thermal conductivity of each phase (see correspondent values in Table 1). The
authors pay attention that the thermal conductivities $k$ of the phases were
selected as constants and are placed aside the $\nabla^{2}$–operator. In the
present paper, this is justified by the fact that the effective thermal
conductivity is analyzed as a function of geometric characteristics of the
powder bed. Thus, the dependencies of $k$, $C_{p}$ and $\rho$ on temperature
are intentionally excluded from analysis and its values is selected for
$T_{0}=293$ K (see Table 1). Obtained effective characteristics could be
utilized as a base for the modeling of the powder media in the continuous
medium approximation when the effective media conductivity is separated in two
components: the effective structural conductivity component and thermal
dependent component resulting
$k_{\textrm{media}}=k_{eff}(\varepsilon,d_{C},\langle d\rangle)\cdot
k^{\textrm{metal}}(T)/k^{\textrm{metal}}(T_{0})$ [14, 15, 20]. This approach
allows one to simply describe the complex averaged medium’s properties with
many dependent parameters at low cost.

The temperature at the boundary $z=0$ was selected as the melting temperature
$T_{m}$. So the initial and boundary conditions of the heat conduction problem
on the particle’s surfaces and domain’s walls are

$\displaystyle T|_{t=0}=T$ ${}_{amb},\quad T|_{z=0}=T_{m},$
$\displaystyle\mathbf{n}\cdot k\mathbf{\nabla}T=0\,\,\,$ at outer boundaries.
(12)

At the interfaces between gas and metal, continuity of the heat fluxes and
temperature was defined

$\displaystyle k^{\textrm{gas}}\,\mathbf{\nabla}T^{\textrm{gas}}$
$\displaystyle=k^{\textrm{metal}}\,\mathbf{\nabla}T^{\textrm{metal}},$
$\displaystyle T^{\textrm{gas}}=T^{\textrm{metal}}\,\,\,$ at inner boundaries.
(13)

Ambient temperature $T_{amb}$ was equal $0.7\times T_{m}$ as far as this value
corresponds to the start temperature of the process of solid state
consolidation according to [50, 51].

###  2.4 Numerical implementation

The direct problem Eqs. (10)–(13) of heat transfer in a two-phase powder bed
was solved using the finite element method (FEM). The size of mesh elements
and time step were selected according to the conditions Eq. (6) and Eq. (7) of
local thermal equilibrium in RVE. The explicit integration time scheme was
used with the backward-differential formula of the 2nd order. This provided
sufficient accuracy and smoothness of the time-dependent solution. We
generated the spatial mesh consisted up to 106 tetrahedral Lagrange finite
elements with the linear $C^{1}$ interpolation functions. This mesh provides
up to $3\times 10^{5}$ degrees of freedom. The powder bed generation was first
conducted in the MATLAB software and then the subdomain classes which
describes particles positions and sizes were invoked in the COMSOL
Multiphysics software [52]. We performed up to 5 simulations with random
particle configurations for each set of parameters: porosity $\varepsilon$,
contact spot diameter $d_{C}$, mean particle diameter $\langle d\rangle$ and
size distributions. The periodic boundary conditions on elongates sides of the
computational domain were tested but no significant effect on the temperature
field was found if the RVE of a sufficient size is constructed (see Sec. 2.1).
Different types of the surface layer on the heated top boundary were tested.
The difference in the efficient heat conductivity of the powder bed was found
negligible. The specific calculations to clarify the role of the dispersion
and size distribution of particles were also performed. Finally, the inverse
heat transfer problem and its fitting were carried out in MATLAB.

###  2.5 Evaluation of the effective thermal conductivity

The inverse problem of heat conduction [53, 54] was solved using the
analytical solution with the complementary error function erfc in order to
determine the effective thermal diffusivity $\alpha_{eff}$ as follows

$\displaystyle\frac{\partial T}{\partial
t}=\alpha_{eff}\nabla^{2}T,\quad\textrm{which leads to the solution}$ (14)
$\displaystyle
T(z,t)=T_{m}\\!-(T_{m}\\!-T_{amb})\,\textrm{erfc}\\!\left(\frac{z}{2\sqrt{\alpha_{ef\\!f}\,t}}\right)\\!\\!.$
(15)

The one-dimensional problem is solved in the semi-infinite approximation
admitting that the numerical solution is received in a relatively long in the
$z$-direction domain. The three-dimensional formulation of Eqs. (10)–(13)
considers the planar heat wave propagation, so temperature was averaged in
$xy$-planes leading to the one-dimensional temperature distribution
$T^{\textrm{FEM}}(z,t)$ (see the two-dimensional projection of the $T$-field
in Fig. 4(b)). Then minimization of the functional
$F(z,t)=\int(T^{\textrm{FEM}}(z,t)\\!\\!-T(z,t))dV$ was carried out. In the
performed simulations, a number of temperature profiles calculated by FEM at
fixed times in the gas-metal domain was analyzed. The calculated effective
thermal diffusivity $\alpha_{eff}$ were minimized running through the
different time frames for each $z$. By this, one can reduce the influence of
the boundary particles and their contribution to the $T(z,t)$-distribution and
relaxation of the heat wave. At the end, $\alpha_{eff}$ was averaged for the
different initial particle distributions at a given value of $\varepsilon$ and
the effective thermal conductivity of the metallic powder is obtained as

$k_{eff}(\varepsilon)=\alpha_{eff}\left(C_{p}^{\textrm{gas}}\rho^{\textrm{gas}}\,\varepsilon+C_{p}^{\textrm{metal}}\rho^{\textrm{metal}}\,(1-\varepsilon)\right).$
(16)

##  3 Results

###  3.1 Role of radiative and convective heat transfer

Heat transfer comprises the diffusive, convective and radiative contributions.
The effect of radiation on heating of metal powder has been estimated
theoretically. According to the Stefan-Boltzmann law, the total emissive power
$q_{\text{radiative}}$ of a surface is

$q_{\text{radiative}}=\epsilon\,\sigma_{\text{SB}}\,(T_{\text{surf
1}}^{4}-T_{\text{surf 2}}^{4}),$ (17)

where $\epsilon$ is the emissivity ($\epsilon=0.3$ for Al-12Si and
$\epsilon=0.7$ for stainless steel), $\sigma_{\text{SB}}=5.67\times 10^{-8}$ W
m-2 K-4 is the Stefan-Boltzmann constant, $T_{\text{surf 1,2}}$ are the
surface absolute temperatures of two neighbour particles in a powder bed. The
heat flux given by conduction between the contacting particles is defined by
Fourier’s law as

$q_{\text{conductive}}=k_{eff}G,$ (18)

where $G$ is an estimate of the temperature gradient in the powder bed with
the effective conductivity $k_{eff}$. The conductive heat flux between the
particles and gaseous atmosphere follows from a balance of heat exchange
between the solid and viscous medium

$q_{\text{convective}}=h\Delta T,$ (19)

where $h$ is the surface heat transfer coefficient and $\Delta T$ is the
temperature difference between the particle surface and gas temperature [54].
In what follows, the convection contribution was calculated in the
approximation of capillary flow at the Nusselt number $\mathrm{Nu}=2$. All
heat fluxes calculated for Al-12Si and CL20ES powders are provided in Fig. 5
accounting for actual thermal gradients achieved in SLM processing.

Figure 5: Different contribution in the total heat flux calculated for (a)
Al-12Si powder and (b) stainless steel CL20ES powder beds with the porosity
$\varepsilon=0.55$. The conductive mechanism Eq. (18) is assumed for solid-to-
solid conductance. The convective flux Eq. (19) controls heat transfer between
solid particles and gas and dominates in capillary flow inside gas pores. The
radiative transfer Eq. (17) proceeds between the particles.

As follows from the result of simulation, the radiative heat flux is by 4
orders of magnitude smaller for Al-12Si and by 3 orders smaller for the
stainless steel than the conductive flux between particles. The obtained
conclusions differs with the results in literature [33, 34] where radiation is
considered as an important physical phenomenon for SLM processing. However,
for partially melted powders the conductive mechanism substantially exceeds
other heat transport mechanisms in the powder bed.

###  3.2 Dependence of the thermal conductivity on porosity

Figure 6:  Dependence of the effective thermal conductivity $k_{eff}$ on
porosity $\varepsilon$ calculated for (a) Al-12Si powder in air at $d_{C}=3$
$\upmu$m (dashed curve), $0.4$ $\upmu$m (dash-dotted curve) and $0.1$ $\upmu$m
(solid curve); (b) stainless steel CL20ES powder in vacuum (dash-dotted
curve), air (dashed curve) and argon (solid curve). This curves were
interpolated based on 30 runs with 3 different random-generated particle
configurations for each porosity value. The numerically calculated points were
smoothed by splines. The dotted lines represent the linear model.

Considering a weak dependence of the effective heat conductivity on convective
and radiative transfer mechanisms (see Sec. 3.1), the study was focused on the
geometrical conductance of the powder. During consolidation of particles in
presence of high temperatures, metallic powders reduce their porosity via
continuous shrinking. Then the contact spot diameter increases. Therefore,
assessment for powder layers with different contacts areas in a wide range of
porosities $\varepsilon$ is highly demanded. At the moderate values of
$\varepsilon$, the generated powder morphology corresponds to industrial
powders (see Sec. 2.2). The porosity $\varepsilon$ is upper-bounded by
$\varepsilon<0.26$ which corresponds to dense packing where drastical
improvement of the contact surface between particles is expected. There is
also a bottom bound at $\varepsilon>0.7$ which corresponds to the percolation
limit _i.e._ to a break of continuous heat transfer through the metallic
phase.

The dependence of the effective thermal conductivity of Al-12Si powder on the
porosity $\varepsilon$ for different contact spot diameters $d_{C}$ calculated
in air is presented in Fig. 6(a). Small contact spots (see the solid curve at
$d_{C}=0.1$ $\upmu$m) correspond to the case of dry-powders where a relative
size of the contact spots is about $d_{C}/\langle d\rangle=0.02$. As follows
from the plot, the curve sharply tends to very small values in the range
$k_{eff}\simeq 2.5..3.2$ W/(m $\cdot$ K) at $\varepsilon>0.4$ which represents
a non-consolidated powder bed before laser SLM processing. Such low thermal
conductivity provides poor heat dissipation and leads to heat accumulation in
the uppermost surface layer. A slight increase of the contact diameter between
particles (see curves $d_{C}=0.1$ and $d_{C}=0.4$ $\upmu$m in Fig. 6(a)) leads
to increase of the heat flux between particles by 16 times.

In general, the dependence of $k_{eff}$ on $\varepsilon$ has almost linear
behaviour for porosities smaller than the percolation threshold. Due to random
distribution of particles in the bed, heat conduction is reduced to only gas-
phase conduction earlier than the percolation limit is achieved. In case of
highly consolidated powder (see dashed curve $d_{C}=3.0$ $\upmu$m in Fig.
6(a)), one gets a curve abutting to the theoretical percolation threshold
which is close to the solution for contacted soft spheres (see [55] and
references therein).

Heat conduction in partially consolidated powder for $d_{C}\geqslant 0.4\upmu$
($d_{C}/\langle d\rangle>0.08$) which correspond to high-density packed powder
(briquette) at $\varepsilon<0.3$ is close to the geometrical limit of heat
conductivity (Fig. 6(a)). Dry powder at $d_{C}/\langle d\rangle\leqslant 0.02$
reaches such high conductance only at $\varepsilon<0.2$. The inflexion point
in the curve at $\varepsilon\simeq 0.25$ matches to the HCP packing limit.

The effect of different gaseous atmospheres is presented in Fig. 6(b). First,
in powders with relatively small porosity or with relatively high bulk thermal
conductivity the presence of air or protective atmosphere in pores is
insignificant. In calculations with the highly conductive Al-12Si powder, the
influence of gas atmosphere on heat conductivity was found negligible.
Therefore, the calculated curves $k(\varepsilon)$ for argon and air coincided.
Second, in a case of powders with low conductivity like stainless steel this
effect becomes valuable and a significant difference between $k(\varepsilon)$
curves follows from computations (see Fig. 6(b)). The $k_{eff}(\varepsilon)$
curve in vacuum differs from ones for argon and air. All curves show the decay
abutting to the theoretical percolation threshold as for Al-based alloys.

Thus the following conclusions can be drawn. Influence of gas atmosphere on
thermal conductivity cannot be neglected for
$k^{\textrm{metal}}/k^{\textrm{gas}}<1000$. This outcome corresponds to the
previous studies [5, 13] and theoretical models as confirmed in what follows
in Fig. 9. The effective heat conductivity of a powder bed processed in argon
and air are very close owing to similar thermal diffusivities of the gas
atmospheres. From a practical standpoint, the calculated
$k_{eff}(\varepsilon)$ functions allow one to quantify the effect of
dynamically changed porosity and the consolidation coefficient during powder
compaction in SLM/SLM processing.

###  3.3 Dependence on the consolidation coefficient

Figure 7: Relative effective thermal conductivity $k_{eff}/k_{eff,0.25}$
calculated at different variance $\sigma$ of particle sizes in the log-normal
size distribution with the mean diameter $\langle d\rangle$. The randomly
generated configurations of particles with a fixed porosity
$\varepsilon\approx 0.4$ was used. The plot is scaled to $k_{eff,0.25}$ which
is the thermal conductivity at $\sigma=0.25$. The dimensionless overlapping
coefficient was kept constant as $\xi=0.125$ in all cases.

The effect of random distribution of particles on effective thermal
conductivity of Al-12Si powder with the average particle size $\langle
d\rangle=4.9$ $\upmu$m in presented in Fig. 7. The plot shows that this effect
is small if the average porosity $\varepsilon$ is fixed. Then the variation of
particle sizes also results in variation of the diameter of the contact area
between neighbor spheres. Thus the small value of variance $\sigma$ used in
the vast majority of calculations is a reasonably representative. In
principle, a high value of variance $\sigma$ in the size distribution allows
one to increase the powder packing density and hence its resulted effective
thermal conductivity. For instance, the decrease in average porosity from
$\varepsilon=0.55$ to $\varepsilon=0.40$ allows to double the effective
conductivity (see Fig. 6(b) for vacuum). Considering the results of Sec. 3.2
one can assume that the basic limiting factors for effective heat transfer in
powders are both porosity and the contact spot diameter. Increase of the size
variance $\sigma$ helps to reach denser packing which facilitates heat flow in
porous structures.

Figure 8: Dependence of the relative thermal conductivity
$k_{eff}/k^{\textrm{metal}}$ on the relative contact spot diameter
$d_{C}/\langle d\rangle$ (see scheme in Fig. 3) calculated for Al-12Si in air.
Here $k^{\textrm{metal}}$ corresponds to the conductivity of bulk metal. The
rectangle points and solid black curve denote the results obtained from direct
numerical simulations at $\varepsilon\approx 0.55$, red dashed curve
corresponds to the Hadley model Eq. (3.4) with substituted interpolation Eq.
(21) of consolidation coefficient (see Sec. 3.4).

Influence of the contact area between particles with a contact spot diameter
$d_{C}$ on an effective thermal conductivity is analyzed in Figure 8. In
accordance to the analytical model of the elementary cell by Kaviany [5],
under conditions of week overlapping with $d_{C}/\langle d\rangle<0.1$ a
significant descent of thermal conductivity of a metal porous structure
occurs. Further decrease of $\xi$ leads to full regression of the contact area
and a transition to the dry powder limit of non-consolidated particles occurs.
At moderate $d_{C}/\langle d\rangle$, a weakly increasing function of the
dimensionless thermal conductivity on the overlapping coefficient is observed
in air. In vacuum, there is no conductive heat transport hence the geometrical
effect of the contact zone is even more emphasized. The range of the
overlapping coefficient $\xi>0.2$ geometrically conforms with the preheated
powder where neck formation has been already completed or with a case of rough
consolidation during SLS and SLM processing on later stages of powder
compaction. We shall especially note that for the same values of $d_{C}$ the
thermal conductivity could differ significantly owing to the different contact
resistance and may be a subject of dynamic change in the SLM process. For
example, the analytical assessment of [5, 9] gives the value of the contact
resistance around $\approx 1/(k^{\textrm{metal}}d_{C})$ for sphere-to-sphere
Hertzian contact in vacuum. The overall effect on the thermal conductance with
surface effects such as oxide film for the above average contact pressure may
be assessed as $\approx 0.8\times 10^{-4}$ W m-2 K-1 for aluminum alloys and
$\approx 0.3\times 10^{-4}$ W m-2 K-1 for stainless steels [10, 56]. These
effects are very important in the region of small contact areas $d_{C}$’s for
dry powders and shall be disclosed in detail in future study in connection to
SLM utilized powders.

In laboratory experiments [3, 57] on sintered partially-melted powders, heat
transport occurs under conditions of randomly distributed particles where
$d_{\textrm{C}}/\langle d\rangle=0.1..0.3$. In the present work, the pre-
processed Al-12Si powder with similar characteristics was experimentally
tested to visually assess the rate of solid-state consolidation, see Fig.
1(b). In SLM processing, significant consolidation is provided by either
preheating of powder or by its plastic deformation during leveling of the
powder layer. At temperatures close to the melting temperature $T_{m}$ the
particles rapidly start to conglomerate. At the beginning, the total flux
through the metal matrix is sufficient to provide initial sintering between
particles. As a result, the role of specific contact zones quickly decreases
thanks to an increasing number of conglomerated particles. Thus at the late
stages of sintering of the metal matrix with low porosity $\varepsilon$ the
effect of the overlapping coefficient and the area of the contact zone decays.
To conclude, according to the present calculations at the moderate porosities
$\varepsilon=0.4..0.6$ the influence of the contact zone area for a preheated
sintered powder for $d_{C}/\langle d\rangle\geqslant 0.2$ is little. In the
range $d_{C}/\langle d\rangle<0.2$, thermal conductivity of the powder
drastically drop to zero as the contact zone becomes smaller.

###  3.4 Comparison to existed models

Figure 9: Comparison between the analytical models of effective heat
conductivity, experimental data by Nozad et al.[58, 59] and numerical
simulations carried out in the present work at $\varepsilon\approx 0.38$ for
powders with spherical particles. The curves depicted in the figure are the
models of Maxwell upper bound (Lykov model), Ofuchi-Kunii-Smith, Zehnder-
Schlünder, Chen-Cheng-Hsu, Maxwell lower bound, Kaviany and Hadley [6, 60, 27,
7, 4, 5, 13] correspondingly. The present results are depicted by the squares
and triangles (for different gas atmospheres) in different dashed boxes for
stainless steel, iron and Al-12Si. The shape and color of the symbols
correspond to different contact spot diameters $d_{C}/\langle d\rangle$ and
gas atmosphere used in simulations. The figure is based on the diagram by
Henriksen [61].

Comparison of our numerical simulations with different analytical models is
provided in Fig. 9. The plotted diagram gives the ratio
$k_{eff}/k^{\textrm{gas}}$ of the effective thermal conductivity scaled to the
thermal conductivity of gas in the $y$-axis. The ratio
$k^{\textrm{metal}}/k^{\textrm{gas}}$ of the thermal conductivities of metal
and gas is plotted in the $x$-axis. The diagram is depicted in the log-log
scale and it was calculated at the fixed porosity $\varepsilon=0.38$.

The Lykov model also known in literature as the Maxwell upper bound model for
thermal conductivity of two-phase media [5, 6] yields the upper geometrical
limit of conductivity. The Maxwell lower bound corresponds to the lower limit
of $k_{eff}/k^{\textrm{gas}}$ where heat transfer is mainly controlled by
thermal conductivity through the gas phase.

The models of Zehnder-Schlünder [7] and Chen et al. [4] are very close to each
other for the given $\varepsilon$ and are plotted by the same line. These
models are of the elementary-cell type (also the Ofuchi-Kunii-Smith [60, 27]
model) and consider heat transfer in powders in the approximation of point
contacts between particles. This approach is valid for dry-powders and it
results in the $k_{eff}$-curves located lower than our data for partially-
melted powder. The model of Kaviany [5] considers finite necks between powder
particles and it allows to describe high-conductive powders. It is worth
mentioning that the semi-empirical Kaviany model is conceptually close to our
approach because both models account for formation of necks with variable
cross-section areas between particles.

The Hadley model suggested in [13] introduces a consolidation coefficient
$\alpha_{H}$ which in fact is a conversion coefficient to match two limiting
cases of the Maxwell upper and lower bound models:

$\displaystyle\frac{k_{eff}}{k^{\textrm{gas}}}=\left(1-\alpha_{H}\right)\\!{\frac{\varepsilon{\it
f_{0}}+\left(1-\varepsilon{\it
f_{0}}\right)\frac{k^{\textrm{metal}}}{k^{\textrm{gas}}}}{1-\varepsilon\left(1-{\it
f_{0}}\right)+\frac{k^{\textrm{metal}}}{k^{\textrm{gas}}}\varepsilon\left(1-{\it
f_{0}}\right)}}+$
$\displaystyle{+\alpha_{H}\frac{2\left(1-\varepsilon\right)\left({\frac{k^{\textrm{metal}}}{k^{\textrm{gas}}}}\right)^{2}+\left(1+2\,\varepsilon\right)\frac{k^{\textrm{metal}}}{k^{\textrm{gas}}}}{1-\varepsilon+\left(2+\varepsilon\right)\frac{k^{\textrm{metal}}}{k^{\textrm{gas}}}}}.$
(20)

where ${\it f_{0}}$ is an empirical constant, which is selected as ${\it
f_{0}}=0.9$ in all our cases. This model allows one to describe both partially
melted and dry-powders in a single coherent approach. At $\alpha_{H}=0.01$,
the media is loose-packed and the curve plotted in Fig. 9 is very close to the
data of Nozad et al. [58, 59] for dry powders. Our computations are presented
in Fig. 9 by red squares and blue triangles in the columns combined for
metallic powders of stainless steel, Al-12Si and pure iron. The parameters of
iron were taken from [62, 41] and the mean diameter was $\langle d\rangle=5$
$\upmu$m.

The results of present calculations in a case of partially-melted particles
conform well to the Hadley model at intermediate and large
$\alpha_{H}\geqslant 0.03$. Further, in consolidated powders with good thermal
contact between particles the dependence of the effective thermal conductivity
$k_{eff}$ on the contact spot diameter $d_{C}$ is sufficiently small (see Fig.
8 in the interval $d_{C}/\langle d\rangle\geqslant 0.2$). This conclusion has
been also confirmed by both the Hadley’s and our models. For smaller
$\alpha_{H}<0.03$, the function $k_{eff}/k^{\textrm{gas}}$ depends on the
contact spot diameter non-linearly. The contact area decreases as square of
the contact spot diameter and hence the heat flux between the particles
drastically deteriorates as was shown in Fig. 8 in the interval $d_{C}/\langle
d\rangle<0.2$. To link our model to the analytical expression we propose a
simple approximation of the consolidation coefficient $\alpha_{H}$:

$\alpha_{H}=1-\exp\left(-A\left({d_{C}}/{\langle
d\rangle}-d_{0}\right)\right),$ (21)

where $A$ is pre-exponential factor and $d_{0}$ is a dry-powder shift. In
Figure 8, one can find fair agreement between the numerical data (black curve)
and the resulted function $k_{eff}/k^{\textrm{metal}}$ analytically calculated
using Eq. (3.4) with substituted Eq. (21) at $A=9$, $d_{0}=0.03$,
$\varepsilon=0.5$ (red dashed curve).

The smooth rise of $k_{eff}$ in Fig. 9 reported by data of Nozad et al.,
Kaviany and Hadley models for highly heat-conductive metals such as pure iron
and Al-based alloys has also been reproduced in our calculations. The black
rectangular points in Fig. 8 correspond to weak thermal contact between
particles at $d_{C}/\langle d\rangle\simeq 0.02$. There is a clear point of
inflection in the $k_{eff}$ curves of the Hadley model. The inflection occurs
at about $k^{\textrm{metal}}/k^{\textrm{gas}}\sim 1000$. This transition in
the $k_{eff}$ is not described well by the elementary-cell models.

One can conclude that good agreement with the Hadley model has been received
in our simulations for consolidated powders. In the dry-powder limit,
correspondence is worse. Therefore, we think that the numerical model has
better accuracy in the account of local morphology and the presented
approximation of the consolidation coefficient will lead to more robust
analysis in wide interval of parameters of partially consolidated powders.

##  4 Conclusions

The effective thermal conductivity $k_{eff}$ as a function of porosity
$\varepsilon$ for Al-12Si and stainless steel powders was calculated in a
series of computer simulation runs. The obtained results account for the
effects of consolidation which can drastically change heat conductivity of
powders in SLS/SLM processing.

The obtained effective heat conductivities for commercial powders are suitable
for rapid modeling of powder consolidation and optimization of selective laser
sintering/melting. The temperature dependence of $k_{eff}$ can be introduced
by multiplication of the normalized $k_{eff}/k^{\textrm{metal}}$ function on
the bulk heat conduction coefficient which is dependent on temperature.

It was found that the function $k_{eff}(\varepsilon)$ is a nonlinear curve
which has three distinctive intervals. At porosity $\varepsilon<0.26$, the
powder layer has dense packing (briquette) where the contact spots between
particles are developed and drastically improvement of heat conductivity is
observed. The calculated $k_{eff}(\varepsilon)$ curves have quasi-linear
behavior $k_{eff}\sim k_{0}(1-\varepsilon)$, Fig. 6. In the intermediate
interval of porosities, strong dependence on the contact spot diameter $d_{C}$
is registered. Hence, the $k_{eff}(\varepsilon)$ curves are shifted to lower
values comparing to the linear model. Also there is a bottom bound at
$\varepsilon>0.7$ in Fig. 6. which corresponds to the percolation limit _i.e._
where a break of continuous heat transfer through the metallic phase occurs.

The effect of gas atmosphere is as follows. The thermal conductivity of
stainless steel powder is affected by heat transfer through the gas phase. At
the same time, Al-12Si powder is less sensitive to heat conduction through
gas. In the paper, the criterion for accounting of the heat conductance in gas
has been suggested as $k^{\textrm{metal}}/k^{\textrm{gas}}<10^{3}$. There is
no substantial difference in $k_{eff}$ between the powders processed in argon
and air.

The effective heat conductivity of partially consolidated particles steeply
increases if the dimensionless diameter of thermal contacts becomes
$d_{C}/\langle d\rangle>0.2$, Fig. 8. The heat flux through a contact spot
depends on its diameter $d_{C}$ as a quadratic function, thus in the range
$0.2<d_{C}/\langle d\rangle<0.8$ a weak dependence of $k_{eff}$ on $d_{C}$ is
obtained. The nonlinear dependence is found at $d_{C}/\langle d\rangle<0.1$
where the degree of consolidation becomes smaller approaching the dry-powder
limit of thermal conductivity.

Based on the performed analysis, we suggested a new analytical fitting Eq.
(21) which links the consolidation coefficient $\alpha_{H}$ of the Hadley
model [13] Eq. (3.4) to the realistic powder parameters which are the diameter
of contact necks $d_{C}$ and mean particle diameter $\langle d\rangle$. The
results of numerical simulations show that the distribution of particle in
size does not affect $k_{eff}$ explicitly. The increase of the distribution
variance leads to a more dense packing with a smaller porosity.

Suggested coefficients $k_{eff}$ were derived to use in the approximation of
continuous media. Thus, it is valid only under the conditions of local thermal
equilibrium. For SLS/SLM processes, the characteristic scale of applicability
of $k_{eff}$ is limited by $L>200$ $\mu$m for macroscopic models due to high
temperature gradients observed in the powder layer.

In summary, the suggested numerical model broadens the description of
transport characteristics of partially melted powders. It substantially
extends the interval of applicability of the analytical models from dry-
powders to fully-consolidated powders. This allows to evaluate more precisely
heat transfer in the powder layer during its compaction in SLM processing.
Also the thermal conductivity of preheated sintered powder can be assessed.

This study was financially supported by Russian Science Foundation, project
19-79-20012.

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