ENERGY ABSORPTION AND PENETRATION
IN SELECTIVE LASER SINTERING: A RAY TRACING MODEL
X.C. Wang & J.P. Kruth
K. U. Leuven - Department of Mechanical Engineering - Division PMA
ABSTRACT An analytical ray tracing model is developed to simulate the energy
absorption and penetration during the selective laser sintering (SLS) of metal
powders. The model is applied to a Fe-Cu powder mixture. It gives an evaluation of
the energy absorption and penetration and an estimation of the sintering zone
dimension. The simulations will help to understand the physical phenomena
involved, to identify the processing window and to optimize the SLS process.
1. INTRODUCTION
Selective laser sintering is a Material Accretion Manufacturing or Rapid Prototyping
(RP) technology (1). It produces parts in a layer-by-layer fashion. The SLS
technology allows a direct coupling with the CAD-model of the product, in which
successive cross sections are calculated, to produce three dimensional parts without
dedicated tools, like dies, as used in conventional sintering. Total production time
and cost can hence be reduced.
K.U. Leuven aims at the development of the SLS process to make metal parts
directly from commercially available powders, without using a polymer binder or a
specially developed metal powder. Some successful applications have been made to
high strength powder mixtures, like Fe-Cu, WC-Co and TiC-Ni. In order to master
well the process, it is necessary to investigate the influence of processing and
material parameters, such as laser power, scan speed, mixture ratio and particle size.
The utilization of a new powder mixture requires extensive testing, which can be
expensive and time-consuming. The development of reliable analytical and numerical
tools to analyze the SLS process is important to reduce the number of tests needed.
The simulations will help us not only to get a better understanding of the physical
phenomena involved (temperature evolution, phase changes, fluid and mechanical
problems, etc.) but also to identify the processing window and to optimize the SLS
process through proper selection of some processing and material parameters.
Although several physical phenomena are involved in the process and a coupling
between them exists, SLS is mainly dominated by its thermal process. The actual
investigation is limited to thermal simulations. An analytical ray tracing model is
developed for evaluating the total energy incoupling (the ratio between the absorbed
and the total input energy) and optical penetration of the laser beam (energy
absorption profile versus the depth into the powder bed) and for estimating the
sintering zone dimension. The model is then applied to a Fe-Cu powder mixture,
irradiated by Nd:YAG or by CO2 laser, two suitable laser sources for direct SLS of
metal parts.
2. SLS PROCESS
The basic material in SLS consists of a mixture of two metal powders: a high melting
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point metal, called the structural powder and a low melting point powder, called the
binder. This mixture is spread as a thin layer, normally between 0.1 mm to 0.5 mm,
on top of a container by means of a deposition system then is heated by a moving
laser beam to the temperature at which the binder melts.
With liquid formation there is rapid initial bonding due to the capillary forces exerted
by the wetting liquid on the solid particles. This is the first stage of liquid phase
sintering (LPS), called the rearrangement stage. The second (solution precipitation)
and third (solid stage sintering) stage cause further densification of the parts. Because
these stages are based on migration of atoms, those densification steps require longer
sintering times.
In selective laser sintering, no pre-compaction is applied to the powder bed. This
allows the reuse of unsintered powders. As a result, the powder bed is very loose.
Another important characteristic is that the sintering time is extremely short in SLS,
because the moving laser beam supplies energy to each particle only for about 1 ms ~
0.1 s. Due to low powder density and short sintering time, only the rearrangement
stage occurs during the laser heating. Once the binder is molten and has flown into
the pores between the unmolten structural powders, the system cools down. No
further densification may take place in such a short time interval. This results in very
low density of the green parts. For getting functional 3D parts, it is indispensable to
perform a post processing by, for example, an infiltration with liquid metal.
3. PHYSICAL PHENOMENA IN SLS
SLS is a complicated process, involving several physical phenomena. These include:




Heat generation and transfer, including the heating of the powder bed and the
cooling of the sintered sample;
Microstructure evolution, including the porosity evolution and phase changes
(melting and solidification of the binder);
Fluid problem (molten binder flowing in the solid lattice);
Mechanical problem (no uniformly distributed thermal strains during the
cooling stage may cause residual stresses and distortions of parts produced).
In these coexisting physical phenomena, the thermal problem is dominant. Knowing
the temperature distribution and evolution is essential to describe suitably the SLS
process. However, the temperature distribution and evolution is influenced by other
phenomena. The different physical phenomena interact at different processing stages
with different importance. As a result, a coupling analysis should be performed. This
may complicate our task. Fortunately, the influence of other phenomena on the
temperature evolution is weak because of the extremely short sintering time. For
example, there is not enough time for the molten binder flowing really in the powder
bed. It shows only a flowing trend before the situation is frozen. At the same time,
this extremely short time does not allow the densification of the parts. Consequently,
the porosity changes rarely during the SLS process. Even the influence of the phase
changes is not so important due to the limited quantity of molten powder. Finally,
without a real important large mechanical deformation in the material, the mechanical
problem may be studied separately using the results of the thermal study as inputs.
The uncoupling of the SLS process will simplify considerably our problem. As a
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simplification and without losing much precision, only the thermal problem is taken
into consideration in this paper.
4. A RAY TRACING MODEL
4.1. Energy Absorption and Penetration
During the selective laser sintering process, the powder mixture is irradiated by a
moving laser beam. This is an energy transformation process, in which the light
energy of the laser beam is converted into thermal energy that causes heating of the
powder bed. It is important to understand the interaction between the laser beam and
the powder bed. A good understanding of this interaction helps us not only to control
more easily the process (so lead to more accurate parts having enhanced mechanical
properties), but also to define a set of requirements for new sintering powders (hence
lead to a easier development of powders more suitable for sintering).
It is evident that not all energy contributes to the heating of the powder bed. So a
parameter measuring "energy absorption" should be defined. On the other hand,
unlike an opaque continuous medium, the powder bed allows a certain penetration of
the light energy of the laser beam though multiple reflection into the powders. In
order to describe how the energy will be absorbed in depth, an "energy penetration"
parameter should be introduced.
The absorption of the powder bed is described by the total energy incoupling. It is
defined as the ratio between the absorbed and the total input energy:
Total Energy Incoupling 
Absorbed energy
 100(%)
Input energy
(1)
The total energy incoupling into the powder bed should be distinguished from the
material absorption coefficient. It accounts for multiple reflection/absorption of the
light in powders and is influenced not only by the laser source through its wavelength
but also by the powder bed itself through the powder material (so its absorption
coefficient), mixture ratio, mean particle sizes and shapes, etc.
The energy penetration of a laser beam is defined by the absorption profile across the
powder bed depth. It measures the optical penetrance of laser light into the powder
bed or, in other words, the transparency of the powder bed to a given light. It gives us
an idea how the laser energy penetrates into the powder bed. The energy penetration
depends also, like the energy absorption, on several processing and material
parameters, of which the wavelength of the laser beam, the powder materials, the
mixture ratio and the mean particle sizes are most important. Since the thermal
conduction in SLS powder beds is very weak due to its very low density, taking this
energy penetration into consideration becomes indispensable to get reliable result.
4.2. Assumptions of the Model
In order to evaluate energy absorption and penetration during the direct selective
laser sintering of metal powders, an analytical ray tracing model is developed. The
simulation model is based on the following assumptions:
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





A mixture of two powders is studied. The particles are perfect spheres. Each
powder has a uniform grain diameter, but the diameters of both powders
differs;
The particles of the two materials are randomly located and distributed in
space;
The laser strikes the powders perpendicularly to the powder bed surface;
The powder particles have a specular reflectivity;
The absorption coefficients of the powders are equal to their solid material
values. The absorptivity is independent of the incident angle and of the
temperature.
The powder bed is put in vacuum. This means that the ray path will not be
influenced and no energy will be lost in the pores of the powder bed.
It should be noted that any of these assumptions is indispensable. If necessary they
can be easily modified or extended without difficulty. For example, some possible
extension of the model can be:





any other particle shapes may be studied if only their geometry can be
described mathematically;
size non-uniformity of each powder may be taken into consideration, as is
often the case in the reality;
entrance angle is easily introduced if necessary;
it is possible to take into account the incident angle dependence and the
temperature dependence of the absoptivity of powder particles;
diffuse reflectivity can be used instead of specular reflectivity.
4.3. Ray Reflection and Energy Absorption
A 2D illustration of the model is shown in Fig. 1.
Figure 1. 2D illustration of light reflection of the ray tracing model
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A number of rays, randomly or regular located, are emitted from the xy-plane z = 0.
They are perpendicular to the powder bed surface. Each ray has a certain amount of
energy, calculated from the laser power, the scan speed and other parameters. At each
impingement on a particle, part of the energy of the emitted ray is absorbed by this
particle and the rest is reflected. This reflection is assumed to follow a specular
reflection law. This procedure continues for each emitted ray until the ray is
considered to "disappear", in case of either being reflected outside of the powder bed
through the z = 0 or z = c surface or its energy becoming negligible after several hits
against particles.
During the simulation, the energy absorbed by each particle is accumulated. At the
end, the energy absorbed by all P1 and P2 particles will be used to calculate the total
energy incoupling and the energy absorbed respectively by both materials.
4.4. Estimation of the Sintering Zone through Laser Beam Scanning Simulations
Before real 3D parts may be fabricated using SLS technology, one should determine
the track width and the sintering thickness of a single sintered line or the thickness of
a single sintered layer. Investigation of the relation between these thickness and
width and the processing parameters, including the laser power and scan speed is
important to chose proper settings and to determine the required scan spacing and the
powder layer thickness which can be sintered.
Due to the large porosity of SLS powder beds, the contact between particles is almost
punctual. This makes it possible to neglect the heat transfer between particles. The
ray tracing model may be applied to estimate roughly the dimension of sintered lines
in SLS.
In the laser beam scanning simulation of a single line sintering, the real laser beam is
represented as a bundle of parallel rays, equally spaced. The energy of each emitted
ray is calculated from the energy distribution in the irradiated zone. This distribution
depends on the power density, scan speed and other parameters. It varies in y
direction (perpendicular to the laser beam scanning direction). In the scanning
direction x, the energy distribution is uniform. The energy distribution for the
uniform distributed cylindrical laser beam is given by:
e


 


4P d 2  4 y 2
vd
2
0
1
d
2
1
y d
2
y
(2)
where P is total laser power, d is the diameter of the laser beam spot and v is the scan
speed of the moving laser beam. The total energy of any emitted ray is calculated by
integrating the energy distribution (Equation 2) around its initial position in x-y plane
according to its representing zone.
During the laser beam scanning simulation, the absorbed energy of each individual
particle is accumulated to get its total absorbed energy (Ei for ith particle). At the
same time, we know the energy necessary to fuse this particle, calculated respectively
for both materials according to the following equation:
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Em

(c p  T  c l )    V
(3)
where cp (KJ/KgK) is the specific heat, ΔT (K) is the temperature rise needed for
melting, cl (KJ/Kg) is the latent melt energy, ρ (kg/mm3) the density and V the
volume of the spherical particle. At the end of the simulation, a simple comparison of
the absorbed energy Ei to Em will determine whether any particle absorbs enough
energy to melt or not. The sintering zone dimension is evaluated from the most sidewise molten particles.
5. VALIDATION AND APPLICATION
5.1. Validation of the Model
In order to check how the simulation results fit the reality, a series of absorption
simulations were realized using different values of the absorption coefficient of
individual particles, varying from zero to one. Figure 2 shows the comparison
between simulations and measurements (2, 3). A good agreement is found.
100%
Incoupling into the Powder Bed
90%
80%
70%
60%
50%
Simulations
40%
Measurements
30%
20%
10%
0%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Absorption of Plain Particle Material
0.8
0.9
1
Figure 2. Experimental validation of the model
5.2. Energy Absorption and Penetration in a Fe-Cu Powder Mixture
The developed ray tracing model is applied to a Fe-Cu powder mixture. This is a
well-described powder mixture for liquid phase sintering with attractive mechanical
properties: the hard Fe lattice is dispersed in a ductile Cu matrix. Fe-Cu system is one
of first powder mixtures that have been used successfully in direct laser sintering to
produce 3D metal parts at PMA. Much care should however be taken for this powder
mixture due to the physical characters of two materials. Their melting points are
relatively close (Fe: Tm = 1535 ºC and Cu: Tm = 1083 ºC) and their absorption
coefficient, on the contrary, are very different (the absorption coefficient of Cu, the
binder which should be melted, is very small compared to that of Fe). If no special
precautions are taken, Fe particles tend to melt before Cu particles do. This results in
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a complete melting, it may shut off the micro channels, which are necessary to allow
the evacuation of air or infiltration of liquid metal, and make it impossible to further
densify the remaining porosities of green parts during post treatments. The molten
parts do not have the desired composite microstructure that ensures good mechanical
properties and may yield high shrinkage and distortion (4).
A Fe-Cu powder bed is constructed with the material parameters corresponding to
that used in our laboratory. The diameters of two powders are respectively 50 m
(Fe) and 30 m (Cu). The mixture ratio of Cu is 30wt%. The powder assemblage is
constructed with 2779 Fe spheres and 4844 Cu spheres.
Nd:YAG laser and CO2 laser are used in the simulations. The absorption coefficients
of Fe and Cu, when irradiated by these two laser sources, are taken from (3). They
are shown in Table 1.
Table 1. Absorption coefficient of Fe and Cu for Nd:YAG laser and for CO2 laser
Materials
Fe
Cu
Absorption coefficients
YAG (λ=1.06 μm)
CO2 (λ=10.6 μm)
0.3
0.035
0.1
0.015
The simulations consist of emitting a series of randomly located rays. For each laser
source, 5 simulations (tests) are realized. The total energy incoupling and the energy
absorbed respectively by the two powders are given in Table 2. We find that
Nd:YAG laser is much more absorbed by the powder. The efficiency of this laser
energy compared to CO2 laser is also proved by the experiments (5).
Table 2. Total energy incoupling (Etotal) and the energy absorbed respectively
by two powders (EFe and ECu)
Nd:YAG Laser (λ=1.06 μm)
Tests
Etotal (%) EFe (%) ECu (%)
1
65.59
53.61
11.98
2
66.16
52.49
13.66
3
65.49
53.84
11.62
4
65.28
53.23
12.05
5
67.20
55.87
11.34
Average
65.96
53.81
12.13
CO2 Laser (λ=10.6 μm)
Tests
Etotal (%) EFe (%)
1
27.99
22.00
2
25.71
20.31
3
26.79
20.88
4
26.89
21.10
5
24.81
19.45
Average
26.44
20.75
ECu (%)
5.994
5.339
5.910
5.791
5.361
5.691
The energy penetration is illustrated in Fig. 3. We compare in this figure the ratio of
the accumulated energy absorbed from the powder bed surface up to a certain depth
versus the total absorbed energy at full depth of 1 mm. We find an important
difference between both laser sources. We observe more than 70 % of the absorbed
energy is concentrated within a depth of 0.2 mm for Nd:YAG laser, but only about 42
% within this depth for CO2 laser. Almost all energy (97.5%) is absorbed within a
depth of 0.5 mm for Nd:YAG laser, but only 82.4% being absorbed within this same
depth for CO2 laser. This more uniform absorption profile of CO2 means that more
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energy will be dissipated in the powder bed depth and less energy will contribute in
binder melting.
Acc. Abs. Energy / Total Abs. Energy
1
0.9
Nd: YAG Laser
0.8
CO2 Laser
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Depth (mm)
Figure 3. Accumulated absorbed energy in function of the powder bed depth:
comparison of two laser sources
5.3. Estimation of Sintering Zone Dimension
The sintered thickness and sintered width of a single sintered line are estimated for
the same Fe-Cu powder bed presented above, irradiated by a Nd:YAG laser. Three
laser powers are studied, 8.1 W, 17.9 W and 28.1 W (these are effective powers,
corresponding respectively to nominative powers of 40 W, 50 W and 60 W). The
scan speed varies from 3 mm/s to 25 mm/s. Figure 4 shows the sintered thickness and
width compared to measurements (4). A general agreement is found even though
simulations give only a rough estimation by neglecting thermal conduction in the
powder bed.
Power = 8.1, 17.9 & 28.1 W - Scan Speed = 3, 5, 7.5, 10, 15, 20 & 25 mm/s
2.00
1.80
Thickness & Width (mm)
1.60
1.40
1.20
1.00
0.80
Thickness Simulations
Thickness Measurements
Width Simulations
Width Measurements
Trendline of Simulaiton Results
0.60
0.40
0.20
0.00
0.00
2.00
4.00
6.00
Energy Density (J/mm2)
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8.00
10.00
12.00
P
)
vd
for Nd:YAG laser: comparison between simulations and measurements
5.4. Processing Efficiency
Figure 4. Sintering thickness and width versus energy density ( E 
The process efficiency of the laser is determined as the fraction of the laser energy
that is theoretically needed to sinter the part and the laser energy that effectively is
used. The minimum energy, that can melt Copper, can be calculated as follows:
Cu
Fe
Emin  mCu .(cCu
p . T  cl )  mFe . (cp . T)
(4)
, Fe
where m Cu , Fe is the mass of Fe and Cu of the powder mixture, cCu
is the heat
p
capacity of Cu and Fe at constant pressure, c Cu
l is the latent melt energy of Cu and T
is the temperature rise for melting Cu. Finally, the process efficiency is calculated:
process 
E min
Pd / v
(5)
where d is the laser beam spot diameter. The process efficiency calculated from
simulation results and measurements (4) are compared in Fig. 5.
Power = 8.1, 17.9 & 28.1 W - Scan Speed = 3, 5, 7.5, 10, 15, 20 & 25 mm/s
40.00
35.00
Calculation Based on the Simulations
Calculation based on the Measuremets
Trendline of Simulation Results
Process Efficiency (%)
30.00
25.00
20.00
15.00
10.00
5.00
0.00
0.00
2.00
4.00
6.00
8.00
10.00
12.00
Energy Indensity (J/mm2)
P
)
vd
for Nd:YAG laser: comparison between simulations and measurements
Figure 5. Process efficiency of the laser versus energy density ( E 
Here again, a good agreement between simulations and measurements is found. From
this figure, we observe a decrease strongly of this efficiency with the increase of the
energy density.
6. CONCLUDING REMARKS
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In order to master well the SLS technology for direct sintering of metal powders and
to optimize the process through the processing and material parameters, each new
material combination requires extensive testing. This can be expensive and timeconsuming. Developing some analytical and numerical tools to simulate the SLS
process should be helpful. However, a precise evaluation of the energy absorption
and penetration is indispensable for getting a reliable simulation of the process. The
simulation results reported in this paper show that the developed model may give a
good estimation not only to the energy absorption and penetration but also to the
sintering zone dimension.
Concerning the two laser sources, it is found that the Nd:YAG laser is more efficient
in the direct selective laser sintering of metal powders: it is absorbed more by the FeCu powder bed and by most other metal powder mixtures and the absorbed energy
concentrated more near the surface. It is this part of energy that will contribute to
binder melting and sintering. While for CO2 laser, the absorbed energy contributes
more to heat that is uselessly dissipated deeper into the powder bed. It may be
concluded that Nd:YAG laser is at the moment the best laser for selective laser
sintering of metal powders. This laser is more efficient, it has a larger processing
window and results in green parts with higher density.
An inconvenience does nevertheless exist for Nd:YAG laser. The concentration of
the absorbed energy near the powder bed surface leads to a larger temperature
gradient in depth. This will cause a larger stress gradient in this direction during the
cooling stage. The produced parts tend to curl more. If the process is not well
controlled, thermal deformation will cause cracks and delaminations. This is more
probable to occur especially for the first layer of powder, which may fail to stick to
the base plate.
ACKNOWLEDGEMENTS
This research is supported by the Belgian national fund IUAP P4/33. The authors
would like to thank all involved in the collaboration in this research project.
REFERENCES
1
J.P. Kruth: 'Material incress manufacturing by rapid prototyping
techniques'. Annals of the CIRP, 40(2) 603-614, 1991
2
N.K. Tolochko, T. Laoui, Y.V. Khlopkov, S.E. Mozzharov, V.I. Titov and M.B.
Ignatiev: 'Absorptance of powder materials suitable for laser sintering', Rapid
Prototyping Journal, 6(3), 2000
3
W.W. Duley: Laser Processing and Analysis of Materials, Plenum press, 1983,
New York
4
J.P. Kruth, B. Van der Schueren, J.E. Bonse and B. Morren: 'Basic powder
metallurgical aspects in selective metal powder sintering'. Annals of the
CIRP, 45(1) 183-186, 1996
5
J.P. Kruth, P. Peeters, Th. Smolderen, J. Bonse, T. Laoui and L. Froyen:
'Comparison between CO2 and Nd:YAG lasers for use with selective laser
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sintering of steel-copper powders', Revue International de CFAO et
d'informatique graphique, 13(4-5-6), 95-112, 1998
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