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A combined theoretical and experimental approach to model polyamide 12
degradation in selective laser sintering additive manufacturing
Feifei Yang1, Xu Chen1
*
1Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, USA
Abstract: Selective laser sintering (SLS) generates complex high-performance parts from micrometer-
diameter powders. In polyamide 12 SLS, a considerable amount of expensive polyamide 12 materials
remains un-joined in the additive manufacturing (AM) process. Such materials, particularly the ones near
the heat-affected zones (HAZ), go through irreversible chemical degradations originated from thermal
oxidations. Despite efforts in understanding the degradation mechanisms of the materials, full modelling of
the complex material degradation remains not well understood. In this work, through a combined theoretical
and experimental approach, we propose a first-instance kinetic model considering the effects of both oxygen
and laser to model the material degradation in polyamide 12 SLS. By mapping the actual material
degradation rates into the oxidation physics and data-driven parameter identification, we obtain the
coefficients of the actual coupled oxygen and laser effects. Through sensitivity analysis, we derive the
fitting equations between the sample degradation rates and the oxidation time. The proposed kinetic model
can predict the oxidation rates of pure or mixed materials using two easily available parameters: materials
density and oxidation time. We show that the laser effects are 4-time stronger than oxygen effects on
polyamide 12 degradation. The predicted oxidation matches on average 89.53% with the actual SLS
degradation rates, in contrast to a 34.48% accuracy from a basic autoxidation model. Besides, the paper
identifies how coupling of oxygen, laser irradiation, and preheating impacts the rate of material degradation.
Key words: Kinetic modelling; Polyamide 12; Thermal oxidation; Coupled oxygen and laser effects;
Selective laser sintering
1. Introduction
Additive manufacturing (AM) is a collective term with unrivalled design freedom to fabricate
functional applications by joining layers of materials on top of each other [1-3]. Selective laser sintering
(SLS) is a popular powder-based AM process with superior potentials to produce products with high
mechanical properties and good thermal stability compared to other 3D additive techniques [4,5]. The
capability to process almost any material, including polymers, metals, ceramics, and many types of
composites, further extends the popularity of SLS [6]. Supporting materials are not needed in SLS as the
powders can directly act as support to the printed parts [7]. Resulting from the high flowability, high melting
*
Corresponding author.
Email addresses: yangff@uw.edu (Feifei Yang), chx@uw.edu (Xu Chen).
enthalpy, and sharp melting peak, polyamide 12 appears to be the most suitable material among the wide-
ranging material scope appliable for SLS [8-10]. Polyamide 12 (and its compounds) takes up approximately
90% of complete industrial consumption [11].
The extensive usage of polyamide 12 powders in SLS results in a large amount of un-sintered powders
after going through complex degradations [12-14]. Previous research revealed that irreversible oxidation
and post-condensation dominate the aging process and change the polymer chemical structures by chain
scission, branching, and chain cross-linking [13,15,16]. The macro-structural chain cross-linking attributes
to an increase in the material molecular weight and a decrease in the melt flow index (MFI) [17,18]. As the
molecular weight increases with the powder aging, melt viscosity also increases and powder flowability
decreases [14,16]. Aging affects little the distribution of powder sizes but leads to the deteriorated thermal
property and reduced surface morphology [13].
Despite the property changes, a considerable amount of un-sintered polyamide 12 residues (80% -
90%) has the potential to be reused for further applications [14]. One of the solutions to the successful reuse
of SLS residue is to fabricate the powder residue into feedstock for other AM processes without
significantly reducing its value. Polyamide 12 powder for SLS is priced at around $150/kg (in 2020
currency). The cost of extrusion-based additive manufacturing (EAM) or fused deposition modelling (FDM)
polyamide 12 filament is approximate $100/kg, while the cost of polyamide 12 pellets for conventional
plastics processing is below $3/kg [19]. It is more economical to process the polyamide 12 powder residue
into filaments for EAM or FDM rather than pellets for conventional plastics processing [19,20]. The
common practice of reusing the polyamide 12 powder residue in SLS is to mix 50% new powders with 50%
reclaimed powders from past experiments [13,19].
Besides, several research works dedicated to better understand the aging mechanisms of polyamide 12
in SLS. Diller TT et al. [21] built computational models at two complexity levels, a one-dimensional model
and a two-dimensional finite element model, to explore the influences of heat transfer on the aging of
polyamide 12 in SLS. Yuan M et al. [22] measured thermal conductivity of fresh and aged polyamide 12
powders to establish a baseline for thermal aging control in SLS. Dadbakhsh S et al. [13] examined new
and aged polyamide 12 powders along with their mixtures to identify the effect and mechanisms of in-
process aging on material thermal and coalescence behaviors in SLS. Chen P et al. [12] investigated the
aging mechanisms and microstructural evolution of polyamide 12 in SLS. Bernard et al. [23] performed
thermogravimetric experiments with mass spectrometric analysis to obtain the kinetic parameters on the
thermal degradation of polyamide 12 in SLS.
Heated and exposed to intensive laser radiations, the nature of material degradation in SLS involves
coupled thermal and laser-induced oxidation reactions. Despite the previous works, the kinetics and the full
modelling of polyamide 12 degradation in the complex SLS remain not well addressed. We propose a first-
instance kinetic scheme considering both the oxygen and laser effects to model material degradation in SLS
through multi-physics modeling and data-driven parametric identification. In this work, we conduct SLS
printing experiments and calculate the actual polyamide 12 degradation rates through Fourier-transform
infrared spectroscopy (FTIR) results and Beer-Lambert’s law. By data-driven parameter identification of
the actual SLS degradation rates into the oxidation model, we obtained the coefficients of actual coupled
oxygen and laser effects in SLS. Through a further sensitivity analysis, we derive the relationship between
the sample degradation rates and oxidation time. The proposed model can predict the degradation rates of
materials using materials density and oxidation time. The new kinetic model applies to not only pure
material but also mixed powders. Furthermore, using the proposed kinetic model, we identified the
influences of the coupled oxygen, laser irradiation, and preheating on the rates of material degradation in
the SLS of polyamide 12. The findings provide new knowledge of quantitative influences of the process
parameters on material degradation and on approaches to reduce oxidation in SLS.
2. Method
Figure 1 presents an overview of the proposed research approach. We discuss the details of each step
in the following sections.
Degradation rates
SLS experiment
Basic autoxidation scheme of polymer
Oxidation mechanism of polyamide 12
Conventional oxidation model of polyamide 12
FTIR
Theoretical Experimental
Comparisons between the model results and experimental results
Oxygen effectsCoupled laser and oxygen effects
Basic autoxidation
model
SLS sample group 1
Coefficients of aging SLS sample group 2
Data-driven parameter
identification
Sample density
Aging time
The proposed model
Model verification
Basic autoxidation
model
Figure 1 The proposed approach to build the kinetic scheme of polyamide 12 aging in SLS considering the coupled
oxygen and laser effects
2.1. Oxidation model
2.1.1. Mechanism of thermal oxidation
Constituted of polymethylenic sequences and the amide group (-NHCO-), polyamide 12 has the
following chemical structure:
–[–CH2–
O
‖
C–N
H
–(CH2)10 –CH2–
O
‖
C–N
H
–(CH2)10 –CH2–
O
‖
C–N
H
–]–n
From the basic autoxidation scheme of polymers [24,25], high temperatures initiate the thermal
oxidation of polyamide 12 materials to form alkyl radicals P• (Eq. 1). Incorporation of oxygen and
abstraction of hydrogen atoms propagate oxidation of polyamide 12 [15]. In the propagation stage (Eqs. 2-
3), alkyl radicals P• combine with O2 to form peroxy radicals PO2•. PO2• captures hydrogen atoms from
polymer substrates to further produce hydroperoxides. Thermal decomposition of the hydroperoxide groups
is the main mechanism of polymer oxidation below 200 °C. Such a process involves a unimolecular mode
(Eq. 4) and a bimolecular mode (Eq. 5) [26,27]. Hydroxyl radicals PO•, alkoxy radicals HO•, and peroxy
radicals PO2• with polymer substrate rapidly interact and form two balance reactions (Eqs. 6-7) [15]. This
process also involves chain scission (S) and hydrogen abstraction.
PH Polyamide 12k1
P• (1)
P• + O2
k2
PO2• (2)
PO2• + PH k3
POOH + P• (3)
POOH k4
PO• + HO• (4)
2POOH k5
PO• + PO2• + H2O (5)
Balance reactions:
POOH k6
2P• + PNH2• + PH=O + H2O + S [-2PH, -CN] (6)
2POOH k7
P• + PO2• + PNH2 + PH=O + H2O + S [-PH, -CN] (7)
Here, parameters ki’s are a series of elementary reaction constants of the thermal oxidation.
The termination reactions of alkyl radicals P• involve coupling or disproportionation (Eqs. 8-9), where
F and X denote double bonds and chain crosslinking (X), respectively. The termination reactions of peroxy
radical pairs are ascribed as Eqs. 10-13. Peroxy radical pairs first react to form the transition cage [PO••
OP]cage with oxygen. The transition cage further generates final products (e.g., POOP, NH(P=O)2, and PNH2)
together with chain crosslinking (X) and scission (S).
P• + P• k8
γ1PP + 1-γ1PH + 1-γ1F + γ1X -1-γ1PH (8)
P• + PO2•k9
γ2POOP + 1-γ2POOH + 1-γ2F + γ2X -1-γ2PH (9)
PO2• + PO2•k10
PO••OPcage + O2 (10)
[PO••OP ]cage
k11
POOP + X (11)
[PO••OP ]cage
k12
NH(P=O)2 + PNH2 + PH=O + S [-CN] (12)
[PO••OP ]cage
k13
2P• + 2PNH2 + 2PH=O + 2S [-2PH, -2CN] (13)
2.1.2. Basic autoxidation model
The fundamental kinetics led to a basic model on thermal oxidation of polyamide 12 materials, defined
as the basic autoxidation model. The solution involves 5 main non-linear differential equations (Eqs. 14-
18) [15] indicating the derivatives of the compound concentrations with respect to time. For instance, the
rate of concentration changes of [POOH] (Eq. 14) equals the formation rates (POOH formed in Eqs. 3 and
9) minus the consumption rates (POOH consumed in Eqs. 6 and 7). We define the coefficient of oxygen
effect as (Eqs. 15, 16, 19), relating close to the oxygen concentrations O2. equals 1 in the basic
autoxidation model.
d[POOH]
dt = -k6fPHPOOH - 2k7fPHPOOH2 + k3PHPO2• + 1-γ2k9fPHP•PO2• (14)
d[P•]
dt = 2k6fPHPOOH + k7fPH[POOH]2- k2O2P•+ k3PHPO2•- 2k8P•2-
k9fPHP•PO2• + 2k13fPHPO••OPcage (15)
d[PO2•]
dt = k7fPH[POOH]2 + k2O2P•- k3PHPO2•-
k9fPHP•PO2•- 2k10PO2•2 (16)
d[PO••OP]cage
dt =k10[PO2•]2- k11 + k12 + k13fPH PO••OPcage (17)
d[PH]
dt = -2k6fPHPOOH- k7fPHPOOH2- k3PHPO2•- 1-γ2k9fPHP•PO2•
- 2k13fPH PO••OPcage (18)
Here, fPH is defined to avoid negative concentrations of substrate. fPH=[PH]/([PH]+ε) and ε=0.01 [15]; the
parameter itself does not significantly influence the oxidative kinetics.
From the mechanistic scheme in section 2.1.1, we also obtain the concentration changes of the
following reactants and products [15]:
d[O2]Consumed
dt = k2O2P•- k10PO2•2 (19)
d[PNH2]
dt = k6fPHPOOH + k7fPH[POOH]2 + k12+2k13fPHPO••OPcage (20)
d[PH=O]
dt =dPNH2
dt (21)
d[NH(P=O)2]
dt = k12[PO••OP]cage (22)
d[C-N]
dt = -dPNH2
dt (23)
dS
dt =dPNH2
dt (24)
dX
dt = γ1k8[P•]2 + γ2k9fPHP•PO2• + k11PO••OPcage (25)
Chain scission (S) occurs simultaneously with the oxidation-related signal diminishment near the
wavelengths of 1369.23, 1159.03, 1062.60, and 948.82 cm-1 [12]. In the basic autoxidation model, we define
the degree of chain scission (S) (Eq. 24) occurring in unit time (s) as the modelling aging rate for
polyamide 12 sample i.
S
(26)
where S is the degree of chain scission in oxidation time . We use Matlab ODE23s to solve the model
when knowing the initial concentrations of the main component and the elementary reaction coefficients ki.
We will introduce the details of these parameters in section 2.2.3.
At a specific temperature (e.g., a pre-heating temperature of the SLS machine, 160 °C), the elementary
reaction coefficients ki remain unchanged. However, the oxygen effect O2 can vary significantly at a
specific temperature when at different atmosphere, and largely affects the rates of material degradation.
Stronger oxygen effects result in faster degradation rates. In SLS, the nature of material degradation
involves the coupled oxygen and laser effects. The laser has even stronger effects than oxygen effects on
material degradation. Thus, the coupled laser and oxygen effects are substantially more significant than the
single oxygen effects on material degradation. However, it is difficult or impossible to get laser effects
using the modelling-only approach. Through experimentation, we shall get the actual material degradation
rates to derive the coefficients of the coupled laser and oxygen effects, referred to as
, through mapping
experimental results to the modelling results.
shows the enhancement effects on material degradation
from oxygen to the coupled laser and oxygen.
O2 is the coupled laser and oxygen effects in SLS.
2.2. Experimentation
2.2.1. SLS printing using polyamide 12 powders
We sintered different polyamide 12 combinations. The SLS machine used is an in-house built open-
configuration SLS AM research testbed with a 100 W Coherent GEM100A CO2 laser and a Scanlab
intelliSCAN 14 scanner (Figure 2). The parameter settings used in the printing experiments are: 160 °C
preheating, 3000 mm/s scanning speed, 18 W laser power, 0.3 mm scan spacing, and 150 µm layer thickness.
Table 1 exhibits the 22 kinds of printed samples with calculated density and oxidation time (time in the
chamber), i=1, 2, …22. In detail, the oxidation time is the sum of the preheating, printing and the post-
heating time. For part samples in this work, the preheating time is 5 minutes, and the printing time is 2
minutes. We change the post-heating time (20 seconds, 60 seconds, 120 seconds and 300 seconds) to obtain
part samples with different oxidation time.
Table 1 SLS printed samples using polyamide 12 powders, the calculated density, and oxidation time
Samples
Density/g·cm-3
Oxidation time/seconds
Parts using 100% new powders
0.9
440; 480; 540; 720
Parts using 70% new and 30% aged powders
0.828
420; 720
Parts using 60% new and 40% aged powders
0.804
420; 720
Parts using 50% new and 50% aged powders
0.78
420; 720
Parts using 40% new and 60% aged powders
0.756
420, 720
Parts using 30% new and 70% aged powders
0.732
420; 720
Parts using 20% new and 80% aged powders
0.708
420; 720
Parts using 10% new and 90% aged powders
0.684
420; 720
Parts using 100% aged powders
0.66
440; 480; 540; 720
* Polyamide 12 new powders are purchased from EOS Corp. Polyamide 12 aged powders are reclaimed from standard SLS
processes on an EOS P 390 machine.
Figure 2 SLS testbed and samples
2.2.2. Measured material degradation rates
In the polyamide 12 FTIR spectra, the dramatically diminished signals of peaks near wavelengths of
1369.23, 1159.03, 1062.60, and 948.82 cm-1 indicate the oxidation of amide groups [12]. We conducted
FTIR tests on the specimens in Table 1 as well as the pure polyamide 12 powder to examine the aging-
related signals using a Nicolet Magna-IR 560 FTIR instrument (wavelength ranges: 6500 cm-1-100 cm-1,
spectral resolution: 0.35 cm-1). The FTIR of powder (new powder) serves as the benchmark against the
degradation comparison. For specimen i, through the FTIR results and Beer-Lambert’s law (Eq. 27) [15],
we calculated the concentrations of the four oxidation-related components Yn (n=1, 2, 3,4) (corresponding
to peaks near 1369.23, 1159.03, 1062.60, and 948.82 cm-1), respectively.
Abs
(n=1,2,3,4)(27)
where Abs
,
and
are, respectively, the absorbance, the coefficient of molar absorptivity, and the
concentrations of the chemical component Yn; and is the thickness of the tested sample i. We obtain the
absorbances from FTIR results, and get the coefficients of molar absorptivity from the new powder
(benchmark sample). For the new powder, we read the tested thickness and absorbances from FTIR, and
calculate the molar concentrations using density and molar mass [28]. Then we get the coefficients of molar
absorptivity using the Beer-Lambert’s law [15], and insert these coefficients in Eq. 27 to calculate
concentrations of chemical component for part samples. We write
as the difference of
between the
benchmarked powder materials and the 3D-printed samples. As there are four peaks for each specimen i,
we write (in mol/L) to denote the average of
,
,
, and
for specimen i. We define the
actual degradation rate (involving both oxygen and laser effects) as the average concentration changes
of the oxidation-related components in unit time (Eq. 28), in mol/(L·s):
(28)
where i is the sample index in Table 1, and is the associated oxidation time.
2.2.3. Comparisons between the actual SLS degradation and the modelling aging rates
Specimen density and oxidation time in Table 1 are important parameters for the basic autoxidation
model. Besides, the initial concentrations of reactants are necessary to run the model. For each specimen i,
we get the initial concentrations of reactants, namely, POOH, PH and C-Nin the basic autoxidation model
(Eqs. 14-25) using the molar concentration formula
c
reactant-R=qi
M (29)
where c
reactant-R (in molar/L) is the initial molar concentration of reactant R (POOH , PH or C-N) in
specimen i; q
(in g/cm3) is the density of specimen i; M (in g/mol) is the molar mass of polyamide 12. In
the basic autoxidation, the initial concentrations of P•, PO2•, [PO••OP]cage, [O2]Consumed, PNH2, PH=O,
NH(P=O)2, S, and X are zero because they are intermediate products. The oxygen concentration, [O2], is
3.6×10-4 mol·L-1 [15] in the air atmosphere. Table 2 lists the elementary reaction coefficients for thermal
oxidation of polyamide 12 at 160 °C [26,27,29]. Inserting the above parameters in the basic autoxidation
model, we get the modelling aging rate for sample i, and compare the result to the actual degradation
rate .
Table 2 The elementary reaction constants for thermal oxidation of polyamide 12 at 160 °C [26,27,29]
Parameter
Value
Parameter
Value
k2 (L·mol-1·s-1)
108
k10 (L·mol-1·s-1)
1.6×1011
k3 (L·mol-1·s-1)
45.8
k11 (s-1)
2.0×108
k6 (s-1)
8.0×10-4
k12 (s-1)
3.4×108
k7 (L·mol-1·s-1)
6.0×10-3
k13 (s-1)
2.2×109
k8 (L·mol-1·s-1)
8.0×1011
γ1 (%)
50
k9 (L·mol-1·s-1)
5.0×1011
γ2 (%)
50
2.3. Kinetic model of polyamide 12 aging in SLS considering the coupled oxygen and laser effects
Figure 3 shows the main proposed procedures to build the kinetic model of polyamide 12 aging
involving the coupled oxygen and laser effects in SLS.
We separated the printed SLS specimens in Table 1 into two sample groups (SLS sample groups 1 and
2). Each group contains SLS samples with different polyamide 12 combinations. The objective is to ensure
that the method can derive the coefficient of the coupled oxygen and laser effects,
, for different
polyamide 12 combinations. For samples i in group 1, we performed sensitivity analysis on the modelling
degradation rates as the coupled laser and oxygen effects
O2 changes, using the basic autoxidation
model. Figure 4 presents the relationship between and
O2 for sample i.
SLS sample group 2
The proposed model
Model verification
Coefficients of aging
Aging rates with time
Basic autoxidation model FTIR spectra
Beer-Lamberts law
Actual degradation rates
Sensitivity analysis
Modelling the aging
SLS sample group 1
Sample density Oxidation time
Figure 3 Proposed procedures to build the kinetic model of polyamide 12 involving the coupled oxygen and laser-induced
aging in SLS
Figure 4 Experimental relationship between and
O2 for sample i
The experimental data suggest a second-order relationship between and
O2 in the tested operation
zone. We thus propose the following second-order correlation mapping
O2
O2 (30)
where , , and are constants for sample i. We will perform parameter identification with R-squared
regression and a full model verification in Section 3. When the modelling degradation rate equals the
actual degradation rate (section 2.2.2), the corresponding
O2 represents the actual coupled laser
and oxygen effects in SLS, defined as
O2.
Replacing with
in the basic autoxidation model, we obtain an updated oxidation model of
polyamide 12 in SLS. In this model, we define the material degradation rates as the updated modelling
degradation rates . We utilize the updated model to conduct sensitivity analysis on and the specimen
oxidation time . Figure 5 shows the experimentally identified relationship between and for sample i.
Figure 5 Experimental relationship between and for sample i
After the initial transient, the relationship between and fits an R-squared cubic polynomial, defined as
Eq. 31.
(31)
where , , and are constants.
The proposed kinetic model contains the basic autoxidation model, the coefficient of the actual coupled
laser and oxygen effects in SLS, and the relationships between the updated modelling degradation rates
and specimen oxidation time . To verify the proposed kinetic model, we apply it to the SLS sample group
2 to compare the updated modelling aging rates and the actual degradation rates .
2.4. Characteristics of the updated modelling degradation rates
The actual coupled laser and oxygen effects in SLS,
O2, and the preheating temperature are
predominant parameters in SLS affecting the material degradation rates. To understand the process further,
we use the proposed kinetic model to identify the influences of
O2 and preheating temperatures
(Table 2 and Table 3) on the updated modelling degradation rates, . These results will be analyzed in
section 3.3.
Table 3 The elementary reaction constants for thermal oxidation of polyamide 12 between 90 and 150 °C [26,27,29]
Parameter
Value
90 °C
100 °C
120 °C
140 °C
150 °C
k2 (L·mol-1·s-1)
108
108
108
108
108
k3 (L·mol-1·s-1)
1.6
2.7
7.7
19.6
30.3
k6 (s-1)
8.0×10-7
2.2×10-6
1.8×10-5
1.6×10-4
4.0×10-4
k7 (L·mol-1·s-1)
4.0×10-5
9.0×10-5
5.0×10-4
1.7×10-3
3.5×10-3
k8 (L·mol-1·s-1)
8.0×1011
8.0×1011
8.0×1011
8.0×1011
8.0×1011
k9 (L·mol-1·s-1)
5.0×1011
5.0×1011
5.0×1011
5.0×1011
5.0×1011
k10 (L·mol-1·s-1)
8.0×109
2.6×1010
6.0×1010
5.0×1010
9.5×1010
k11 (s-1)
2.0×108
2.0×108
2.0×108
2.0×108
2.0×108
k12 (s-1)
3.4×108
3.4×108
3.4×108
3.4×108
3.4×108
k13 (s-1)
3.2×108
4.7×108
7.9×108
1.2×109
1.8×109
γ1 (%)
100
95
80
55
55
γ2 (%)
100
95
80
55
55
3. Results and discussions
3.1. Comparisons between the modelling aging and the actual SLS aging rates
As introduced in section 2.2.3, we run the basic autoxidation model and get the modelling degradation
rates for each specimen i in Table 1, presented in the following section. As for the actual degradation,
Figure 6 exhibits the FTIR results of the SLS samples in Table 1, which we shall now extrapolate into
measured degradation rates . The horizontal axis is the wavenumber, and the vertical axis is the
absorbance. Our focus here is to get the absorbance differences of oxidation-related wavelengths at 1369.23,
1159.03, 1062.60, and 948.82 cm-1. Then using the Beer-Lambert’s law [15], we can calculate the
concentrations of the oxidation-related components for each sample. The differences of the concentrations
between different samples represent the different oxidation states.
(a) Polyamide 12 new powder and parts (b) Polyamide 12 parts using mixed powder (420 s)
(c) Polyamide 12 parts using mixed powder (720 s) (d) Polyamide 12 aged powder and parts
Figure 6 FTIR test results of different polyamide 12 powders and different 3D-printed part samples with different
oxidation time
Figure 7 compares FTIR curves of SLS samples at the oxidation-related wavelengths of 1369.23,
1159.03, 1062.60, and 948.82 cm-1. The diminishment or disappearance of peaks at oxidation-related
components indicates the material degradation and oxidization (Figure 7a and 7i). The peaks decrease more
when the aging time increases (Figure 7b, 7c, 7d, 7e, 7f, 7g, and 7h). As introduced in section 2.2.2, we
calculated the concentration changes (mol·L-1) of oxidation-related components for each sample during
specific aging durations, and further obtained the actual degradation rates in mol·L-1·s-1 using the Eqs.
27-28.
(a) New powder and parts using new powders (b) Part using 70% new and 30% aged powders
(c) Part using 60% new and 40% aged powders (d) Part using 50% new and 50% aged powders
(e) Part using 40% new and 60% aged powders (f) Part using 30% new and 70% aged powders
(g) Part using 20% new and 80% aged powders (h) Part using 10% new and 90% aged powders
(i) Aged powder and parts using aged powders
Figure 7 Comparison of FTIR peaks at the wavelengths of 1369.23, 1159.03, 1062.60, and 948.82 cm-1 when testing
different polyamide 12 samples
Figure 8 compares the modelling degradation rates and the actual degradation rates . Nontrivial
but unsurprising, the actual degradation rates of polyamide 12 are much larger than the modelling
degradation rates. This phenomenon exists in all samples, including those using pure new polyamide 12
powders (Figure 8a), new-aged mixed powders (Figure 8b), and pure aged powders (Figure 8c). This core
finding indicates that the coupled oxygen and laser age the material much faster than the case with oxygen
only. It is thus necessary and important to build the kinetic model of polyamide 12 aging in SLS considering
the coupled oxygen and laser effects.
(a) Parts using new polyamide 12 powders
(b) Parts using mixed polyamide 12 powders
(c) Parts using aged polyamide 12 powders
Figure 8 Comparisons between the modelling aging rates and the actual SLS aging rates for different printed samples
using polyamide 12 powders
3.2. Building the kinetic model of polyamide 12 aging in SLS considering the coupled oxygen and
laser effects
3.2.1. Determining the coefficients of the actual coupled laser and oxygen effects in SLS,
Table 4 shows the selected SLS sample group 1 and the associated time of oxidation. After performing
the sensitivity analysis using the basic autoxidation model, we conducted curve fitting between the
modelling degradation rates and the coupled laser and oxygen effects
O2 to an R-squared second-
order polynomial (Eq. 30). Figure 9 exhibits the results of sensitivity analysis and the fitting equations.
Table 4 SLS sample group 1
Sample
Time of oxidation/seconds
Parts using 100% new powders
440, 480
Parts using 70% new and 30% aged powders
420
Parts using 60% new and 40% aged powders
420
Parts using 50% new and 50% aged powders
420
Parts using 40% new and 60% aged powders
720
Parts using 30% new and 70% aged powders
420
Parts using 20% new and 80% aged powders
420
Parts using 10% new and 90% aged powders
720
Parts using 100% aged powders
440, 480
(a) 440 seconds of oxidation (part using new powders) (b) 480 seconds of oxidation (part using new powders)
(c) 420 seconds of oxidation (part using 70% new powders) (d) 420 seconds of oxidation (part using 60% new powders)
(e) 420 seconds of oxidation (part using 50% new powders) (f) 720 seconds of oxidation (part using 40% new powders)
(g) 420 seconds of oxidation (part using 30% new powders) (h) 420 seconds of oxidation (part using 20% new powders)
(i) 720 seconds of oxidation (part using 10% new powders) (j) 440 seconds of oxidation (part using aged powders)
(k) 480 seconds of oxidation (part using aged powders)
Figure 9 Sensitivity analysis and the fitting equations between and
O2 to an R-squared second-order polynomial
In each fitting equation, letting the modelling degradation rates, , equal to the actual degradation
rates , we obtain the actual coupled laser and oxygen effects in SLS,
O2, and the coefficients of
(Table 5). Here, we obtain the updated oxidation model, including the basic autoxidation model and
. The values of
indicate that the coupled laser and oxygen effects are about 4 times more than
the case with only oxygen (=1), and the laser effects are on average 4.4 times stronger than oxygen
effects on polyamide 12 degradation.
Table 5 The calculated coefficients of the coupled oxygen and laser effects,
, in the SLS process
Sample
Fitting curves between modelling degradation
rates () and the coupled oxygen and laser
effects (
O2)
The actual
degradation
rates in SLS
/mol·L-1·s-1
The actual
coupled laser and
oxygen effects in
SLS,
O2
/mol·L-1
Coefficients of
the actual
coupled laser
and oxygen
effects in SLS,
/mol·L-1
Figure 9a
μ=6.563×10-6+0.029O2-2.475(O2)2
5.105×10-5
1.804×10-3
5.010
Figure 9b
μ=6.283×10-6+0.030O2-2.338(O2)2
4.916×10-5
1.632×10-3
4.533
Figure 9c
μ=7.071×10-6+0.029O2-3.043(O2)2
4.944×10-5
1.753×10-3
4.868
Figure 9d
μ=5.353×10-6+0.031O2-3.272(O2)2
5.109×10-5
1.861×10-3
5.170
Figure 9e
μ=5.853×10-6+0.029O2-3.002(O2)2
4.228×10-5
1.453×10-3
4.037
Figure 9f
μ=5.632×10-6+0.016O2-0.578(O2)2
3.418×10-5
1.919×10-3
5.332
Figure 9g
μ=5.916×10-6+0.029O2-3.118(O2)2
4.992×10-5
1.878×10-3
5.216
Figure 9h
μ=5.840×10-6+0.029O2-3.142(O2)2
4.486×10-5
1.622×10-3
4.504
Figure 9i
μ=1.504×10-6+0.014O2-0.729(O2)2
2.532×10-5
1.622×10-3
4.505
Figure 9j
μ=3.900×10-6+0.034O2-4.761(O2)2
3.371×10-5
1.022×10-3
2.838
Figure 9k
μ=3.517×10-6+0.036O2-4.749(O2)2
3.363×10-5
9.740×10-4
2.705
3.2.2. Determining the relationship between the updated modelling degradation rates and
oxidation time
This subsection identifies the relationships between the updated modelling degradation rates and
oxidation time using the updated oxidation model. First, we conducted a sensitivity analysis on
concentration changes of oxidative components as oxidation time increases (Figure 10). The observation
is that the sample using 100% aged powders has the slowest rates of concentration changes when
seconds. This is largely due to that the aged powders develop a lot of oxidized components from the thermal
history.
Figure 10 Sensitivity analysis on concentration changes of the oxidative components as oxidation time increases using
the updated oxidation model
We divided the concentration changes by oxidation time to get degradation curves, and fit the curves
to a series of cubic-polynomial (Eq. 31). Figure 11 shows the fitting equations between the updated
modelling degradation rates and oxidation time . In the sensitivity analysis curve, the model output
goes up quickly from zero to the maximum and then goes down within seconds. The reason is that a strong
thermal impetus initiates and simultaneously accelerates the degradation reaction at a time close to zero. At
this stage, the impetus dominantly controls the reaction and continuously increases the degradation rates
until arriving at the maximum point. However, when the reaction runs normally, the basic parameters, e.g.,
initial concentrations of components, elementary reaction coefficients, laser and oxygen effects, take
control of the reaction. At this stage, the influences of the initiation impetus on degradation rates diminish
rapidly and disappear gradually.
(a) Part using 100% new powders (b) Part using 70% new powders
(c) Part using 60% new powders (d) Part using 50% new powders
(e) Part using 40% new powders (f) Part using 30% new powders
(g) Part using 20% new powders (h) Part using 10% new powders
(i) Part using 100% aged powders
Figure 11 Fitting equations between the updated modelling degradation rates and oxidation time
3.2.3. The proposed kinetic model
For different polyamide 12 powder combinations, Table 6 lists the coefficients of the actual coupled
laser and oxygen effects in SLS,
, and the fitting equations between the updated modelling
degradation rates and oxidation time . The proposed kinetic scheme of polyamide 12 aging in SLS
considering the coupled oxygen and laser effects includes the basic autoxidation model,
and fitting
equations between and . From there, we can predict the sample degradation in SLS through powder
combination and oxidation time.
Table 6 The coefficients of the actual coupled laser and oxygen effects in SLS,
, and the fitting equations between
the updated modelling degradation rates and oxidation time
Sample
/mol·L-1
The fitting equations between the updated modelling
degradation rates and oxidation time
Part using 100% new powders
4.772
= 1.268×10-4-3.015×10-7·t + 3.250×10-10·t2-1.244×10-13·t3
Part using 70% new powders
4.868
= 1.265×10-4-3.008×10-7·t + 3.246×10-10·t2-1.244×10-13·t3
Part using 60% new powders
5.170
= 1.259×10-4-2.884×10-7·t + 3.050×10-10·t2-1.155×10-13·t3
Part using 50% new powders
4.037
= 1.149×10-4-2.841×10-7·t + 3.123×10-10·t2-1.209×10-13·t3
Part using 40% new powders
5.332
= 1.228×10-4-2.908×10-7·t + 3.450×10-10·t2-1.547×10-13·t3
Part using 30% new powders
5.216
= 1.195×10-4-2.784×10-7·t + 3.263×10-10·t2-1.449×10-13·t3
Part using 20% new powders
4.504
= 1.127×10-4-2.759×10-7·t + 3.335×10-10·t2-1.513×10-13·t3
Part using 10% new powders
4.505
= 1.096×10-4-2.639×10-7·t + 3.150×10-10·t2-1.416×10-13·t3
Part using 100% aged powders
2.772
= 1.096×10-4-2.639×10-7·t + 3.150×10-10·t2-1.416×10-13·t3
Inserting the modelling related parameters of SLS sample group 2 (Table 7) into the proposed kinetic
model, we predicted the degradation rates of these samples. Figure 12 compares the predicted degradation
using the proposed kinetic model and the measured actual SLS degradation of sample group 2. Small
deviations between the predicted and the actual degradation results exist. Figure 12 presents average
deviations of 9.43% between and , respectively, exhibiting a substantial improvement compared to the
results in Figure 8. The proposed kinetic model is capable to predict the SLS degradation rates of polyamide
12 accurately.
Table 7 SLS sample group 2
Sample
Oxidation time/seconds
Parts using 100% new powders
540, 720
Parts using 70% new and 30% aged powders
720
Parts using 60% new and 40% aged powders
720
Parts using 50% new and 50% aged powders
720
Parts using 40% new and 60% aged powders
420
Parts using 30% new and 70% aged powders
720
Parts using 20% new and 80% aged powders
720
Parts using 10% new and 90% aged powders
420
Parts using 100% aged powders
540, 720
Figure 12 The comparisons between the predicted degradation rates and the actual SLS degradation rates of the SLS
sample group 2 (Parts using polyamide 12 powders of different combinations)
3.2.4. Discussions
Table 8 presents the actual SLS degradation rates from experimentation, the modelling degradation
rates from the basic autoxidation model, and the updated modelling degradation rates from the
proposed kinetic model. The modelling degradation rates have large deviations compared to the actual
SLS degradation rates , while the updated modelling degradation rates are close to . The predicted
degradations from the proposed kinetic model match on average 89.53% with the actual SLS degradation
rates , in contrast to a 34.48% accuracy from a basic autoxidation model.
Table 8 Comparisons between the actual SLS degradation rates , the modelling degradation rates from the basic
autoxidation model, and the updated modelling degradation from the proposed kinetic model
Sample
The actual SLS
degradation rates
/mol·L-1·s-1
The basic autoxidation model
The proposed kinetic model
Degradation
rates
/mol·L-1·s-1
Deviation/%
Degradation
rates
/mol·L-1·s-1
Deviation/%
Part/100% new powder/540 s
4.726×10-5
1.347×10-5
71.505
3.913×10-5
17.177
Part/100% new powder/720 s
3.021×10-5
1.023×10-5
66.129
3.174×10-5
5.054
Part/70% new powder/720 s
3.065×10-5
9.819×10-6
67.964
2.971×10-5
3.065
Part/60% new powder/720 s
3.386×10-5
9.538×10-6
71.832
2.971×10-5
10.156
Part/50% new powder/720 s
3.128×10-5
9.648×10-6
69.162
2.432×10-5
22.260
Part/40% new powder/420 s
3.4974×10-5
1.577×10-5
54.894
3.954×10-5
13.058
Part/30% new powder/720 s
3.354×10-5
9.545×10-6
71.542
2.907×10-5
13.319
Part/20% new powder/720 s
2.634×10-5
6.678×10-6
74.647
2.533×10-5
3.823
Part/10% new powder/420 s
4.100×10-5
1.560×10-5
61.943
3.554×10-5
13.311
Part/100% aged powder/540 s
2.560×10-5
1.253×10-5
51.069
2.894×10-5
13.044
Part/100% aged powder/720 s
2.395×10-5
9.568×10-6
60.045
2.372×10-5
0.936
3.3. Characteristics of the updated modelling degradation rates
3.3.1. Influences of the coupled laser and oxygen effects on
3.3.1.1. Degradation characteristics in presence of decreasing oxidations
To identify the degradation trend here, we reduce
O2 for different part samples in Table 9.
Inserting the
O2 into the proposed kinetic model, we obtained curves between the updated
modelling degradation rates and oxidation time (Figure 13). Figures 13a, 13b, and 13c are the results
for, respectively, SLS 3D-printed part using 100% new powders, part using 50%-50% new-aged powders,
and part using 100% aged powders. The curves in different colors point out the nonlinear
relationship between
O2 and the degradation rate. The black curves in Figure 13a-c are the
benchmark curves with the original
O2.
For different samples in Figure 13, when
O2 reduces, the updated modelling degradation rates
increase from zero to the maximum quickly, then decrease with time. For a specific sample (e.g., Figure
13a, the new-SLS part), the maximum of drops as
O2 decreases. The new-SLS part (Figure 13a)
always has the largest degradation rate, while the aged-SLS part (Figure 13c) has the smallest . For the
mixed (Figure 13b) and aged-SLS parts (Figure 13c), curves with
O22 (7.267×10-4 mol/L for
mixed and 4.989×10-4 mol/L for aged) and
O25 (2.907×10-4 mol/L for mixed and 1.996×10-4
mol/L for aged) result in a large drop of compared to the benchmark curves, especially at the
peak degradation points. However, further curves (curves with
O210,
O220 , and
O2100) show small changes in comparison to the curve with
O25. After about 1200 s of
oxidation (=1200 s), the rates of degradation all approach the steady state. This convergence is much
faster when the oxidation effect is reduced. In addition, when reusing powders, degradation significantly
slows down when oxidation is reduced by a factor of 5, and remains afterwards (Figure 13bc).
Table 9 The decreasing
O2
Sample
The actual
coupled laser
and oxygen
effects in SLS,
O2
/mol·L-1
O2/
2
O2/
5
O2/
10
O2/
20
O2/
50
O2
/100
Unit: mol·L-1
Part using 100%
new powders
1.718×10-3
8.589×10-4
3.435×10-4
1.718×10-4
8.589×10-5
3.435×10-5
1.718×10-5
Part using 50%
new powders
1.453×10-3
7.267×10-4
2.907×10-4
1.453×10-4
7.267×10-5
2.907×10-5
1.453×10-5
Part using 100%
aged powders
9.978×10-4
4.989×10-4
1.996×10-4
9.978×10-5
4.989×10-5
1.996×10-5
9.978×10-6
(a) Part using 100% new powders (b) Part using 50%-50% new-aged powders
(c) Part using 100% aged powders
Figure 13 Curves between the updated modelling degradation rates and oxidation time with decreasing
O2 for
different part samples
3.3.1.2. Degradation characteristics in presence of increasing oxidations
Table 10 presents the designed experiments with increasing
O2 for different part samples.
Applying the increasing
O2 to the proposed kinetic model, we obtain the curves between
the updated modelling degradation rates and oxidation time (Figure 14). Figures 14a, 14b, and 14c are
respectively for SLS 3D-printed part using 100% new powders, part using 50%-50% new-aged powders,
and part using 100% aged powders. The curves in different colors represent differently increased
O2. The black curves in Figure 14a-c are the benchmark curves with the original
O2.
In Figure 14, increases from zero to the maximum quickly, then decreases with time. For any
specific sample (e.g., Figure 14a, the new-SLS part), the maximum of rises as
O2 increases.
Having the same increasing degree for
O2 (e.g.,
O2×10), the new-SLS part always has the
largest , while the aged-SLS part has the smallest . For new, mixed, and aged-SLS parts, curves with
O2×2 and
O2×5 lead to obvious increases of . However, the other curves (curves with
O2×10,
O2×20,
O2×50,
O2×100) differ little compared to the curve with
O2×5. The result indicates that further increasing
O2 does not influence significantly.
Till 1200 s, all the curves are at or close to reaching the steady state.
Table 10 The increasing
O2
Sample
The actual
coupled laser
and oxygen
effects in SLS,
O2
/mol·L-1
O2
×2
O2
×5
O2
×10
O2
×20
O2
×50
O2
×100
Unit: mol·L-1
Part using 100%
new powders
1.718×10-3
3.435×10-3
8.589×10-3
1.718×10-2
3.435×10-2
8.589×10-2
1.718×10-1
Part using 50%
new powders
1.453×10-3
2.907×10-3
7.267×10-3
1.453×10-2
2.907×10-2
7.267×10-2
1.453×10-1
Part using 100%
aged powders
9.978×10-4
1.996×10-3
4.989×10-3
9.978×10-3
1.996×10-2
4.989×10-2
9.978×10-2
(a) Part using 100% new powders (b) Part using 50%-50% new-aged powders
(c) Part using 100% aged powders
Figure 14 Curves between the updated modelling degradation rates and oxidation time with increasing
O2 for
different part samples
3.3.1.3. Comparisons
Figure 15 compares at 1200 s for different samples to reveal material degradation rates at the steady
state. In Figure 15, the purple bars are the benchmark @1200 s data with the original
O2. When
O2 decreases (increases), the @1200 s decreases (increases) quickly first. Further decreasing
(increasing)
O2 has little effects on @1200 s.
Table 11 compares 1200 s between the original
O2 and the decreased/increased
O2 for different samples. In contrast to the case with the original σi-SLS
ol O2, 1200 s with
σi-SLS
ol O2/100 decreased by, respectively, 89.02%, 88.77%, and 81.90% for the new, mixed and aged-SLS
parts. On the other hand, 1200 s with σi-SLS
ol O2×100 increased by, respectively, 181.78%, 197.35%,
and 183.42% for the new, mixed and aged-SLS parts.
Figure 15 Comparisons of at 1200 s for different samples to compare material degradation rates at a more stable state
Table 11 Comparisons of 1200 s between the original
O2 and the decreased/increased
O2 for different
samples
Samples
Part using new powders
Part using 50%-50% new-
aged powders
Part using aged powders
The decreasing
O2
Percentages of 1200 s decreasing when
O2 decreasing
O2/100
89.02
88.77
81.9
O2/50
89.73
88.77
80.53
O2/20
87.13
86.45
88.89
O2/10
80.38
82.86
79.73
O2/5
71.96
76.27
70.54
O2/2
43.83
43.82
42.4
The increasing
O2
Percentages of 1200 s increasing when
O2 increasing
O2×2
81.41
82.13
39.87
O2×5
180.91
193.45
164.06
O2×10
183.34
198.76
183.6
O2×20
182.53
198.23
184.14
O2×50
181.13
196.76
182.41
O2×100
181.78
197.35
183.42
3.3.2. Influences of the preheating temperature on
Figure 16 presents the curves ( curves) between the updated modelling degradation rates and
oxidation time with different preheating temperatures for (a) Part using 100% new powders, (b) Part
using 50%-50% new-aged powders, and (c) Part using 100% aged powders. In Figure 16a-c, the decreased
preheating temperatures lower the curves. When at the same temperature (e.g., 150 °C, 140 °C), the
new-SLS part has the largest (Figure 16a), while the aged-SLS part has the smallest (Figure 16c). As
the preheating temperature decreases, the peaks in the curves diminish (from 160 °C to 140 °C) and
disappear gradually (from 120 °C to 90 °C). Therefore, the peaks in the curves are likely caused by
the high temperature. Besides, when the preheating temperatures are below 120 °C, are nearly zero,
indicating that a low storage temperature below 120 °C can effectively reduce material degradation.
(a) Part using 100% new powders (b) Part using 50%-50% new-aged powders
(c) Part using 100% aged powders
Figure 16 Curves between the updated modelling degradation rates and oxidation time with different preheating
temperatures for different part samples
Figure 17 shows comparisons of at 1200 s between (a) Samples with different preheating
temperatures, and (b) Preheating temperatures for different samples. At 90 °C, 1200 s approaches to
zero, leading to almost no degradation for the material at this temperature (Figure 17b). 1200 s
decreased evenly with decreasing preheating temperatures (Figure 17a). At high temperatures (150 °C –
160 °C), the differences of 1200 s between different samples are large; those differences reduce quickly
at lower temperatures below 120 °C (Figure 17b).
(a) Samples with different preheating temperatures (b) Preheating temperatures for different samples
Figure 17 Comparisons of at 1200 s (1200 s between (a) Samples with different preheating temperatures, and (b)
Preheating temperatures for different samples
4. Conclusions
In SLS, a considerable amount of expensive polyamide 12 powders remains un-sintered but reusable
after going through severely irreversible chemical degradations. The degradation originates from the
thermal energy controlled by the coupled oxygen and laser effects. Through experimentation, and by fitting
the actual SLS degradation rates to the basic autoxidation model, we obtained the coefficients of coupled
oxygen and laser effects. A further sensitivity analysis suggests the existence of a polynomial fitting
between the sample degradation rates and oxidation time. From there, we propose a new kinetic scheme for
SLS degradation of polyamide 12 composed of the basic autoxidation model, the coefficients of coupled
oxygen and laser effects, and the relationships between the sample degradation rates and oxidation time.
The new model can predict the oxidation rates of pure or mixed (different degradation levels) polyamide
12 using two easily available parameters: materials density and oxidation time. The predicted degradations
from the proposed kinetic model match on average 89.53% with the actual SLS degradation rates, in
contrast to a 34.48% accuracy from a conventional aging model. We found that the laser effects are 4-time
stronger than oxygen effects on polyamide 12 degradation. Furthermore, we identified the influences of the
coupled oxygen and laser effects in SLS and preheating temperatures on the degradation rates. The findings
provide a first-instance knowledge of quantitative material degradation related to the estimated parameters,
and insights to reduce degradation in SLS. This work established a novel effective model to obtain the
kinetic scheme of polyamide 12 degradation to aid future studies of materials degradation and reuse in the
SLS process.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships
that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported in part by National Science Foundation Award 1953155.
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