Modeling of Selective Laser Sintering of Viscoelastic Polymers

Modeling of Selective LaserSintering of Viscoelastic Polymers

Modellierung des Selektives Lasersintern vonviskoelastischen Polymeren

Der Technischen Fakultät derFriedrich-Alexander-Universität Erlangen-Nürnberg

zur Erlangung des Doktorgrades

Doktor-Ingenieur

vorgelegt von

Dipl.-Phys. Fuad Osmanlic

aus Gradačac

Als Dissertation genehmigtvon der Technischen Fakultät derFriedrich-Alexander-Universität Erlangen-Nürnberg

Tag der mündlichen Prüfung: 25.09.2019Vorsitzender des Promotionsorgans: Prof. Dr.-Ing. Reinhard Lerch

Gutachter: Prof. Dr.-Ing. habil. Carolin KörnerPD Dr.-Ing. Julia Mergheim

ToEnes and Indira

Acknowledgment

Many people have made this work possible and contributed directly or indirectlyto it. The following acknowledgment makes no claim to completeness.

First of all I would like to thank my supervisor Prof. Dr.-Ing. habil. CarolinKörner for all her support during my PhD period and beyond. She has given methe freedom I needed to unfold. I would also like to thank PD Dr.-Ing. JuliaMergheim for the quick and thorough review of my thesis as a secondary referee.Furthermore I like to thank Prof. Dr. Michael Stingl for his work as an externalexaminer and Prof. Dr. habil. Dirk W. Schubert for chairing my defense. Thisresearch also benefited from the support of German Research Foundation (DFG).

I would also like to thank all my former colleagues at the Chair of Materials Scienceand Engineering for Metals, Friedrich-Alexander University Erlangen-Nürnberg(FAU), for the unforgettable time. It was a privilege to work in a family envi-ronment with so many talented and dedicated people. In particular I would liketo thank Vera Jüchter, Alexander Klassen, Harald Helmer, Matthias Schwankel,Franziska Warmuth, Andreas Bauereiß and Thorsten Scharowsky for their persis-tent willingness to discuss challenging opinions independently of place, time andcircumstances. To all unwilling witnesses of these discussions, you’re welcome.

I would like to thank all colleagues who worked with me within the Collabora-tive Research Center 814 Additive Manufacturing. Especially Katrin Wudy, LydiaLanzl and Tobias Laumer who supported me with experimental data and discus-sions.

My family supported me from an early age, they encouraged my talents and theywere willing to sacrifice at any time to make my path possible. Thank you somuch for your continued faith in me. Finally, I would like to thank the person whosupported me most emotionally, practically and professionally. She was alwaysready to help me with her knowledge, talents, abilities and patience. Thank youVera.

Abstract

Selective laser sintering of polymers is a process to produce complex componentslayer-by-layer by selective melting of a powder bed with a focused laser beam. Theadvantages of crucible-free processing, freedom of design, material diversity andresource efficiency make this process unique and particularly interesting for indus-trial, academic and consumer applications. Although a large number of productionmachines are commercially available, there are still gaps in the understanding ofthe relevant consolidation mechanisms at powder level. This is mainly due to thepredominant lengths and time scales, which complicate an experimental approach.

Subject of this thesis is the development and application of a simulation frame-work for modeling selective laser sintering of viscoelastic polymers to provide aninsight at powder scale. The governing equations for thermo- and hydrodynamicsare derived and a numerical scheme is presented to solve the equations using theLattice Boltzmann Method, including viscoelasticity described by the Oldroyd-Bconstitutive equation. The new aspects of the derived solver are then validatedwith analytical and benchmark problems. Additionally, a laser absorption modelis introduced for the interaction between the laser radiation and the powder, ac-counting for the absorption, reflection and refraction. All the derived methods arecombined to simulate the selective laser sintering of polyamide 12.

The absorption characteristic within one powder particle and a stochasticpowder bed is investigated in terms of the absorbed energy distribution and pen-etration depth. The consolidation of viscoelastic powder particles is analysed.Furthermore, the influence of the process parameters beam power, scanning veloc-ity and scanning strategy on the melt pool shape of a single line, the mean meltpool depth of a single layer and the relative density of several layers are discussed.In addition, the influence of powder aging and viscoelasticity within the relevantlimits is examined. Finally, the relevant factors are identified and their impact onthe consolidation of the powder discussed.

I

Zusammenfassung

Das selektive Lasersintern von Polymeren ist ein Verfahren zur Herstellung kom-plexer Bauteile in einem schichtweisen Aufbau durch selektives Schmelzen einesPulverbettes mit einem fokussierten Laserstrahl. Die Vorteile der tiegelfreien Ver-arbeitung, die Gestaltungsfreiheit, die Materialvielfalt und die Ressourceneffizienzmachen diesen Prozess einzigartig und besonders interessant für industrielle undakademische Anwendungen. Obwohl eine große Anzahl von Produktionsmaschi-nen kommerziell verfügbar ist, gibt es noch Lücken im Verständnis der relevan-ten Mechanismen für die Konsolidierung auf Pulverebene. Dies ist vor allem aufdie vorherrschenden Längen- und Zeitskalen zurückzuführen, die einen experi-mentellen Zugang erschweren.

Ziel dieser Arbeit ist die Entwicklung und Anwendung einer Simulation zurModellierung des Selektives Lasersintern von viskoelastischen Polymeren, um einenEinblick in den Prozess auf Pulvermaßstab zu geben. Die relevanten Gleichungenfür Thermo- und Hydrodynamik werden hergeleitet und ein numerisches Verfahrenzur Lösung der Gleichungen mit der Lattice Boltzmann-Methode vorgestellt, ein-schließlich Viskoelastizität, beschrieben durch das Oldroyd-B Modell. Das neuentwickelte Modell wird mit analytischen und numerischen Ergebenissen aus derLiteratur validiert. Zusätzlich wird ein Laserabsorptionsmodell für die Interaktionzwischen dem Laser und dem Pulver vorgestellt, das die Absorption, Reflexionund Brechung berücksichtigt. Alle entwickelten Modelle werden kombiniert, umdas selektive Lasersintern von Polyamid 12 zu simulieren.

Die Absorptionscharakteristik innerhalb eines Pulverpartikels und eines stochastis-chen Pulverbettes wird in Bezug auf die absorbierte Energieverteilung und Ein-dringtiefe untersucht. Das Konsolidieren von viskoelastischen Pulverpartikeln wirdanalysiert und der Einfluss der Prozessparameter Strahlleistung, Scangeschwindigkeitund Scanstrategie auf die Schmelzbadform einer einzelnen Linie, die Schmelzbadtiefeeiner einzelnen Schicht und die relative Dichte mehrerer Schichten untersucht. DerEinfluss der Pulveralterung und der Viskoelastizität innerhalb der relevanten Gren-zen wird untersucht und schließlich werden die relevanten Faktoren identifiziertund ihre Auswirkungen auf die Konsolidierung des Pulvers diskutiert.

II

Contents

1. Introduction 11.1. Additive manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Selective laser sintering . . . . . . . . . . . . . . . . . . . . . . . . . 31.3. Polyamide 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4. State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5. Scope of work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. Theory 102.1. Kinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1. Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2. Oldroyd-B model . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3. Numerical methods . . . . . . . . . . . . . . . . . . . . . . . 15

2.2. Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. Verification 233.1. Analytical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1. Simple shear flow . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2. Transport of passive scalar field . . . . . . . . . . . . . . . . 263.1.3. Oscillating velocity field under uniform motion . . . . . . . . 28

3.2. Benchmark problems . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.1. 4 to 1 planar contraction . . . . . . . . . . . . . . . . . . . . 303.2.2. Four-roll mill . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4. Laser absorption model 414.1. Ray trace model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3. Irradiation of a laser onto a plane surface . . . . . . . . . . . . . . . 46

III

Contents

5. Simulation setup 495.1. Material properties of PA12 . . . . . . . . . . . . . . . . . . . . . . 50

5.1.1. Hydrodynamic properties . . . . . . . . . . . . . . . . . . . 525.1.2. Thermal properties . . . . . . . . . . . . . . . . . . . . . . . 54

5.2. Cell conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3. Powder bed generation . . . . . . . . . . . . . . . . . . . . . . . . . 595.4. Beam properties and scanning strategy . . . . . . . . . . . . . . . . 62

6. Powder-Laser-Interaction 656.1. Laser absorption in single particle . . . . . . . . . . . . . . . . . . . 656.2. Laser absorption in powder bed . . . . . . . . . . . . . . . . . . . . 696.3. Attenuation coefficient of PA12 . . . . . . . . . . . . . . . . . . . . 726.4. Laser absorption of a Gaussian power distribution . . . . . . . . . . 746.5. Sintering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7. Modeling additive manufacturing of PA12 837.1. Exposure with a stationary beam . . . . . . . . . . . . . . . . . . . 837.2. Single line melting . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.2.1. Temperature field . . . . . . . . . . . . . . . . . . . . . . . . 867.2.2. Melt pool shape . . . . . . . . . . . . . . . . . . . . . . . . . 887.2.3. Melt pool depth . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.3. Single layer melting . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.3.1. Temperature field . . . . . . . . . . . . . . . . . . . . . . . . 937.3.2. Melt pool depth . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.4. Melting of several layers . . . . . . . . . . . . . . . . . . . . . . . . 1017.5. Scan strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.5.1. Scan length . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.5.2. Hatching spacing . . . . . . . . . . . . . . . . . . . . . . . . 110

8. Conclusion 115

9. Bibliography 119

Appendices 128

IV

Contents

A. List of Symbols 129A.1. Latin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.2. Greek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

B. Notation and operators 135

V

1. Introduction

1.1. Additive manufacturing

Additive manufacturing (AM) is a process to produce 3D geometries by gradualaddition of material using the additive shaping principle, one of the three basicprinciples of manufacturing aside from formative and subtractive shaping. AMas a term is mainly used by the scientific and industrial community. In public itis more commonly known as 3D printing, which is defined as the fabrication ofobjects through the deposition of material using a print head, nozzle, or anotherprinter technology [1] and hence is a subclass of processes within AM. This isreflected in the fact that a Google search (Jul. 2018) of 3D printing yields aboutten times more results than additive manufacturing. The same ratio also holds forGoogle search requests as shown in figure 1.1 [2].

One of the first descriptions of the additive manufacturing process as we knowit today was given by H. Kodoma in 1981 [3]. For roughly 30 years this fieldwas mainly known to researchers and professionals. At the beginning of 2011 thepublic perception started rapidly to grow, which can be correlated to the first widedistribution of affordable 3D printers for polymers. It is interesting to note that thesearch requests for 3D printing reached their maximum in 2013. Since then theyare at a constant level, within some scattering. On the other hand, the requests foradditive manufacturing are still rising. One possible explanation is that the deeperunderstanding and thereby the wording used by professionals is still growing inthe user base. It also reflects the nature of the additive manufacturing process,easy to learn but hard to master.

The advantages of AM are appealing to both, consumer and original equip-ment manufacturer (OEM). With its crucible-free freedom of design, material di-

1

CHAPTER 1. INTRODUCTION

Figure 1.1.: Google search requests progression for 3D Printing (black, left axis)and Additive Manufacturing (red, right axis) normalized to the max-imum number requests of 3D Printing [2].

versity and resource efficiency, it is possible to locally produce personalized customparts thereby creating the opportunity to decentralize production [4]. Some evenexplore if a fully circular economy could be realized with AM [5], reducing wasteto a minimum [6]. To achieve these goals, the produced parts need to competewith prices and quality of products manufactured by conventional processes. Asof today, AM is most successful in the field of rapid prototyping, rapid tooling,replacement parts and reverse engineering with applications in the health, auto-motive, aerospace and fashion industry.

Material classes that can be processed by AM include metal, polymer, ceramicand composite materials. Each material has its unique properties and thereforediffers in its application and joining process. Chemical and thermal reaction bond-ing are the two fundamental mechanisms to fuse polymers. Processes utilizingchemical bonding are binder jetting, material jetting, stereolithography and sheetlamination, while thermal bonding is used in material extrusion, material jettingand powder bed fusion (PBF). Selective laser sintering (SLS), a sub process ofPBF, is one of the most likely processes to play a major role in future industrialproduction due to its potential to manufacture complex parts with excellent ma-terial properties. First applications in automotive include personalized parts by

2


BMW for its Mini Cooper cars and spare parts for air duct and heating unit controlby Daimler. Airbus is also exploring the potential of AM by producing over 90 %of the structural parts of an unmanned aerial vehicle from polyamide 12 powderusing SLS [7].

1.2. Selective laser sintering

Selective laser sintering of polymers is a process for the layer by layer productionof components by selective melting of a powder bed with a focused laser beam.Figure 1.2 illustrates the basic steps of SLS.

Figure 1.2.: Preprocessing: One slice of a 3D model, cutting plane indicated bythe red line, with a zoom into an exemplary scanning strategy.

At the beginning of every AM process, a CAD-model of the desired part iscreated. Even though SLS is known for its design freedom, it still has some limits.The CAD model needs to fulfill certain criteria, like the minimum structure size ofthe model cannot exceed the minimum resolution of the process, enclosed volumesmust be avoided if trapped powder is to be removed and support structures needto be considered at overhangs. The part is then sliced by cutting the 3D modelwith plane perpendicular to the build direction. Each cutting plane represents apowder layer and contains the contour of the area to be melted. This data can beused to determine the melting strategy in terms of beam path, scanning velocityand beam power.

3


Figure 1.3.: SLS: Four major steps of the SLS process containing lowering theplatform, applying powder, heating the powder and selective melting.

After the preprocessing is finished the process can be started. First the buildchamber is heated by infrared heaters to generate a homogenous temperaturedistribution. Then the building platform is lowered by a given layer thickness,typically in the range of 100 µm and a layer of heated powder is applied by aroller or blade onto the platform. The new powder layer is preheated but not hotenough, so that it also has to be heated to the desired ambient temperature, whichis chosen just below the melting point of the used polymer.

As soon as the required temperature has been reached, the selective meltingof the powder by a laser begins. The scanning path of the laser is calculated duringthe preprocessing. Although theoretically any arbitrary path would be possible,only a few parametrized scanning strategies have been well established. The laseris deflected by a mirror and beam power and scanning velocity are set so that theirradiated powder absorbs enough energy to be melted. Optimal process parame-ters are those parameters that provide sufficient energy to establish a connection

4


to the underlying layer without overheating the melt. Once selective melting of thepowder is finished, the steps can be repeated and start from beginning by loweringthe platform. This way a component forms layer by layer from a 3D model.

Table 1.1.: SLS devices with the largest build chamber by each manufacturer withmaximum beam power PB,max and maximum scanning speed vB,maxsorted by the build rate rB. Data is taken from the corresponding datasheet provided by the manufacturer.

Manufacturer Name PB,max [W ] vB,max[ms

]Volume

[mm3] rB

[lh

]EOS EOS P 500 70 10 500 x 330 x 400 6.6

Prodway ProMaker P4500 X 100 15.8 400 x 400 x 450 4.0

Frason Farson 403P 100 15.2 375 x 375 x 430 4.0

3D Systems ProX SLS 6100 100 12.7 380 x 330 x 460 2.7

Sentrol SP250 100 2 250 x 250 x 300 0.4

Ricoh AM S5500P 100 15 500 x 500 x 480 n.a.

Sondasys SL 01 100 12 350 x 350 x 600 n.a.

There are numerous commercially available devices for SLS. The most impor-tant manufacturer and their machines with the largest build chamber are listed intable 1.1. Considering the EOS P 500, designed for industrial mass production,relevant process time scales can be estimated. A build rate of up to 6.6 l

hand the

given volume suggest a total build time of 10 h, with a layer thickness of 100 µm.This would result in a layer time of approximately 10 s. Although experienceshows that the build rate given by the manufacturer is optimistic, the order ofmagnitude is conclusive. The smallest time scale is given by the irradiation of thelaser during melting. A beam diameter of 400 µm and a maximum scanning speedof 10 m

sresult in an interaction time of 0.04 ms. Therefore, the relevant time

scales of the SLS process extend over eight orders of magnitude. This makes itrather challenging to find the optimal process parameter especially on short timescales. Analogous considerations can be made for the spatial scale, which rangesform the diameter of the powder particles to the size of the build chamber.

5


1.3. Polyamide 12

An additional layer of complexity is added by the used material. This work willfocus on the processing of polymers and, within the vast diversity of polymers,especially concentrate on polyamide 12 (PA12). The derived methods and con-clusions can be applied to other polymers and even material classes. PA12 is ahigh-performance polymer and one of the most common materials used in AM [8].One example of the increasing demand is the announcement of Evonik to doubleits production capacity for PA12 by investing € 400 million in a new plant inGermany [9].

Polyamide 12 is a semi-crystalline thermoplastic with a rather large hysteresisbetween melting and crystallization. Figure 1.4 illustrates a typical trace of athermal analysis, e.g. by Differential Scanning Calorimetry (DSC), for PA12.Starting from a solid phase the polymer is heated and the amount of energy permass needed to heat up the specimen over temperature is recorded. The minimumduring the transition to liquid phase marks the melting temperature Tm, whilethe maximum during cooling is the crystallization temperature Tc. It is commonto adjust the build chamber temperature TB during the heating step in SLS sothat it is in the interval between Tc and Tm and close to Tm. Therefore, the energydeposited by the laser only needs to be sufficient to overcome the enthalpy of fusion∆Hf .

As soon as the polymer powder is liquid it starts to consolidate due to sur-face tension and gravity, which are opposed by the viscosity and elasticity of theliquid. The balance between the consolidation forces and the vicoelasticity of thefluid during the time window given by the melting process determines the me-chanical properties of the produced part. While on the one hand, a long melt lifetime allows the powder particles to fuse, it is on the other hand, not desirablefor dimensional accuracy, since the melt can infiltrate the surrounding powder.Furthermore, depending on the process parameters used to increase melt pool lifetime, it can lower the build rate and increases the peak temperature in the melt.

Numerous experimental studies have been conducted to investigate the non-linear correlation between process parameters and material properties. Among

6


Figure 1.4.: Schematic of a typical trace of a thermal analysis, e.g. by DifferentialScanning Calorimetry (DSC), for PA12.

others, the influence of the beam power, scanning speed and scan strategy wereidentified as relevant parameters [10, 11, 12]. Furthermore theoretical approachesto model the process were performed since the early stages of SLS [13]. Thefollowing section will give an overview of the models developed.

1.4. State of the art

A first thermal model was presented by Nelson et al. [14] in 1993. The powder bedwas treated as an one dimensional semi infinite solid and the heat transfer waseither calculated by finite-difference (FD) or by finite element method (FEM).The rate of coalescence of the powder was approximated using Frenkel equation[15]. Despite the simplicity of the model it was shown that it is able to predictthe sintering depth of Polycarbonate powder in SLS for a wide range of processparameters with 20 % accuracy. The numerical results showed that the sinteringdepth is most sensitive to the relative density of the powder bed, the powder bedtemperature, the scan speed and laser power. In 1998 Williams et al. [13] used asimilar model to investigate the influence of numerous process parameters. Their

7


results demonstrated that scanning strategy has significant effects on the SLSprocess response independent of the applied energy density. A two dimensionalfinite element model was developed by Tontowi and Childs [16] in 2001, whichtakes latent heat effects for crystalline polymers into account. They comparedexperimental and simulation results of the density of sintered parts by mainlyvarying powder bed temperature.

A first model for the 3D simulation of the sintering of a single track in SLS wasdeveloped by Gabriel et al. [17] in 1999. It accounts for both thermal and sinteringphenomena. To simplify the problem, they define a local coordinate system atthe center of the laser beam. With the moving frame of reference it is possibleto rewrite the governing equations to describe a steady state problem, which issolved by FEM. To calculate the evolution of the relative density of the powderbed as a function of time and temperature, the Scherer [18, 19] and the Mackenzieand Shuttleworth [20] models have been used. In 2009 Dong et al. [21] developeda three dimensional FEM for SLS, using the commercial code Abaqus. Theycalculated temperature and density distribution in sintered amorphous polymerpowders, depending on process parameters including the laser scan velocity, laserpower, laser beam diameter and the powder bed temperature. An even moresophisticated model was intrudiced by Liu et al. [22] in 2012. They used a microscale 3D finite element model, where the powder particles where modeled as cubes,which are placed in equal distance to each other over several layers. The powderswere directly exposed to laser irradiations with a Gaussian function of heat flux.Two main mechanism where identified for the sintering of the powder bed. Thefirst mechanism is that the laser beam not only heats the first layer of powderparticles by irradiation but also the second layer of the powder to make themmelt. The second is that the falling down of the melting liquid from the top layerfills the gabs between the particles.

In 2017 Xin et al. [23] presented one of the most complete three dimen-sional FEM simulation frameworks for SLS. It includes thermal diffusion, a modi-fied Monte Carlo ray-tracing method for laser absorption and a Discrete ElementMethod (DEM) to predict the physical behavior of discrete particles. Therefore itaccounts for radiative heat transfer, heat conduction, sintering and granular dy-

8


namics and the effect of scattering on the laser-particle interaction. It was foundthat the scattering of the laser beam has a strong effect on the distribution of theabsorbed energy.

Even though theoretical approaches to model the process were performed sincethe early stages of SLS, still no simulation tool is available considering thermo-and hydrodynamics of viscoelastic polymers at powder scale capable of model-ing melting with several scan lines over several layers. Such simulations wouldallow insight at a process scale not accessible via experiments and improve thefundamental understanding of SLS.

1.5. Scope of work

Subject of this thesis is the development and application of a simulation frame-work for modeling selective laser sintering of viscoelastic polymers at powder scale.First, the derivation of the governing equations for thermo- and hydrodynamicsand the corresponding numerical methods for their solution will be presented.These methods are then validated with analytical and benchmark problems. Inaddition, a laser absorption model will be introduced to account for the inter-action between the laser and the powder. The methods derived will be used tosimulate the SLS process. Starting at the absorption within one powder particlethe simulation setup is gradually expanded until the entire process is modeled,including the melting of several layers investigating the influence of beam power,scanning speed and scanning strategy. Finally, the potential and limitations of thepresented framework and the results of the conducted parameter studies will bediscussed.

9

2. Theory

This chapter presents a brief introduction on the derivation of the governing equa-tions for fluid dynamics of viscoelastic fluids using kinetic theory and thermo-dynamics. The derived analytical expressions will be followed by the numericalmethods used to solve them, namely the lattice Boltzmann method (LBM), arather young numerical method suitable for complex dynamic flows. The shownderivation is not comprehensive, but offers the possibility to follow the essentialsteps and ideas used in this work. The nomenclature is structured as follows:Latin and Greek letters are scalars, letters with arrows are vectors and bold largeletters are second order tensors. A definition of the used operators can be foundin appendix B.

2.1. Kinetic theory

Kinetic theory is one of the most successful theories in applied science and engi-neering with a wide range of applications. Its success is based on the deduction ofequations for macroscopic observables based on simple microscopic models. Thissection will give a brief overview on how to derive the governing equations to de-scribe viscoelastic fluids. For the most parts this will follow the ideas outlinedin the textbook by Huilgol and Phan-Thien [24]. A compact summary is alsopublished by the author of this work [25].

2.1.1. Hydrodynamics

In general, a liquid consists of atoms and molecules. One basic model is to representthese particles with hard spheres, with mass m and velocity ~ξ. Then a particledistribution function f

(~x, ~ξ, t

)can be defined which will give the total mass when

10

CHAPTER 2. THEORY

integrated over the volume d~x and the velocity space d~ξ at time t. The first twomoments of f in velocity space can be identified with the density ρ and the meanvelocity ~v.

ρ (~x, t) ≡∫f(~x, ~ξ, t

)d~ξ (2.1)

~v (~x, t) ≡ 1ρ (~x, t)

∫~ξ f

(~x, ~ξ, t

)d~ξ (2.2)

Demanding conservation of mass, e.g. no chemical reaction, no fission or fusionis considered, the total derivative with respect to time of f only changes due tocollisions. These collisions are represented by the collision operator Ω, which takesthe change of f due to inter-particle collisions into account, while conserving massand momentum.

df

dt= ∂tf + d~x

dt· ∇f + d~ξ

dt· ∇~ξf = Ω (2.3)

The time derivatives can be replaced with

d~x

dt= ~ξ (2.4)

d~ξ

dt=

~fextm

(2.5)

where ~fext is an external force field, e.g. gravity, electric or magnetic force andm is the mass of the particle. Equation (2.3) can be rewritten to the Boltzmannequation.

∂tf + ~ξ · ∇f +~fextm· ∇~ξf = Ω (2.6)

Since only elastic collisions between two particles are considered, the first twomoments of the collision operator must be identical to zero to ensure conservationof mass and momentum. Bhatnagar, Gross and Krook [26] introduced a simpleexpression for Ω, assuming that the collision between particles drives the massdistribution function f to its equilibrium state f eq at a rate of the molecular

11

CHAPTER 2. THEORY

collision time τ .

Ω ≡ −1τ

(f − f eq) (2.7)

The Maxwell–Boltzmann distribution can be used as an analytical expression forfeq. A numerical representation will be given in section 2.1.3. By taking momentsof the microscopic description, the more commonly known macroscopic equationsfor fluid dynamics can be derived. The 0th moment of the Boltzmann equation(eq. (2.6)) leads to the continuity equation in the incompressible limit:

∇ · ~v = 0 (2.8)

Additionally, by taking the 1st moment of equation (2.6) the general form of theNavier-Stokes equation is recovered

ρ (∂t~v + (~v · ∇)~v) = −∇p+∇ ·T + ~fext (2.9)

with p being defined as pressure, which can be connected by the ideal gas law tothe density assuming an adiabatic process. When describing viscoelastic fluids itis common to separate the stress tensor T into a Newtonian TN and a viscoelasticTP contribution.

T = TN + TP (2.10)

In an incompressible Newtonian fluid the stress tensor is related to the shear rateby

TN = µN(L + LT

)(2.11)

L ≡ (∇~v)T (2.12)

where µN is the dynamic viscosity of the Newtonian fluid. The incompressibleNavier-Stokes equation with the extra stress is derived by substituting T in equa-tion (2.9) by the expressions given in equations (2.10) and (2.11).

ρ (∂t~v + (~v · ∇)~v) = −∇p+ µN∆~v +∇ ·TP + ~fext (2.13)

12

CHAPTER 2. THEORY

In this work the Oldroyd-B model is used to solve for the time evolution of theviscoelastic stress tensor TP .

2.1.2. Oldroyd-B model

An important group of viscoelastic fluids consist of liquids containing polymermolecules. One of the common approaches to model these fluids is to simplify themolecules as elastic dumbbells, consisting of two beads connected by an unbendablespring, dissolved in a Newtonian fluid as illustrated in figure 2.1.

Figure 2.1.: Dumbbells dissolved in a Newtonian fluid. The dumbbells consist oftwo beads connected by an unbendable spring. The fluid-dumbbellinteraction is only considered between the fluid and the beads due tofriction at the surface of the beads. The solvent flow accelerates thebeads as the spring contracts them. The connection of the beads isdescribed by the vector ~q.

A configurational distribution function Ψ(~q, ~x, t) can be defined as probabilityto find a dumbbell within the volume d~x with the connection vector ~q at a giventime t. To calculate Ψ(~q, ~x, t) the Fokker–Planck equation can be used, whichdescribes the time evolution of a probability density function [27]:

∂tΨ = −~v · ∇Ψ−∇~q · (L · ~q) Ψ + 2εthcf∇~q ·

(∇~qΨ + 1

εth~fcΨ

)(2.14)

where ~fc is the connecting force between the two beads, which can an arbitraryfunction of the length between the two beads. εth = kbT is the thermal energy with

13

CHAPTER 2. THEORY

kB being the Blotzmann constant and cf is the friction coefficient of the solvent.The first term of equation (2.14) accounts for advection due to solvent flows. Thesecond term considers the change of Ψ(~q, ~x, t) due to hydrodynamic drag, e.g. shearand rotation. The third term represents diffusion based on Brownian motion. Thefourth term is the relaxation because of the connecting force between the beads.Equation (2.14) is a second order differential equation which, in general, can benumerically solved for Ψ(~q, ~x, t) with a given velocity field. To calculate the stressesfrom the configurational distribution function, Kramers form for the viscoelasticstress tensor can be used [27]:

TP = −nP⟨~q ~f c

⟩+ nP εthI (2.15)

with 〈.〉 being the ensemble average with respect to the phase space given byΨ(~q, ~x, t). Following the same ideas as in the previous section, the macroscopicevolution equation for the viscoelastic stress tensor can be derived. Multiplyingequation (2.14) with Q = ~q~q and integrating over configurational space, the upper-convected Maxwell (UCM) model can be obtained:

TP + λ1∇TP= µPD (2.16)

D = L + LT (2.17)

where λ1 is the relaxation time and µP the zero shear viscosity of the viscoelasticfluid. The upper convected time derivative is defined as:

∇T= ∂tT + (~v · ∇) T− (L ·T + T · LT ) (2.18)

and describes the rate of change of a tensor within a small volume that is rotatingand stretching with the fluid. Using the incompressible viscoelastic stress tensor(eq. (2.10)), the total viscosity µ0 and the retardation time λ2, the UCM modelcan be transformed to the Oldroyd-B constitutive equation:

T + λ1∇T = µ0

(D + λ2

∇D)

(2.19)

14

CHAPTER 2. THEORY

with

µ0 = µN + µP (2.20)

λ2 = µNµ0λ1 (2.21)

In summary, an Oldroyd-B fluid is fully described by the following system ofequations:

∇ · ~v = 0 (2.22)ρ (∂t~v + (~v · ∇)~v) = −∇p+ µN∆~v +∇ ·TP + ~fext (2.23)

TP + λ1∇TP = µPD (2.24)

2.1.3. Numerical methods

Due to the complex nature of the SLS process it is not possible to solve thegoverning equations with the given boundary conditions analytically, therefore, anumerical approach is necessary. In this work the rather new lattice Boltzmannmethod (LBM) is used, which evolved out of lattice gas automata [28]. Its keyfeature is that it solves the microscopic Boltzmann equation for the particle dis-tribution function f(~ξ, ~x, t), where ~ξ is the microscopic velocity at (~x, t). With themoments of these distribution functions the macroscopic properties of the fluidcan be recovered [29]. LBM is known for its quality to calculate free surface flowsin complex geometries [30] and its performance on parallel computer architecture[31].

The Boltzmann equation is solved on a lattice by discretization of continuousvelocity space to a set of velocity vectors ~ei. All shown derivations in this workwill focus on a lattice in two dimensions with nine discrete velocities (D2Q9) asshown in figure 2.2 (left).

15

CHAPTER 2. THEORY

Figure 2.2.: Illustration of discrete velocities (left) and dumbbell connection vec-tors (right) on a D2Q9 lattice [32].

For a D2Q9 lattice ~ei are defined as:

~ei =

(0, 0) i ∈ 0(1, 0) , (0, 1) , (−1, 0) , (0,−1) i ∈ 1, 2, 3, 4(1, 1) , (−1, 1) , (−1,−1) , (1,−1) i ∈ 5, 6, 7, 8

(2.25)

With the discrete velocity space, discrete particle distribution functions fi(~x, t)at each lattice site ~x at a time t can be defined. Using fi(~x, t), the macroscopicproperties of the fluid can be calculated:

ρ(~x, t) =8∑i=0

fi(~x, t) (2.26)

~v(~x, t) = 1ρ(~x, t)

8∑i=0

fi(~x, t)~ei (2.27)

T(~x, t) =8∑i=0

fi(~x, t)~ei~ei (2.28)

The Boltzmann equation can then be solved on a lattice with the numerical scheme[29], using the BGK-approximation [26]:

fi(~x+ ~ei, t+ δt) = fi(~x, t)−δt

τN + 0.5δt (fi(~x, t)− f eqi (~x, t)) (2.29)

where δt = 1 is the time step, τN = 3νN is the relaxation time of the Newtonian

16

CHAPTER 2. THEORY

fluid and νN is its kinematic viscosity. All quantities are given in lattice unites.It can be shown that the continuity equation and the Navier-Stokes equations canbe recovered by a Chapman-Enskog expansion from equation (2.29) [28].

The numerical scheme is divided into two steps. The first step is the streamingof the distribution functions.

fi(~x+ ~ei, t) = fi(~x, t) (2.30)

Here, the distribution functions propagate to the neighboring lattice sites theypoint at. Since the collision operator preserves mass and momentum, the macro-scopic quantities given by equations (2.26) to (2.28) can be calculated already afterthe streaming step. The second step is

fi(~x, t+ δt) = fi(~x, t)−δt

τN + 0.5δt (fi(~x, t)− f eqi (~x, t)) (2.31)

in which the collision operator is applied and the distribution functions relax totheir equilibrium state f eqi , given by equation (2.39).

Analogous to the discrete velocity space, the configurational space ~q of theconnecting vector of the dumbbells can be represented by a set of discrete connec-tion vectors ~qi as shown in figure 2.2 (right).

~qi = ~ei (2.32)

Therefore, a discrete configurational distribution function Ψi(~x, t) can be defined.To calculate the time evolution of Ψi(~x, t) on a lattice, J. Onishi et al. [32] proposea numerical scheme to solve the Fokker–Planck equation (eq. (2.14)) :

Ψi(~x, t+ δt) =Ψi(~x, t) + δΨi

− δt

τP + 0.5 δt (Ψi(~x, t)−Ψeqi )

+ τPτP + 0.5 δt Φi δt

(2.33)

with τP = λ1 being the viscoelastic relaxation time. Equations (2.29) and (2.33)are solved within one time step. The first term on the right hand side of equation(2.33) is the discrete distribution function of the previous time step. This distribu-

17

CHAPTER 2. THEORY

tion is then modified by δΨi, which accounts for the convection of the dumbbellsdue to solvent flow. The third term represents the relaxation of Ψi(~x, t) towardsits equilibrium function Ψeq

i , given by equation (2.37), within one time step. Thelast term includes Φi, which takes elongation and rotation because of solvent flowsinto account [33]. This acts as the source term increasing or decreasing viscoelasticstress. The source term due to solvent shear rates can be calculated with:

Φi = ωiH2

ε2th

(Qi −

εthH

I)

: (L · 〈Qi〉) (2.34)

where H is the Hookean constant of the connecting spring and 〈.〉 is the ensem-ble average defined as the sum over the configurational space ~q, weighted by thedistribution function Ψ, and is applied component wise.

Qi = ~qi~qi (2.35)

〈Qi〉α,β =∑j ~qj,α~qj,βΨj∑

k Ψk

(2.36)

The construction of equation (2.34) is conducted so that the density number ofdumbbells nP and the isotropy is maintained, hence, recovering the correct dy-namics up to the second order of Q [32]. The discrete equilibrium configurationaldistribution functions Ψeq

i are trivial to define, since they correspond to fully re-laxed dumbbells and are equal to the weights of the lattice ωi. In the case of aD2Q9 lattice:

Ψeqi = ωi =

4/9 , i ∈ 01/9 , i ∈ 1, 2, 3, 41/36 , i ∈ 5, 6, 7, 8

(2.37)

The macroscopic polymeric stress tensor can be calculated at each lattice site with:

TP = −nP8∑i=0

~qi ~fci Ψi + nP

8∑i=0

~qi ~fci Ψ

eqi (2.38)

Once the stress tensor is calculated, it can be taken into account in the discreteequilibrium particle distribution functions f eqi , which are used in the collision op-

18

CHAPTER 2. THEORY

erator in equation (2.31).

f eqi = nN ωi

[1 + 3~ei · ~v + 9

2 (~ei · ~v)2 − 32~v

2]

+ ωi ψi (2.39)

where the number density nB = 1 in lattice unites and with [32]:

ψi = 92

(Qi −

13I)

: 1εth

TP (2.40)

Equation (2.39) differs by the lattice weights ωi in front of ψi from the originalequilibrium function purposed by J. Onishi et al. [32], which is presumably theresult of a typo. The weighting factor is necessary to recover the macroscopicstress tensor T inserting equation (2.39) into (2.28) to match equation (2.10):

T(~x, t) =8∑i=0

fi(~x, t)~ei~ei = TN(~x, t) + TP (~x, t) (2.41)

Figure 2.3.: Illustration of the transport model: The net mass flow between twoneighboring cells is given by the difference of the two discrete particledistribution functions pointing in the direction of the neighboring cell.The dumbbells propagate in the direction of the mass flow and areweighted by the mass change [25].

To calculate the transport of the dumbbells, it is assumed that their diffusiontime is orders of magnitude higher than the diffusion time due to convection of theNewtonian solvent. Therefore, a transport model can be derived for δΨi takingonly the net mass transport of the solvent into account. The mass flow between two

19

CHAPTER 2. THEORY

cells can be calculated by the difference of two neighboring distribution functionspointing in the opposite direction, as illustrated in figure 2.3.

δmi = fi (~x+ ~ei, t)− fi (~x, t) (2.42)

Where the subscript i is the inverse direction of i. The dumbbell transport δΨi canbe calculated at each lattice site using the transport model proposed by Osmanlicet al. [25]:

δΨi =8∑

k=0

δmk

ρ (~x+ ~ek, t)Ψi (~x+ ~ek, t) , δmi ≥ 0

δmk

ρ (~x, t) Ψi (~x, t) , δmi < 0

(2.43)

The transport model (eq. (2.44)) presented by J. Onishi et al. [32] differs fromthe model used in this work in two essential features.

δΨi,Onishi =8∑

k=0

[−Ψi (~x, t)

fk (~x, t)ρ (~x, t) + Ψi (~x+ ~ek, t)

fk (~x+ ~ek, t)ρ (~x+ ~ek, t)

](2.44)

First, a zero net flow between two cells (δm = 0), when using the transportmodel of this work, also means that no dumbbells are exchanged. Second, thetransport of the dumbbells is only possible in one direction within one time step,while the model in equation (2.44) is bidirectional. Both properties lead to thesuppression or minimization of unwanted diffusion due to numerical errors. Adetailed analysis will be given in chapter 3. To solve the macroscopic equations(2.22) to (2.24) using the presented numerical scheme, the following calculationsmust be conducted sequentially for each time step:

1. Stream the distribution functions (eq. (2.30)) to the neighboring cells.2. Evaluate the density ρ (eq. (2.26)) and velocity ~v (eq. (2.27)).3. Propagate the dumbbells δΨi (eq. (2.43)).4. Calculate the elongation and rotation due to the solvent flows Φi (eq. (2.34)).5. Calculate the new configurational distribution functions Ψi (eq. (2.33)).6. Reconstruct the polymeric stress tensor TP (eq. (2.38)).

20

CHAPTER 2. THEORY

7. Calculate the equilibrium particle distribution function f eqi (eq. (2.39)).8. Calculate the new particle distribution functions fi (eq. (2.31)).

2.2. Thermodynamics

To calculate the thermal field the convection-diffusion equation is solved,

∂ρT

∂t+∇ · (~vρT ) = ∇ · (αρ∇T ) +Q (2.45)

where T is the temperature, Q is a heat source and α is the thermal diffusivity ofthe material being defined as

α = λαcpρ

(2.46)

with λα being the thermal conductivity and cp being the specific heat capacity. Theenthalpy of fusion ∆Hf is take into account by defining a modified heat capacityc∗p:

c∗p (T ) =

cp , T < Ts

cp + ∆HfTm−Ts , Ts ≤ T ≤ Tm

cp , T > Tm

(2.47)

where Ts is the solidification temperature and Tm is the melting temperature defin-ing the phase transformation limits.

The used Lattice Boltzmann scheme was first presented by R. Zhang et al.[34]. To solve equation (2.45) a third set of distribution functions Ti is intro-duced, in addition to the hydrodynamic and viscoelastic distribution functions.The evolution in time and space is given by:

Ti (~x+ ~ei, t+ δt) = T (~x, t) +(

1− 1τT

)Θi (~x, t) (2.48)

21

CHAPTER 2. THEORY

with

Θi (~x, t) = 3~ei − ~v (~x, t)ρ (~x, t) ·

∑j

~ejfj (~x, t) (Tj (~x, t)− T (~x, t)) (2.49)

T (~x, t) = 1ρ (~x, t)

∑i

fi (~x, t)Ti (~x, t) (2.50)

where τT is the relaxation time for the scalar distribution functions. When usedto calculate the thermal field, τT can be expressed by the thermal diffusivity as

τT = 3 α + 0.5 (2.51)

The hydrodynamic distribution functions fi, the density ρ and the velocity ~v arecalculated as discussed in the previous sections. One main advantage of this solveris that it utilizes the so-called BGK regularized collision operator [35, 36] andthereby improves the stability of the numerical scheme by only accounting forthe first order non-equilibrium moments. Equation (2.45) can be recovered byconducting a Chapman-Enskog expansion. Ti propagates, similar to the dumbbelldistribution functions, along with fi. Therefore, the energy distribution Ei ismaintained during advection.

Ei = fi Ti (2.52)

While in general, this scheme can be used for any passive scalar, in this workit will be only used to calculate the thermal field. All solver validations in thefollowing chapter are conducted at isothermal conditions neglecting this solver,since the passive scalar has no impact on the hydrodynamics.

22

3. Verification

This section is divided into analytical and benchmark problems. The analyticaltest cases are chosen so that the order of accuracy of the viscoelastic solver and thetransport model can be calculated individually and combined. Furthermore, theperformance of the algorithm is analyzed in two common benchmarks for complexviscoelastic flows, where no or only partial analytical solutions are available. Theresults of the following section are published by the author [25, 37]. All quantitiesare given in lattice units and µP is set to 0.6. The connection force between thedumbbells is given by Hooke’s law with the Hooken constant H as:

~f ci = H~qi (3.1)

3.1. Analytical tests

Three analytical test cases are considered to validate the implemented algorithm.The simple shear flow setup is one of the most common tests in ComputationalFluid Dynamics (CFD). Its analytic solution is well known and the results areindependent of the used transport model. Therefore, it is ideal to check the orderof accuracy of the viscoelastic solver. The second test case is the advection of aGaussian distribution of a passive scalar field in a uniform velocity field, withoutconsidering viscoelasticity. The passive scalar field has no influence onto the ve-locity field and hence, only the transport model can be studied. As the third testcase an oscillating velocity field under uniform motion is chosen. In this setup allthe parts of the presented algorithm are active including the viscoelastic solverand the transport model.

23

CHAPTER 3. VERIFICATION

3.1.1. Simple shear flow

In the simple shear flow setup the velocity field is imposed onto the simulationdomain and is given by:

vx = γy (3.2)vy = 0 (3.3)

With γ being the shear rate. The transport model is neglected, since the flowparallel to the velocity gradient is zero. With the given velocity field and the gov-erning equations (2.22 - 2.24), solving for the steady state case is straightforwardand the components of the viscoelastic stress tensor can be written as:

TP,xx = −2µPλ1γ2 (3.4)

TP,xy = TP,yx = −µP γ (3.5)TP,yy = 0 (3.6)

The dumbbell distribution functions are initialized with their equilibrium values,given by equation (2.37), and the simulation is stopped when no change in timeis observed. Figure 3.1 shows the numerical results and the analytical solutionsfor TP,xx and TP,xy for three different relaxation times and different shear rates.TP,yy is found to be zero within machine accuracy. The analytical and numericalresults are in excellent agreement. Neglecting the terms with time derivativesresults in the steady state solution. To investigate the order of accuracy in time,the evolution of TP in a start up shear flow is necessary. Starting from a zerostress state, it can be shown that the time dependent components of the stresstensor are given by [38]:

TP,xx(t) = −2µPλ1γ2(1− e−t/λ1

)+ 2µP γ2te−t/λ1 (3.7)

TP,xy(t) = TP,yx(t) = −µP γ(1− e−t/λ1

)(3.8)

TP,yy(t) = 0 (3.9)

It is simple to validate that for t → ∞, the steady state analytical solutions arerecovered (eq. (3.4 - 3.6)). In figure 3.2 the numerical and analytical results for

24


Figure 3.1.: Simulation results and analytical solution of the steady state TP,xx(left) and TP,xy (right) in simple shear over the shear rate γ. [25]

three different relaxation λ1 are shown. The simulation is terminated after tentimes λ1. Good agreement is found between the analytical and numerical results.To check the order of accuracy, λ1 is varied, while the Weissenberg number Wi

Figure 3.2.: Simulation results and analytical solution of the time evolution of TP,xx(left) and TP,xy (right) in start up shear flow over time. [25]

25


is kept constant, which is a dimensionless number describing the relation of therelaxation time and a characteristic time scale, e.g. given by the shear rate γ.

Wi = γλ1 (3.10)

This is equivalent to scaling the time discretization ∆t. The simulation is initializedwith relaxed stresses and is evaluated after one λ1. The error Err to investigatethe order of accuracy is calculated as:

Err =

√1N

N∑∣∣∣T lbP,αβ − TP,αβ∣∣∣2 (3.11)

Where N is the number of lattice cells evaluated, T lbP,αβ is the result of the simu-lation and TP,αβ is the analytic result. The evaluation of Err is conducted overthe whole simulation domain. Figure 3.3 shows Err over λ1 for two Weissenberg

Figure 3.3.: Scaling of Err with λ1 of TP,xx and TP,xy in start up shear flow fortwo Weissenberg numbers. The solid line indicates second oder ofaccuracy. [25]

numbers and indicates second order of accuracy in time, thereby confirming thetheoretical derivation.

3.1.2. Transport of passive scalar field

To test the transport model, the advection of a Gaussian distribution of a passivescalar field in a uniform velocity field in 1D is calculated. For this setup the

26


viscoelastic solver is neglected and, therefore, only the performance of the transportmodel for a passive scalar field is investigated. The Gaussian is defined as

g (x) = 1√2πσ2

exp(−(x− x0)2

2σ2

)(3.12)

with σ = 50 and x0 = 10σ in a domain with the size of N0 = 1500. The uniformvelocity field propagates with a constant velocity of vx = 0.01. Figure 3.4 (left)shows the distribution at initial state and after 50000 time steps. The peak valueof the Gaussian is lowered and the width increased. Numerical dispersion can beobserved and the numerical diffusion coefficient D is given by [37]:

D = 12 (1− vx) · vx (3.13)

D approaches 0 for small velocities, which is a desired property since the LBM isvalid in that regime. The Péclet number Pe relates advection to diffusive transportand can be derived for this case as:

Pe = vxD

= 21− vx

(3.14)

The stability limit for the maximal velocity using a BGK collision operator isvx 0.5 [39], which means that Pe > 1. Therefore the advection transportalways dominates over numerical dissipation. To investigate the order of accuracy,the parameters are slightly changed. The Gaussian is initialized with σ = 5 · N0

200

and the velocity is set to vx = 200N0· 10−3 with the domain size N0 scaling from 200

to 6400, while keeping the Reynolds number

Re = vxN0

µ0(3.15)

constant. After 1000 time steps the error is evaluated according to the methodsdescribed in section 3.1.1. As can be seen in figure 3.4 (right) the results for Errare on the slope for first order of accuracy. However, the absolute error is smallcompared with the finite differences methods of Lax-Wendroff [40] and Succi et al.[41] and the lattice based method developed by Onishi et al. [32]. A comprehensiveanalysis of all three methods and the one presented in this work can be found in[37]. It should be noted that not only the absolute error is small, also no oscillations

27


Figure 3.4.: Left: Advection of a passive scalar field in a uniform velocity field in1D. Right: Scaling of error of a advected Gaussian distribution after1000 time steps with N0 and the slope for first order of accuracy [37].

in the scalar field could be observed.

3.1.3. Oscillating velocity field under uniform motion

With an oscillating velocity field under uniform motion the transport model of thedumbbells together with the viscoelastic solver can be investigated. The velocityfield is modified with a sine function in y-direction, while moving in x-directionwith a constant velocity:

vx = const. (3.16)vy = γ sin (x− vxt) (3.17)

Therefore, periodic boundary conditions can be imposed, with the domain lengthin x-direction L is set to L = 2π. The relaxation time is λ1 = 1000, Wi = 1 andthe simulation duration is set to 50λ1, after which the equilibrium state is reached.The steady state solutions for the components of the viscoelastic stress tensor can

28


be derived from the results of the simple shear flow (eq. (3.7) - (3.9))

TP,xx = 0 (3.18)TP,xy = TP,yx = −µP γ cos (x− vxt) (3.19)TP,yy = −2µP γ2λ1 [cos (x− vxt)]2 (3.20)

The simulation time ts is kept constant, i.e. the number of time steps, while thevelocity vx is changed, which determines the distance in x-direction s0 = vxts thatis covered within one run. In figure 3.5 the analytic and numerical results of TP,xyfor different s0 are compared, which corresponds to different velocities vx. Here,the number of cells within L is set to Nx = 200. Again excellent agreement is foundbetween the analytical and numerical results. Results of similar quality are foundfor TP,yy. The number of cells Nx within one period L is changed to investigate

Figure 3.5.: Analytical and simulation results of TP,xy for an oscillating velocityfield under uniform motion. Every tenth point of the simulation isplotted. [25]

the order of accuracy in space. The error of TP,xy and TP,yy over Nx is plotted infigure 3.6 for three different s0, which corresponds to different vx. The steepestslope is found for s0 = 0.0L (vx = 0) and all tested velocities indicate secondorder accuracy. This is remarkable, because the viscoelastic solver is second order

29


accurate in space and time, but the transport model is not. There are two mainreasons for this behavior. First the absolute error of the transport model is smallcompared to the error of the viscoelastic solver and second, numerical diffusion Dapproaches zero for small velocities (see sec. 3.1.2).

Figure 3.6.: Scaling of Err of TP,xy and TP,yy over the spatial resolution Nx for dif-ferent distances s0, which corresponds to different vx since the numberof time steps is kept constant, in an oscillating velocity field under uni-form motion. The solid line indicates second order of accuracy. [25]

3.2. Benchmark problems

Two benchmark problems are considered, namely the 4 to 1 planar contractionand the four roller mill. Both cases include contraction and elongation flows andexhibit unique features that can only be observed for viscoelastic fluids. Eventhough these problems are considered as complex flows, they are simple enough tospecify exact boundary conditions and to compare key features qualitatively andquantitatively with results found in literature.

3.2.1. 4 to 1 planar contraction

The 4 to 1 planar contraction (Fig. 3.7) consists of two parallel planes that contractwithin a step to one fourth of its height. The flow direction is from the wider

30


Figure 3.7.: Schematic illustration of a half plane of the 4 to 1 planar contraction.The flow direction is from left to right. The silent vortex is in thelower and the lip vortex at the entrance corner. [25]

channel towards the narrow channel. It is an often used benchmark problem totest numerical schemes for viscoelastic fluids [42, 43, 44, 45]. Even though theconfiguration is rather simple, the resulting flow behavior is complex and can notbe described analytically. One very interesting feature, which is only observed forviscoelastic fluids in simulations and experiments, is a lip vortex at the entrancecorner at high Weissenberg numbers Wi and a silent vortex [46, 47].

Wi = λ1v

h(3.21)

where v is the mean downstream velocity and h is the half width of the downstreamas illustrated in figure 3.7. Numerical methods that can reproduce the lip vortexof the 4 to 1 planar contraction solving the Oldroyd-B constitutive equation [45,48], suggest that a minimum numerical resolution is required until the size of thevortices is independent of the mesh used. Therefore a sensitivity analysis regardingmesh refinement is conducted. Since these studies require a lot of computationaltime, the problem is assumed to be symmetric along the center line and thereforeonly a half plane is simulated. The missing incoming distribution functions fi atthe center line with the directions i = 7, 4, 8 at the boundary are replaced bytheir mirrored distributions from the simulation domain as illustrated in figure 3.8.

31


Figure 3.8.: Missing incoming distribution functions fi at ~x at the mirror line wherethe directions i = 7, 4, 8 are replaced by their mirrored distributionfunctions i = 6, 2, 5 at ~x+ ~e4.

Thereby modifying the stream step at the boundary with:

f7(~x, t) = f6(~x+ ~e4, t) (3.22)f4(~x, t) = f2(~x+ ~e4, t) (3.23)f8(~x, t) = f5(~x+ ~e4, t) (3.24)

A fully developed parabolic velocity flow field is imposed at the in and out bound-ary. The distance of the boundary to the contraction is set to 20h in each direction.The velocity at the out flow is given by:

vx = −4v0

hy(yh− 1) (3.25)

vy = 0 (3.26)

With the given velocity field and using equation (2.24), the steady state compo-

32


nents of the polymer stress tensor can be calculated as:

TP,xx = −2µPλ1 (∂yvx)2 (3.27)TP,xy = TP,yx = −µP∂yvx (3.28)TP,yy = 0 (3.29)

TP is needed to specify the boundary conditions for the configurational distributionfunctions Ψi. Taking advantage of the symmetry of flow field and assuming steadystate

∂x −→ 0 (3.30)∂t −→ 0 (3.31)

together with equation (2.38), the initial and boundary distribution functions canbe calculated:

Ψ0 = ω0 (3.32)Ψ1 = Ψ3 = ω1 (3.33)

Ψ2 = Ψ4 = TP,xx2nPH

+ ω2 (3.34)

Ψ5 = Ψ7 = −TP,xx − TP,xy4nPH+ ω5 (3.35)

Ψ6 = Ψ8 = −TP,xx + TP,xy4nPH

+ ω6 (3.36)

The incoming configurational distribution functions at the in flow boundary arecalculated in similar fashion. At the walls at lower boundary of the simulationdomain a no slip boundary condition is applied. The dimensionless parameters forthe mesh refinement study are set to β = 1/9, Wi = 1.6 and Re = 0.06 and arerelated to the simulation parameters as,

Re = ρvh

µ0(3.37)

β = µpµ0

(3.38)

with the half width of the downstream being set to h = 1.

33


Table 3.1.: Mesh properties used in this study and resulting characteristic silentvortex length xR compared with values from literature. The result aregiven in quantities of h.

Mesh Cells ∆x xR xR [43] xR [44] xR [45]M1 10000 0.1 1.60

1.26 1.4 1.3M2 62500 0.04 1.44M3 250000 0.02 1.37M4 1000000 0.01 1.27

Figure 3.9.: Streamlines of the 4 to 1 contraction with Wi = 1.6 and Re = 0.06using the meshes listed in Tab. 3.1. M3 shows the silent vortex whichis separated from the lip vortex at the entrance corner. [25]

Table 3.1 shows the number of cells and the spatial resolution ∆x with respectto h of the used meshes. The resulting flow fields are illustrated in figure 3.9. Nolip vortex can be observed in mesh M1. The silent vortex is extended to the

34


entrance corner. With M2 the lip vortex is observable but is still included intothe silent vortex. Finally in M3 two separated vortices can be seen, where nosignificant change in size and feature is found with increasing the resolution inM4. The found characteristic silent vortex length xR of M3 and M4 (Tab. 3.1)are in good agreement with the findings of [45, 43, 44]. It is to be noted that thework of Xue et al. [45] and Phillips and Williams [44] do not offer the values ofthe silent vortex length in a table, therefore they have to be extracted from thegiven graphs.

The meshes M3 and M4 are dense enough to resolve the lip vortex isolatedfrom the silent vortex. Therefore, M3 is chosen for the following parameter studyto minimize the computational costs. By increasing the relaxation time λ1, whilekeeping the Reynolds number constant, the Weissenberg number is increased. Itis known from literature that the size of the lip vortex should also grow [45]. Tocheck if the presented algorithm is capable of reproducing this behavior, Wi isincreased from 0.0, which equivalent to Newtonian flow, to 3.2.

From the resulting flow fields, shown in figure 3.10, it can be seen that the lipvortex mechanism is enhanced with higher Weissenberg numbers. Furthermore atWi = 3.2 and Re = 0.06 the lip vortex is enveloped by the silent vortex. Bothfindings are in agreement with Xue et al. [45].

The relative distribution of the components of the viscoelastic stress tensorTP for Wi = 1.6 and Re = 0.06 are given in Fig. 3.11. The maximum stressesof TP,yy in the region of the entrance corner indicate high shear rates, which isplausible since the gradient of the velocity in x-direction should be at its maximumat that point. The components TP,xx and TP,xy are higher at the downstreamwall. The computed distribution qualitatively matches the results of [48]. Thepresented results show that the developed algorithm is capable of reproducing thekey features of a viscoelastic fluid in a 4 to 1 contraction.

35


Figure 3.10.: Streamlines for different Weissenberg numbers at Re = 0.06. Wi =0.0 corresponds to the velocity field of a Newtonian fluid. Startingfrom Wi = 1.6 a lip vortex can be observed. [25]

Figure 3.11.: Relative distribution of the components of the viscoelastic stress ten-sor TP for Wi = 1.6 and Re = 0.06. [25]

36


3.2.2. Four-roll mill

The so called four-roll mill, designed by Taylor in the early 1930s [49], is an exper-imental setup to create two-dimensional linear flows with a stagnation point be-tween rotating cylinders. This configuration consists of four rolls arranged arounda central point, with each roll having the same diameter and the center of eachroll has the same distance to the stagnation point. The rolls rotate so that aflow is created towards the central point along one symmetry line and an out flowperpendicular to it. This creates a stationary point in which the complex shearconditions onto particles or bubbles can be studied [50, 51]. B. Thomases et al.[52] introduced a simplified theoretical representation of this flow by replacing therotating cylinders with an external body force ~f .

fx = 2 sin x cos y (3.39)fy = −2 cosx sin y (3.40)

With the given force a [0, 2π]×[0, 2π] cell with periodic boundaries is created. Withthis configuration it is possible to investigate the flow field numerically, withoutimplementing rotating rigid bodies or introducing new boundary conditions formoving walls. Furthermore, it allows a simpler analytical analysis of the setup.Figure 3.12 illustrates the resulting flow field.

The Weissenberg number in this configuration is given by:

Wi = λ1LF

µN(3.41)

where L is the length of the domain and F is the magnitude of the force.

Fig. 3.13 shows the vorticity ω

ω = ∂xvy − ∂yvx (3.42)

and the distribution of the viscoelastic stress tensor TP forWi = 0.3 andWi = 5.0at small Reynolds number at equilibrium state with β = 0.5. It is observed thatfor Wi = 5.0 a second set of vortices build up between the main vortices, whichare induced by the external force field, and the diagonal components tr (TP ) of

37


Figure 3.12.: Illustration of the velocity field in a four-roll mill setup.

the viscoelastic stress tensor peak at the center. The presented distributions arein good agreement with the results shown by Thomases et al. [52] and Su et al.[53]. Due to the use of a force field instead of rotating cylinders, a local solutionfor TP at zero Reynolds number in dimensionless units can be found [52]

TP,xx = 11− 2ε + C |y|

1−2εε (3.43)

TP,yy = 11 + 2ε + C |y|

1+2εε (3.44)

where ε is the effective Weissenberg number given by

ε = ∂xvxWi (3.45)

The velocity gradient ∂xvx is evaluated at the stagnation point from the simulationresult by finite differences.

Fig. 3.14 shows the calculated values of the simulation of TP,xx and TP,yy

at the center with their analytic results. An overall good agreement is foundfor ε < 0.4. For higher ε, the numerical results start to deviate from the localsolution. It is straightforward to show that TP,xx (eq. 3.43) approaches infinity for

38


Figure 3.13.: Vorticity ω (left), tr (TP ) and TP,xy for Wi = 0.3 and Wi = 5.0 atsmall Reynolds number at equilibrium state. [25]

Figure 3.14.: Simulation results of TP,xx and TP,yy at the stagnation point comparedwith their local solutions. [25]

39


ε→ 0.5, which must cause numerical errors for a given finite spatial and temporalresolution. The results of this chapter show that the presented Lattice Blotzmannscheme for viscoelastic fluids is capable of reproducing features observed in complexflows at reasonable computational costs.

40

4. Laser absorption model

In Selective Laser Sintering (SLS) of plastics a laser is used to selectively melt par-ticles in a powder bed. The interaction between the laser and the powder createsdynamic thermal conditions, which determine the resulting material and mechani-cal properties of the produced parts. Therefore, it is important to understand andmodel the absorption of the laser in a polymeric powder bed. The light emittedby a laser is coherent at a specific wavelength, thus it is self-evident to implementa ray tracer to model the absorption. The numerical scheme presented in thissection is rather simple compared to state of the art ray tracing algorithms [54,55, 56], but it has been developed with the focus on the use with a volume of fluidmethod (VOF) in 2D. The results of this chapter have been partially published bythe author [57].

4.1. Ray trace model

The basic premise of the absorption model is to trace the path of photons. Theserays are emitted by a coherent light source with a single wavelength, e.g. a laser.Both reflection and refraction are considered, while neglecting diffuse reflection.The energy of the ray is absorbed along its path through matter according to theBeer-Lambert law. The intensity distribution I in a medium for a single wavelengthis given by

I (x) = I0 e−µax (4.1)

where x is the path and µa is the absorption coefficient for a unique wavelengthin a given medium. Beer-Lambert law is valid when moving along a straight linein a homogeneous medium. At the intersection of two different media, reflection

41

CHAPTER 4. LASER ABSORPTION MODEL

and refraction need to be taken into account. Figure 4.1 illustrates the conditionsat an interface between medium A and B. One part of the incoming ray, with thenormalized direction vector ~k, is reflected and is propagated in medium A, whilethe other is transmitted into medium B.

Figure 4.1.: An incoming ray ~k is reflected ~kR in medium A with refractive indexnA and transmitted ~kT into medium B with refractive index nB. [57]

The reflection angle α has the same magnitude as the angle of the incomingray with respect to the surface normal ~n.

α = cos−1(−~k · ~n

)(4.2)

The refraction angle β can be calculated by using Snell’s law

sin (β) = nBnA

sin (α) (4.3)

with nA being the refractive index of the medium A. Applying the rotation matrixto the normal vector, using the corresponding angle γ, the direction of the reflected~kR and refracted ~kT ray can be calculated.

~k(γ) =cos γ − sin γ

sin γ cos γ

~n (4.4)

The fraction of energy for each of the new rays is determined by dividing theremaining energy of the incoming ray EI according to Fresnel equations. Assumingunpolarized light and non magnetic media the energy portion for the reflected ray

42


ER can be calculated as

ER = R EI (4.5)

R = 12

∣∣∣∣∣nA cosα− nB cos βnA cosα + nB cos β

∣∣∣∣∣2

+∣∣∣∣∣nA cos β − nB cosαnA cos β + nB cosα

∣∣∣∣∣2 (4.6)

Thus, the energy for the transmitted ray ET is given by

ET = (1−R) EI (4.7)

Knowing the starting point, the direction vector and the energy of each ray, theenergy distribution along its path can be calculated using equation (4.1).

4.2. Implementation

In the previous section an absorption model was described, that, given the neces-sary geometrical information and the energy distribution of the light source, couldbe applied to any simulation method. However, this work will focus on the im-plementation within the volume of fluid method (VOF) in two dimensions. It isstraightforward to extend the scheme to three dimensions.

Simulating complex geometries, e.g. the interaction of a beam with a powderbed in additive manufacturing is a very challenging task. The VOF method hasbeen successfully applied in this field [58, 59, 60]. A given geometry at initialstate is discretized into cells within the simulation domain. A volume fraction ϕis assigned to each cell according to its occupation level by that geometry, with itsvalue between 0 and 1 (Fig. 4.2).

In general, the information of the exact shape of the geometry is lost afterinitialization. The two main advantages of this method are that it can track locallyfree surface movement and the possibility to reconstruct the surface up to a certainaccuracy, without knowing the global shape of the object. Therefore, the algorithmis ideal for parallel computing. For the absorption model it is necessary to knowthe position of the interface and its surface normal ~n. Due to simplicity and tosave computational time the surface in this work is defined when a ray propagates

43


Figure 4.2.: Left: Arbitrary geometry. Middle: Volume of fluid representation ona 5x5 grid. The exact surface is indicated by the dashed line. Thegray scale shows the volume fraction of the geometry within one cell.Right: Reconstruction of the surface in one cell using eq. (4.10). [57]

from a cell with ϕ = 0 to a cell with ϕ > 0 and vice versa. The normal vector~n within one cell is estimated at its center ~x as the sum over the displacementsvectors ~xi,j of the neighboring cells weighted by their inverse volume fractions.

~n∗ (~x) =2∑

i,j=−2(1− ϕ (~x+ ~xi,j)) ~xi,j (4.8)

~n = ~n∗

|~n∗|(4.9)

~xi,j = i

(10

)+ j

(01

)(4.10)

where ~en is the unit vector in Cartesian coordinates. A linear reconstruction ofthe surface within a cell can be calculated with the template sphere method [61].

Given the initial position of the light source ~x0 and its energy distribution, aray is cast with the normalized direction vector ~k and the energy E (~x0). The pathof the ray is discretized into equal steps of length ∆l and the ray is propagatedincrementally with m being the step number. At each position ~xm the energyabsorbed by the medium ∆E is calculated and added to the corresponding cell,

44


Initializem = 0, ~x0, E (~x0), ~k

Propagatem = m + 1

Calculate~xm, E (~xm), ∆E (~xm)

ReflectionER, ~kR

TransmissionET , ~kT

~x0 = ~xm

E (~xm) < Eminor

~xm /∈ ΘTerminate ray

Change inmedium?

yes

no

no

yes

yes

Figure 4.3.: Absorption model flow chart.

45


using the absorption coefficient of the medium and the covered length.

~xm = ~x0 +m ∆l ~k (4.11)∆E (~xm) = E (~xm−1)

(1− e−µa∆l

)(4.12)

At each step the remaining Energy Em is updated by

E (~xm) = E (~xm−1)−∆E (~xm) (4.13)

When the path of the ray intersects an interface, the initial ray is stopped and twonew rays are cast, accounting for reflection and refraction at the intersection. Therays are also terminated when reaching the boundary of the simulation domainΘ or their energy is below a given threshold Emin. Figure 4.3 illustrates the flowchart of the implementation of the absorption model.

4.3. Irradiation of a laser onto a plane surface

A first validation test for the developed model is the irradiation of a laser beamonto a plane surface. Comparison with experimental data on the absorptance,reflectance and transmittance of PA12 powder layers is given in section 6.3. Inthis simple setup the analytic solution of the energy distribution in a half planewith different incident angles is compared with the numerical results. The laserpower distribution P (x) is assumed to be Gaussian, with the total power P0 andx0 being the center of the beam. The beam diameter is chosen to be 4σ = 400 µm,since this is the typical beam width of lasers used in additive manufacturing.

P (x) = P01

σ√

2πe−

12(x−x0

σ )2

(4.14)

Figure 4.4 shows the relative intensity distribution calculated for an irradiationof a laser onto a half plain. The incoming beam is initialized with one rayper cell at the upper boundary of the simulation domain. Two values for theincident angle γ of 0 and 45 for three different values of spatial resolutions∆x = 40 µm, 20 µm, 5 µm are shown. The attenuation coefficient is set toµα = 104 1

mand the step length of the ray tracer is ∆l = 0.01 ∆x. A qualitative

46


Figure 4.4.: Relative intensity distribution of a laser irradiating onto a flat surfacewith two incident angles γ of 0 (upper row) and 45 (lower row) forthree different spatial resolutions ∆x = 40 µm, 20 µm, 5 µm (leftto right). [57]

good agreement is found with the expected distribution given by the Beer-Lambertlaw (eq. (4.1)).

Furthermore, figure 4.5 compares the relative intensity distribution in thebeam center along the normal direction for five resolutions and three incidentangles with its analytical solution. The radiation intensity distribution in a halfplain can be calculated as a product of the given power distribution at the surface(eq. (4.14)) with the Beer-Lambert law (eq. (4.1)) along its refraction direction.

I (x, y, β) = P0

σ√

2πe− 1

2

(x∗−x0σ

)2

e−µαy∗ (4.15)

To account for refraction a projection of the distribution is applied, which is given

47


by:

x∗ = x−√y∗2 − y2 (4.16)

y∗ = y

cos β (4.17)

with β being calculated by Snell’s law (eq. (4.3)) using the given incident angle γand the refractive index of the medium. Even for low resolutions with ∆x = 80 µm,which corresponds to five rays within the beam diameter, very good agreement isfound.

Figure 4.5.: Comparison of the analytical (eq. (4.15)) and numerical relative in-tensity distribution in the beam center along the normal direction forfive resolutions and three incident angles. [57]

It should be noted that the spatial resolution in the following chapters is fixedto ∆x = 5 µm, which is more than sufficient to resolve the energy distribution ofa beam with a diameter of 400 µm.

48

5. Simulation setup

Additive manufacturing of polymers in a powder bed is a rather new process.Modeling the interaction of a laser beam with particles, the consolidation of thepowder bed and the thermal conditions is challenging. There are no establishedsoftware tools which can map the process at powder scale yet. The methodspresented in the previous chapters to model hydrodynamics of viscoelastic fluids,thermodynamics and the laser absorption model will be combined in the followingto simulate the process.

Powder bed based additive manufacturing consists of three basic steps. First,a powder layer is applied with a defined layer height hl. The second step is toheat this layer until a homogeneous temperature TB within the building chamberis reached. In the third step, the new layer is selectively molten by a movingheat source, e.g. by deflecting a focused laser beam over the desired area. Thisprocess is repeated until the designed structure is produced. To model the process,the presented Lattice Boltzmann scheme is extended by a volume of fluid method,based on the framework provided by C. Körner et al. [62, 30] for free surface flows.The simulation domain is placed as two dimensional slice through the powder bed.At the lower boundary the temperature is set to TB, while the right and leftboundary is periodic. Figure 5.1 (left) shows the domain at initialization.

To ease the discussion, a Cartesian coordinate system is used with the simu-lation domain in the x-z-plane at y = 0. A solid full dense area is placed at thebottom and on top the first powder layer is applied, which will be described infurther detail section 5.3. The beam is deflected perpendicular to the simulationdomain in x-y-plane and the energy is absorbed, thereby heating and melting thepowder particles. The simulation is set to TB after the beam interaction is fin-ished and the maximum temperature in the whole domain drops below the melting

49

CHAPTER 5. SIMULATION SETUP

Figure 5.1.: Build up of four layers. Left: Initial state with a solid area at thebottom and the first powder layer. Middle: Two layers are moltenand on top the powder of the third layer is applied. Right: After fourlayers the simulation is stopped. A detailed discussion of the scanningstrategy is presented in section 5.4.

point Tm. On top of the solidified surface a new powder layer can be applied andthereby the process is modeled layer by layer. Figure 5.1 illustrates the results ofsubstantially melting four layers.

It can be seen that the effective powder layer height h∗l is higher than thedefined layer height. This is due the consolidation of the molten powder bed witha relative density ρrel being lower than one. h∗l can be calculated by:

h∗l = hlρrel

(5.1)

5.1. Material properties of PA12

The predictive character of any model relies on its input parameters. The materialproperties used in this work to model polyamide 12 are the result of a thoroughliterature survey and are summarized in table 5.1. These quantities can be catego-rized into hydrodynamic, thermal and optical properties, which will be discussedin section 6.3. The hydrodynamic, thermal and optical properties are scaled toenforce numerical stability, while keeping capillary (eq. (5.5)) and bond number(eq. (5.7)) constant. The implications will be discussed in section 5.1.1.

50


Table 5.1.: Material properties of polyamide 12 as found in literature. The val-ues for the simulation are scaled to enforce numerical stability whereneeded, while keeping capillary (eq. (5.5)) and bond number (eq. (5.7))constant.

Poperty Sym. Value literature Value LBM Ref.

Density ρ 700− 1000 kgm3 840 kg

m3 [63, 64, 65]

Zero shearviscosity

µ0 390− 5000 Pa · s 0.001 Pa · s [63, 66]

Surface tension γ 0.03− 0.04 Nm

9.7 · 10−8 Nm

[67, 66]

Relaxation time λ1 - 0.01− 100 s

Viscoelastic ratio β - 0.0− 0.9

Meltingtemperature

Tm 177− 185 C 185 C [63, 64, 68]

Solidificationtemperature

Ts 175− 185 C 182 C [63, 64, 68]

Thermalconductivity solid

λs 0.08− 0.28 Wm K

0.08 Wm K

[64, 65]

Thermalconductivityliquid

λl 0.3 WmK

0.3 WmK

[64, 65]

Specific heatcapacity

cp 2670 Jkg K

2670 Jkg K

[64]

Enthalpy offusion

∆Hf (5− 11) · 104 Jkg

(5− 11) 104 Jkg

[69, 64, 68, 70]

Attenuationcoefficient

µα 0.013 1µm

0.013 1µm

sec. 6.3

Refraction index nPA12 1.7 1.7 sec. 6.3

Capillary number Ca 1 1 sec. 5.1.1

Bond number Bo 2 · 10−3 2 · 10−3 sec. 5.1.1

51


5.1.1. Hydrodynamic properties

The input parameters needed for the hydrodynamic solver are the fluid densityρ, the zero shear viscosity µ0, the surface tension γ, the relaxation time λ1 andthe viscoelastic ratio β. One of the main challenges in processing PA12 powder isthe change in material properties due to aging and reuse of the powder [63, 69].Such effects can increase the zero shear viscosity of the fluid at 200C, measuredby rotational rheometry, from 390 Pa · s for virgin powder to 5000 Pa · s for usedpowder, as reported by Haworth et al. [66]. Haworth identifies the changes inmolecular weight and chain structure arising from polycondensation effects, dueto the powder bed temperature during the process, as a possible explanation. Areduction in shear viscosity is also observed for powders that are not pre-dried.This aging and drying effects will be neglected and only the zero shear viscosityof the virgin powder will be considered in the following studies.

The relaxation time in lattice units of the Newtonian fluid τN , used in equation(2.29), can be calculated from the given properties as:

τN = ∆t∆x2

(1− β) µ0

ρ(5.2)

To ensure numerical stability, upper and lower limits for τN are needed, due to theimplicit characteristic of the Lattice Boltzmann Method. The limits are given by[71, 72]:

0.5 > τN > 0.02 (5.3)

There are two options to keep τN within these limits. One option is to adjust thespatial and temporal resolution and the other is to either scale the viscosity or thedensity. The aim of this work is to model the process at powder scale, therefore,the spatial resolution is required to be smaller than the mean particle diameter,with d3,50 = 60 µm (sec. 5.3). Hence, it is set to ∆x = 5 µm. Furthermore, it isnecessary to resolve the interaction of the beam with the simulation domain, asdescribed in section 5.4. The maximum scanning velocity vB,max used in this workis 10 m

swith a beam diameter of dB = 400 µm. This results in a minimum beam

52


interaction time ti,min of

ti,min = dBvB,max

= 40 µs (5.4)

Therefore, a temporal resolution of ∆t = 4 µs is chosen to ensure the beam issampled within the simulation domain at least ten time steps.

Since the simulation resolution is fixed due to the length scale of the powderand the time scale of the beam interaction, the remaining alternative to keep τNwithin its numerical limits is to scale the material properties. Here, the zero shearviscosity is chosen to be scaled while keeping the density constant. The resultingvalues are given in table 5.1. By doing so, it is also important to scale the drivingforces for the fluid motion in order to make sure that the absolute and relativevelocity is calculated correctly. First, the characteristic length lc and the time scaletc needs to be estimated. lc is determined by the particle diameter, the layer heightor the beam diameter and can roughly be set to lc = 100 µm. tc is estimated bythe time scale of the powder sintering (sec. 6.5), which is in the order of magnitudeof tc = 1 s. One of the driving forces implemented in this model is the surfacetension. The capillary number Ca describes the ratio between the force due tointernal viscous and surface tension forces and is given by:

Ca = µ0vcγ≈ 1 (5.5)

where vc is the characteristic velocity and can be calculated as

vc = lctc

(5.6)

The capillary number for this system is approximately one. Therefore, it is ex-pected that viscous forces will balance the surface tension. The surface tension isscaled so that Ca is kept constant. The second driving force for fluid movementand, therefore, for powder consolidation is the gravity force. The ratio betweenthe surface tension and the gravity force is given by the Bond number Bo,

Bo = ∆ρgl2cγ≈ 2 · 10−3 (5.7)

53


where g is the acceleration due to gravity and ∆ρ is the density difference betweenthe fluid density and the surrounding density, which is the density of the atmo-sphere and is assumed to be significantly smaller than the fluid density. Therefore,the density difference can be simplified to ∆ρ = ρ. Since the Bond number is twoorders of magnitude below one, the forces due to surface tension are dominant.However, the force due to gravity will still be considered in the model. It shouldbe noted that the Reynolds number Re is not kept constant but small comparedto one for the non-scaled and scaled material properties.

Re = ρvclcµ0 1 (5.8)

So, for both cases the model is in the same flow regime.

The viscoelastic properties relaxation time λ1 and viscoelastic ratio β arecomplex to measure and could not be found in the literature for polyamide 12.Hence, a wide range for those two parameters will be considered in the followingstudies to investigate their effect on the consolidation of the powder bed. The ratioof the relaxation time and the characteristic time scale is defined by the Deborahnumber De.

De = λ1

tc(5.9)

If De is much smaller than one, the observed behavior will be similar to a New-tonian fluid with the viscosity of µN . For a Deborah number higher than one thefluid will show rubber-like characteristics with the viscosity of µ0. The viscoelasticeffect is expected to be most prominent in the transient regime where De ≈ 1 andβ close to unity.

5.1.2. Thermal properties

The thermal properties of PA12, which are needed as an input for the thermody-namic solver, are melting temperature Tm, solidification temperature Ts, thermalconductivity λ, specific heat capacity cp and the enthalpy of fusion ∆Hf . It shouldbe noted that the solidification temperature is not the crystallization temperatureTc of the polymer. The physical meaning of Ts is that below this temperature a

54


cell will be considered as solid and therefore no fluid movement will be calculated.

With the given material properties, the ratio of the advective energy transportcompared to diffusive transport can be estimated using the Péclet number Pe.

Pe = lcvcρcpλ

≈ 7 · 10−2 (5.10)

Even though the thermal conductivity is rather low for polymers, still Pe is smallerthan one. Hence, the diffusive transport is dominant and needs to be consideredby solving the convection-diffusion equation (eq. (2.45)).

The diffusive time scale tα can be calculated as

tα = l2ccpρ

2λ ≈ 4 · 10−2 s (5.11)

which is smaller than the characteristic time scale but larger than the typical timeof irradiation of a powder particle. The Péclet number and the diffusive timescalecan increase by orders of magnitude when considering powder particles and lengthscales bigger than their diameter. At that regime the effective thermal diffusivityof a powder bed is much lower than its bulk properties due the small contact areabetween the particles. So, while diffusive transport is dominant within the particle,it can become less relevant in the powder bed.

Similar to the effect of aging and recycling of powder on the viscosity, it isalso reported that the enthalpy of fusion can vary due to this effect [64, 68, 69, 70].The change can be as high as a factor of two ranging from ∆Hf,min = 5 · 104 J

kg

for aged or used powder to ∆Hf,max = 11 · 104 Jkg

for new powder. Schmid etal. [68] suggests that the high enthalpy of fusion is favored because it is beneficialfor the surface roughness, due to the reduction of thermal bleeding of neighboringparticles and is therefore designed by the powder manufacturer to increase thequality of the produced parts. On the other hand, Drummer et al. [69] argue thatthe surface roughness is not only dependent on the thermal properties and thatthe viscosity needs to be considered. It is also concluded that polycondensationcould be the main driving force for the drop in the enthalpy of fusion.

This can have a significant impact on the energy per mass ∆H needed to heatthe powder starting from ambient temperature in the building chamber TB to a

55


temperature T1 higher than the melting point Tm.

∆H =∫ T1

TBcp dT + ∆Hf ≈ cp (T1 − TB) + ∆Hf (5.12)

To estimate the relative change in energy per mass δH between the maximum∆Hmax and minimum energy per mass ∆Hmin needed, an ambient temperatureof TB = 172C and T1 = 200C is assumed, which are common values for thisprocess.

δh = ∆Hmax −∆Hmin

∆Hmin

= 0.48 (5.13)

Such a high ratio means that the energy needed to melt the same mass can increaseby nearly 50 % if purely aged powder is used. This effect can be compensated byconstantly adding a defined amount of new powder. Since this effect is expectedto have a huge impact on the process parameters needed to melt dense parts, bothlimits will be considered in the following studies, where needed.

5.2. Cell conversion

To account for the phase transition in the SLS process, the simulation domain isdivided into square cells of equal size. Each cell is assigned to one of the fourstates:

• Solid cells: Solidified cells where only thermal diffusion is calculated.

• Fluid cells: Cells fully filled with fluid and no neighboring atmosphere cells.

• Interface cells: Partly filled cells between fluid and atmosphere cells. Thefree surface boundary condition is applied in these cells.

• Atmosphere cells: Cells filled only with gas and no neighboring fluid cells.Hydrodynamics and thermodynamics is not solved for these cells. Atmo-sphere cells have a constant pressure patm, since the volume of the surround-ing atmosphere is considered much larger than the simulation domain.

While solid, liquid and atmosphere cells account for the phase state, interfacecells represent the boundary condition for the free surface between liquid and

56


atmosphere cells. Figure 5.2 illustrates as an example the fill fraction of each cell,the temperature distribution and the phase state after the interaction of the beamwith the powder and the substrate.

Figure 5.2.: Left: The gray scale indicates the fill fraction of each cell consideredin the VOF. Middle: Temperature distribution after the beam crossedthe simulation domain. Right: Phase state, the assigned colors illus-trate different cell types.

At initialization cells are either solid, with a fill fraction of ϕ > 0 or theyare atmosphere. During the interaction with the beam the material heats up andmelts as its temperature rises over the melting temperature Tm. At that point solidcells are converted into either fluid or interface cells depending on their occupationlevel and their neighboring cells. As soon as those cells cool down and drop belowthe solidification temperature Ts the cells are converted back to solid cells. Theoccupation level of interface cells can change due to movement of the free surface.The driving forces for this movement, which are considered in this work, are thepressure gradient, viscoelastic stress and gravity as described in chapter 2.

Additionally, the surface tension is taken into account by using the numericalscheme described by E. Attar et al. [73], which also describes how the change infill fraction is tracked between the cells. If the occupation level of an interface cellexceeds ϕ > 1 the cell is converted into a fluid cell and all neighboring atmospherecells are initialized as interface cells, with the excess fill fraction divided equallyamong them. On the other hand, if the interface cell is empty ϕ < 0, it is initial-ized as an atmosphere cell and all the neighboring fluid cells are converted into

57


interface cells by dividing the missing fill fraction among them. When convertinga cell the quantities temperature, momentum and viscoelastic stress are kept fromthe primary cell, while for the initialization of new cells these quantities are esti-mated as an average of the non atmosphere neighboring cells. Figure 5.3 shows asimplified flow chart for cell conversion.

FluidSolid Interface

T > TsT > Tm T > Ts

ϕ < 1 ϕ > 1NeighboringAtmosphereto Interface

ϕ < 0NeighboringFluid toInterface

Atmosphere

noyes

yes

no

yes

yes

no

yes

yesno

nono

Figure 5.3.: Cell conversion flow chart.

In all cells, except atmosphere, thermal diffusion is taken into account and noheat transfer coefficient is considered between the different cell types. Atmospherecells are assumed to be perfectly isolating, therefore the boundary condition atthe interface is a heat flux of zero in the normal direction of the surface [61].Additionally, hydrodynamics, including viscoelasticity, is calculated in fluid andinterface cells, with temperatures over the melting temperature.

58


5.3. Powder bed generation

For the powder bed generation the particles are represented by spheres, morespecifically circles in two dimensions. Each particle is defined by its diameter d andthe position of its center. To generate a realistic powder bed, the particle diameterdistribution density q, which is approximated by a log-normal distribution,

q(d)

= 1√2πσd

exp(−(ln(d)− µd)2

2σ2

)(5.14)

is fitted to experimental data, with σ and µd being the fit parameters. The dimen-sionless particle diameter d is normalized to the µm scale. The correlation withthe median particle size is given by:

d3,50 = exp (µd) (5.15)

One common distribution for PA12 powders used in additive manufacturing isgiven by Drummer et al. [74], with a median particle size of d3,50 = 60 µm.Figure 5.4 shows equation (5.14) fitted to the volume density distribution and thecorresponding number density distribution. The fit parameters are given in table5.2.

Figure 5.4.: Volume particle diameter distribution density q (eq. (5.14)) fitted toexperimental data [74] and the corresponding number density distri-bution.

59


Initializen = 0

d = rand (dmin, dmax)

q(d)<

rand (0, 1)

xp = rand (xmin, xmax)yp = ymax

xp, yp =minimizePot(xp, yp, d)

yp + d > hldelete(xp, yp, d)

n+ = 1n > nmax

ρrel < ρ∗rel

Terminate

xp, yp, d =randParticle()

delete(xp, yp, d)

yes

no

no

no

no

yesyes

Figure 5.5.: Powder bed generation flow chart.

60


Table 5.2.: Fit parameters and sampling range for the particle diameter distribu-tion density q.

σ µd dmin dmax d3,50

0.25 4.1 10 µm 150 µm 60 µm

Figure 5.5 shows the flow chart of the powder bed generation. The powderbed is generated by randomly sampling a diameter between dmin = 10 µm anddmax = 150 µm. The diameter is accepted with the probability q

(d), which

is the distribution q normalized to its integral between zero and infinity. Thex-coordinate of the particle center is randomly initialized within the simulationdomain limits xmin and xmax and placed at the top. The potential energy ofthe particle is minimized until a local equilibrium is found. If the new positionis below the adjusted layer height hl the particle is placed into the simulationdomain, otherwise it is removed. The algorithm is repeated till the maximumnumber of tries is reached nmax. Finally, particles are randomly removed from thegenerated powder bed to adjust the relative density ρrel, which is calculated as theoccupation level normalized to the considered area. Figure 5.6 illustrates threegenerated powder beds with different relative densities.

Figure 5.6.: Three generated powder beds with different relative densities.

61


5.4. Beam properties and scanning strategy

The laser beam is modeled as moving source in the x-y-plane with a two dimen-sional Gaussian power distribution given by:

P (x, y) = PB1

2πσ2B

e−(

(x−xB)2+(y−yB)2

2σ2B

)(5.16)

where PB is the laser total power and σB is one fourth of the beam diameterdB = 400 µm. The position of the center is given by xB and yB. At each timestep the position of the beam needs to be updated. In general, it is possibleto model any arbitrary path, but in this work only a sequence of scan lines willbe considered, since this is the common scanning strategy in modern productionmachines. The center position of the beam can be defined by a starting and endpoint and a given scanning velocity vB along its path.

Figure 5.7 illustrates the so called hatch scanning strategy. The beam followseach single scan vector with an alternating direction. This scan path can beparametrized by the length of the vector l, the width w and the hatching spacinghs. The number of hatch lines nh is given by:

nh = w

hs(5.17)

where nh is an integer. The energy is absorbed according to the model describedin chapter 4.

To calculate the energy fraction EI for the initialization of each ray, the powerdistribution needs to be integrated over the surface covered by this ray at positionxr for each time step.

EI =∫ xr+0.5∆x

xr−0.5∆x

∫ 0.5∆t·vB

−0.5∆t·vBP (x, y) dy dx (5.18)

With the spatial resolution of ∆x = 5 µm and a time discretization of ∆t = 4µs,

62


Figure 5.7.: Hatch scanning strategy: The beam follows each single scan vectorwith an alternating direction. The simulation domain is placed in thex-z-plane in the middle of the scan vector at y = 0. The length ofthe vector l, the width w of the scan area and the hatching spacinghs are indicated. The color map illustrates the energy distribution forscanning the first three and a half vectors.

the length scale in x- and y-direction is smaller than the beam diameter,

∆x dB (5.19)∆t · vB dB (5.20)

therefore, the calculation of EI can be approximated by

EI ≈ P (xr, 0) ∆x ∆t · vB (5.21)

Figure 5.8 shows the temperature field of the first layer at the first, fourth andeighth passing of the beam, with the parameters given in table 5.3. The intensitydistribution of the beam within the simulation plane is indicated by the black line.The resulting temperature field for such an interaction will be discussed in moredetail in the following sections.

63


Table 5.3.: Beam parameter used in figure 5.8.

PB vB hs l w

15 W 3ms

250 µm 2mm 2mm

Figure 5.8.: Temperature field of the first layer at the first, fourth and eighthpassing of the beam. The intensity distribution of the beam withinthe simulation plane is indicated by the black line.

64

6. Powder-Laser-Interaction

Before modeling the full process of additive manufacturing it is crucial to under-stand the physics of the interaction of one or more powder particles with the laserbeam. The interaction with the beam and the consolidation of the powder bed aremost relevant for the resulting properties of the bulk material. Therefore, startingfrom a single particle, the absorption characteristics of the powder bed is studiedin this chapter. Some of the results have already been published by the author[57]. Furthermore, the sintering of two particles and the influence of viscoelasticityis investigated.

6.1. Laser absorption in single particle

The absorption in a single particle is modeled by initializing only one particlewith radius R1 = 25 µm and expose it with a laser beam with a uniform powerdistribution. Hence, it is possible to investigate the effects due to geometry andrefraction index without the need to consider a Gaussian distribution. Figure6.1 shows the distribution of the energy absorbed by the particle in linear andlogarithmic scale for three refractive indices nmed with an penetration depth ofλ0 = 100 µm, which is defined as the inverse of the attenuation coefficient µα

λα = 1µα

(6.1)

and a refractive index of natm = 1 of the surrounding atmosphere. For nmed = 1the rays are not refracted and simply pass through the particle on a straight line.The linear and logarithmic scale appear rather similar due to the curvature of theparticle. The energy distribution for nmed = 1.7 is significantly different, sincethe rays are refracted towards a focal point, which is outside of the particle. In

65

CHAPTER 6. POWDER-LASER-INTERACTION

logarithmic scale the path of backscattered rays can be observed, which increasesthe total energy absorbed by the particle. The focal point can even move into thepowder particle if the refractive index is high enough, as illustrated in figure 6.1with nmed = 3.4.

Figure 6.1.: The distribution of the energy absorbed by a single particle in linear(top) and logarithmic (bottom) scale for three refractive indices nmedwith λ0 = 100 µm and natm = 1. The color map is normalized tothe maximum value for each graph. From left to right: nmed = 1 nofocal point; nmed = 1.7 focal point outside of particle; nmed = 3.4 focalpoint within the particle.

Clearly, the powder particles can be considered as lenses and, therefore, theirfocal length fl can be calculated using the general lens maker equation for thicklenses:

1fl

= (nmed − 1)(

1R1− 1R2

+ (nmed − 1)dnmedR1R2

)(6.2)

where d is the distance of the two surfaces along the lens axis and R1 and R2

are the radii of the first and the second surface, respectively. The first surface is

66


defined by the first interface intersected by the incoming ray. The sign conventionfor radii is so that the radii of a thick convex lens have the opposite sign. Usingthe relations for a sphere

R2 = −R1 (6.3)d = 2 |R1| (6.4)

and shifting the origin of the coordinates from the mid point of the lens to thefirst surface, equation (6.2) can be simplified to:

fl =

(

nmed2(nmed−1) + 1

)R1, nmed < 2

nmed(nmed−1)R1, nmed ≥ 2

(6.5)

Here two cases need to be considered. If the refractive index of the medium islower than nmed < 2, the focal point is outside and for higher values it is inside ofthe particle. For the latter case only the refraction at the first interface needs tobe accounted for. The absorbed energy along the mid axis for the three discussedrefractive indices is shown in figure 6.2. For nmed = 3.4 a maximum is observed

Figure 6.2.: Absorbed energy along the mid axis of a particle with radius R1 =25 µm for three refractive indices.

within the particle and its position agrees well with its focal length of 35 µm

calculated by equation (6.5). The position of the focal point for nmed = 1.7 is

67


calculated just below the second surface.

fl (nmed = 1.7) = 55µm (6.6)fl (nmed = 3.4) = 35µm (6.7)

The peak values of the absorbed energy distribution seem to be rather high com-pared to the base line given by the straight passing rays for nmed = 1. It is notexpected to observe such pronounced peaks in real powder, since the particlescommonly used in additive manufacturing are not perfect spheres and do have asurface roughness, impurities and inner structures, which act as additional scatterpoints. Still, the shape is round and the overall absorbed energy by a single par-ticle should be captured correctly. Figure 6.3 shows the calculated total absorbedenergy En of a single particle for a range of refractive indices normalized to theenergy absorbed E1 with nmed = 1.

Figure 6.3.: Total absorbed energy En of a single particle for a range of refractiveindices normalized to the energy absorbed E1 with nmed = 1, λ0 =100 µm and R1 = 25 µm.

For small nmed the absorbed energy increases until it reaches a maximum atapproximately nmed = 1.7. En is in general a complex function of the surface shapeand the penetration depth, which are held constant, and the refractive index. Byincreasing the refractive index also the length of the path of a ray is increasedand thereby the energy absorbed along it. The upper limit for the length is given

68


by the particle diameter. On the other hand also the proportion of the reflectedenergy at the first and the second surface increase with the refractive index. Theabsolute value of the reflected energy at the first surface is larger than its valueat the second surface and thereby exceeds the effects of energy gain for higherrefractive indices.

6.2. Laser absorption in powder bed

Analogous to the previous section a uniform radiation source is chosen to studythe absorption characteristics in a powder bed as a function of relative densityρrel and the refractive index of the medium. The powder bed is generated asdescribed in section 5.3 and the penetration depth is set to λ0 = 100 µm, which isapproximately double the size of the mean particle diameter and also in the rangeof the penetration depth expected for polymers.

Figure 6.4.: Trace of ray cast from the top of the simulation domain propagatingtowards a powder bed. From left to right the number of rays is in-creased. Left ray coller map: blue: incoming; red: first refraction;gray: first reflection; black: secondary and tertiary refraction and re-flection [57]

Figure 6.4 (left) illustrates the trace of one single ray cast from the top ofthe simulation domain propagating towards a powder particle. At the intersectionof the interface and the incoming ray two new rays are cast. The reflected ray

69


points away from the powder bed and propagates until it reaches the boundaryof the simulation domain. The refracted ray transverses the powder particle anddeposits its energy according to eq. (4.12). At the intersection, new rays are castaccounting for reflection and refraction. The traces of two different spacings forthe casting distance ∆r are shown in Figure 6.4 (middle and right). The refractiveindex of the atmosphere is set to natm = 1, while the refractive index of themedium is nmed > 1. Therefore, the ray is refracted towards the normal vector ofthe surface. As shown in section 6.1, the path of rays is refracted so that roundpowder particles will focus the beam and create hot spots in the powder particles.

The intensity distributions in three powder beds with different relative den-sities irradiated by a light source with a uniform power distribution are shown onthe left hand side in figure 6.5. On the right the mean intensity Im is plottedover the powder bed depth. Im is the absorbed energy averaged over the width,accounting only for cells with a fill fraction of ϕ > 0.

In the plane surface setup, discussed in section 4.3, the maximum of the ab-sorbed radiation is found at the interface. Here, the intensity peaks after approx-imately one particle diameter within the powder bed, which is the length neededto raise from zero to the given relative density of the powder bed. In the presentedexamples this length is between 50 µm and 100 µm, which correlates well with amedian particle size of d3,50 = 60 µm. To characterize the absorption in a powderbed, the effective penetration depth λeff is calculated by fitting Beer-Lambert law(eq. (4.1)) onto the tail of Im after the peak. The correlation between λeff andρrel for three different refractive indices is shown in figure 6.6.

Only the data points for nmed = 1 are in good agreement with the simpleestimation of the effective absorption length to be indirectly proportional to therelative density.

λeff = 1ρrel

λ0 (6.8)

This correlation is simple since the rays are not refracted and, therefore, propagatealong a straight line and the probability for a ray to be in a particle is given by ρrel.If nmed > 1 the rays are refracted and the length of the path to reach the samedepth is increased. Furthermore, a higher proportion of the energy of the ray is

70


Figure 6.5.: Left: Three powder beds with their relative absorbed intensity distri-butions. Right: The mean intensity Im over the powder bed depth forthe corresponding relative densities with nmed = 1.7 and λ0 = 100 µm.The blue line indicates the median particle diameter d3,50 = 60 µm.[57]

absorbed in a single particle due to reflection, as discussed in the previous section.Figure 6.6 shows that these effects can cause an effective penetration depth belowthe penetration depth in bulk material.

The consequence is that the energy distribution within a powder bed can notbe simply estimated by equation (6.8), which must be taken into account whenmodeling the powder bed as a continuum. This is especially relevant for powders,e.g. PA12 as shown in the next section, where the penetration depth is largerthan the mean particle size λ0 > d3,50. A significant fraction of the total energy istransmitted beyond the first particle layer. Therefore, more interfaces are crossed.At each interface the ray is reflected and refracted, which increases the path of the

71


Figure 6.6.: The correlation between λeff and ρrel for three different refractiveindices. The dot-dashed line indicates penetration depth λ0 = 100 µmin bulk material. [57]

ray to reach the same depth and thereby decreases the effective absorption length.This results in a higher energy density at the powder bed surface, which can leadto higher peak temperatures during sintering. High peak temperatures can narrowthe processing window due to degradation of the polymer.

6.3. Attenuation coefficient of PA12

The attenuation coefficient of bulk PA 12 could not be found in literature andis determined by measuring the transmission of laser irradiation in thin filmswith different thickness. The sample preparation and the measurements wereconducted by K. Wudy within the framework of the Collaborative Research Cen-ter 814 (CRC814). The specimen slices were cut by a microtome from PA12 bulkmaterial, produced from the same powder (PA2200, EOS GmbH) used for additivemanufacturing by injection molding. The resulting film thicknesses were measuredwith a measuring gauge and are in the range of 3 µm to 81 µm. Figure 6.7 showsthe position of the 25 measuring points over the cross section of one specimenvia microscopic mapping. The range of the considered infrared spectra is between400 cm−1 and 4000 cm−1.

The transmission spectrum was recorded by the infrared spectrometer Nicolet6700 (Thermo Scientific, Germany). For each specimen at each of the 25 points, the

72


Figure 6.7.: Sampling area for infrared spectroscopy mapping and positions forinfrared spectra [57].

transmission at a wavelength of 10.6 µm or wavenumber 943 cm−1 was analyzed,the standard wavelength of lasers in commercially available machines for selectivelaser sintering of polymers. A total of 33 thin film specimens were measured. Theaverage value of the transmitted intensity of 25 measurement positions for one filmare shown in figure 6.8 together with a fit of the Beer-Lambert law (eq. (4.1)).The attenuation coefficient, as determined by the fit, is µα = 0.013 1

µm, which is

equivalent to a penetration depth of λα = 77µm.

Figure 6.8 shows the comparison of the relative transmission and reflectionin a PA12 powder bed between experimental data [75] and the simulation, usingthe presented absorption model. The measurement was conducted using two inte-gration spheres and a CO2 laser with a wavelength of 10.6 µm. With this setupit is possible to measure the reflectance, the transmittance and the absorptanceof the powder bed between the integration spheres. The simulation is initializedanalogous to section 6.2. The refractive index of the medium is nmed = 1.7 andwas chosen so to reproduce the measured reflectance of the powder. The atten-uation coefficient is set to µα = 0.013 1

µm, as determined by the experiment in

the thin film. The portion of the reflection is calculated by summing up the en-ergy of each ray reaching the simulation domain boundary above the powder layersurface. Each ray below the given layer thickness is terminated and its energyis added to the portion of the transmitted energy. A good agreement was foundwith the experimental findings, while a slight overestimation of the transmission

73


Figure 6.8.: Left: Relative transmission in a thin foil of PA12 with different filmthicknesses together with a fit of the Beer-Lambert law (eq. (4.1)).Right: Comparison of the relative transmission and refraction in aPA12 powder bed between experimental data [75] and the simulation[57].

in the simulation results can be observed. This may be caused due to the neglectof diffusive scattering and the assumption of perfect spherical particles with noimpurities or surface roughness.

6.4. Laser absorption of a Gaussian powerdistribution

State of the art industrial plants for selective laser sintering of polymers use aCO2 laser with a Gaussian power distribution, defined by equation (5.16). Thebeam diameter is set to dB = 400 µm which is defined as dB = 4σB. The opticalproperties of PA12 are µα = 0.013 1

µmand nmed = 1.7, as discussed in section

6.3. Figure 6.9 shows the distribution of the absorbed energy in a powder bed

74


irradiated by a laser with a Gaussian power distribution in linear and logarithmicscale.

Figure 6.9.: Distribution of the absorbed energy in a powder bed irradiated by alaser with a Gaussian power distribution in linear and logarithmic scalenormalized to their maximum value with parameters µα = 0.013 1

µm

and nmed = 1.7. The half beam width is indicated by the dashed lines.

The powder particles refract the rays towards a focal point right outside ofthe particle, which is in good agreement with the findings of the previous sections.This point is most likely to be in the surrounding atmosphere, but if two particlesare on top of each other it can also be within the material. A second more pro-nounced effect is visible in the powder bed, the distortion of the Gaussian intensitydistribution due to refraction in the first layer of the particles. While the beampower distribution is clearly visible for the irradiation of a plane surface (fig. 4.4),it is not in the powder bed.

The powder bed morphology has also an impact on the total absorbed energyas a function of the incident angle. A comparison between the absorbed energynormalized to the total energy E0 in a powder bed and a plane surface for differentincident angles γ is shown in figure 6.10. While the energy absorbed in the caseof the plane surface is given by equation (4.7), it is constant within some small

75


scattering regardless of the incident angle for a powder bed. An exception is ofcourse the irradiation of the powder bed at γ = 0.5π, since in this case the beampropagates parallel to the powder surface.

Figure 6.10.: Absorbed energy normalized to the total energy E0 in a powder bedand a plane surface for different incident angles γ with parametersdB = 400 µm, µα = 0.013 1

µmand nmed = 1.7. The analytic curve is

given by equation (4.7).

The reflection and refraction of the beam due to the powder bed morphologycreates an intensity distribution along the powder bed depth that can not besimply estimated from the power distribution of the light source. Together withthe complex correlation of the powder bed density, refraction index and effectiveabsorption depth, it shows the importance of modeling the absorption with aray tracer model. Additionally, the total absorbed energy is independent of theincident angle.

6.5. Sintering

The fusion of fully or partially molten particles is considered as sintering in thiswork, in contrast to solid phase sintering which is a diffusion process. The drivingforce for the consolidation of the powder bed is the surface tension, as discussedin section 5.1.1. The surface energy is minimized by inducing a flow betweenthe particles and thereby reducing the surface to volume ratio. The minimum

76


is reached for a single sphere or a circle in two dimensions, respectively. Themost simple setup to study the fusion characteristic of viscoelastic particles is thesintering of two equally sized particles as illustrated in figure 6.11. The two circles,with the initial radius r0, sinter to a single particle with the final radius R0 =

√2 r0

for an incompressible fluid. The particles are shown at an intermediate sinteringstate with the radii r and the sinter neck half width ws.

Figure 6.11.: Illustration of the sintering of two equally sized viscoelastic particles,shown at an intermediate sintering state with the radii r and thesinter neck half width ws.

An analytical model for the coalescence of two equal spherical droplets wasfirst introduced by Frenkel [15] and was later further modified by Eshelby [76].The model is based on biaxial extension flow of a Newtonian fluid and is derivedby balancing the force of the surface tension by viscous dissipation and is onlyvalid for initial sintering phase [77].

wsr

=√γ t

µ r0(6.9)

A more sophisticated approach was presented by R.W. Hopper [78] describ-ing the coalescence of the cross section of two infinite cylinders driven by capillaryforces. He assumed that the closed boundary of the two cylinders can be rep-resented by an inverse ellipse. An inverse ellipse, also called hippopede, is thetrace of an ellipse inverted with respect to a circle in its center. The analytical

77


expression for the surface is given by

x (θ) = R0(1− j2)√

1 + j2 (1 + 2j cos 2θ + j2)(1 + j) cos θ (6.10)

y (θ) = R0(1− j2)√

1 + j2 (1 + 2j cos 2θ + j2)(1− j) sin θ (6.11)

where 0 ≤ θ < 2π and with j being the parameter changing the shape of the closedsurface, which is in the range of 0 ≤ j ≤ 1.

By solving the Navier-Stokes equation with the given expression for the bound-ary and assuming planar flow, a relation between the time of sintering t and theparameter j can be derived [79].

γt

µR0= π

4

∫ 1

j2

1k√

1 + k K (k)dk (6.12)

with K (k) being the complete elliptic integral of the first kind given by

K (k) ≡∫ 0.5π

0

1√1− k sin2 θ

dθ (6.13)

Equation (6.12) can be solved numerically to calculate the shape parameter j fora given time t. By inserting j into the inverse ellipse, the shape of the surface canbe calculated and the half width ws of the sinter neck extracted.

Pokluda et al. [80] developed a more general form of the Frenkel-Eshelbymodel by introducing an expression for the temporal evolution of the radius r,which made it possible for Bellehumeur et al. [81] to incorporate viscoelasticityusing the steady-state upper-convected Maxwell constitutive model. The obtainednon-linear differential equation is able to predict the slow-down of the sinteringrate due to viscoelastic stresses. However, due to the assumption of steady-stateviscoelastic stress, thereby neglecting the build up phase of the viscoelastic stress,the model needs an unrealistically large relaxation time to fit experimental data[77].

Several numerical studies were conducted to model sintering. J.W. Ross etal. [82] calculated the evolution of the sintering neck in an infinite line of cylin-ders using the finite element method. Others [83, 84] used the boundary element

78


method to account for curved geometries. Furthermore, multiple investigations[85, 86, 87, 88] in two and three dimensions were carried out assuming the prob-lem to be axisymmetric and using the finite element method. More recent workby C. Balemans et al. [89] modeled the fusion of two particles in two dimensionsdescribing the viscoelastic behavior by the Giesekus and XPP constitutive models.In the limits of Newtonian flow they found good agreement between their modeland the solution described by Hopper [79].

Figure 6.12.: Evolution of the dimensionless sinter neck radius over time comparedto the analytic solution of Hopper [79] for different β. The discontinu-ities in the traces of the numerical results are due to cell conversions.

Figure 6.12 shows the evolution of the dimensionless sinter neck radius wsrover

time for two viscoelastic ratios β = 0.5 and β = 0.9 and three Deborah numbersDe = [0.01, 1.0, 100.0]. The characteristic time scale of the sintering process canbe estimated with

tc = µR0

γ≈ 1 s (6.14)

Therefore, the three Deborah numbers correspond to relaxation times of λ1 =[0.01 s, 1.0 s, 100.0 s]. The results where obtained by the Lattice Boltzmann Methodfor viscoelastic fluids described by the Oldroyd-B model presented in this work.Additionally, the analytic solution of equation (6.12) is plotted accounting for the

79


dynamic viscosity of the polymer and the Newtonian fluid µmax = µP + µN andonly for the Newtonian viscosity µmin = µN . For De = 0.01 the numerical resultsare in good agreement with Hoppers solution using µmax for both viscoelastic ra-tios. The relaxation time is much shorter than the time scale of the process andthereby the viscous forces of the polymer contribution are fully established fromthe beginning.

For De = 100 it is the other way around. The process time is much shorterthan the relaxation time and, therefore, the polymer viscosity can not contributeto the fluid viscosity to balance the surface force. These two curves set the limitsbetween fully rubber like and Newtonian characteristic of the fluid. A transientbehavior is observed for De = 1. Here, the initial sintering rate is high and thesinter neck grows comparable to De = 0.01, but as soon as the viscoelastic stressbuilds up the sintering rate is slowed down and transits to the rubber like behavior.To model this transition it is not sufficient to assume an effective viscosity or to usea viscosity purely dependent on the shear rate, without considering the relaxationtime. On the other hand, if the relaxation time of the used material is not inthe range of the process time it is possible to model the flow characteristic withthe viscosity corresponding to the viscoelastic regime. In general, polymers usedin industrial applications consist of complex molecules with multiple relaxationtimes.

Figure 6.13.: Velocity field of two fluid equally sized particles fusing with β = 0.9and De = 1 at t = 0.04 s.

80


The velocity field of two equally sized fluid particles fusing with β = 0.9 andDe = 1 is given in figure 6.13 at t = 0.04 s. Similar to the four-roll mill setupdescribed in section 3.2.2, a stagnation point in the center of the flow field iscreated with incoming flow perpendicular to the sinter neck orientation creatingan elongation flow along the neck axis.

In figure 6.14 the relative distribution of the components of the viscoelasticstress tensor TP for β = 0.9 and De = 1 at different times during the fusionprocess are shown. The maximum values for the diagonal components are alongthe elongation. Those components reach their maximum values after an initialbuild up phase and thereby slow down the neck expansion due to surface forces.The shown distributions are in good agreement with the results presented by C.Balemans et al. [89].

81


Figure 6.14.: Relative distribution of the components of the viscoelastic stress ten-sor TP for β = 0.9 and De = 1.

82

7. Modeling additive manufacturingof PA12

In this chapter all the previously presented methods and material properties willbe combined to model the process of selective laser sintering of polyamide 12 in apowder bed. The method used here is a thermal viscoelastic Lattice BoltzmannMethod for free surface flows combined with a ray trace model for the absorptionof a laser in a powder bed. First, the temperature field in a bulk of a few powderparticles induced by the exposure of a stationary beam will be investigated, beforeexamining the influence of a moving beam source in a single line and superposethese lines to melt single layers. At last, the impact of beam parameters andscanning strategy on the consolidation of several layers will be discussed.

7.1. Exposure with a stationary beam

It is rather difficult to measure accurate temperature profiles in SLS, especiallyunder the beam spot. The time scale of the interaction between the laser andthe powder is within milliseconds and the beam diameter is in the order of afew hundred micrometers. A very interesting experimental setup to acquire thetemperature during melting is presented by L. Lanzl [90]. A high speed DSC chipwas paired with a freely configurable CO2-laser, with similar properties as the laserused in SLS machines. By placing a few particles onto the chip and heating themfor a defined exposure time texp, it was possible to get an accurate temperatureprofile right underneath the particles.

To generate a quantitative comparison, an analogous setup is modeled. Theinitial temperature is set to TB = 170C, the beam power is PB = 1.7 W and thebeam is held stationary with one σB offset to the simulation domain to account for

83

CHAPTER 7. MODELING ADDITIVE MANUFACTURING OF PA12

averaging effects. Figure 7.1 shows as an example of the temperature and phasefield for an exposure time of texp = 0.006 s for three different time steps. In thebeginning the temperature is low and the distribution follows the absorbed energydistribution of the laser beam as discussed in section 6.4. As the time passesthe temperature field gets more homogenized due to thermal diffusion. Here, thesubstrate, in contrast to the experiment, is made of the same material as thepowder.

Table 7.1.: Parameters used for the calculations shown in figure 7.1.

PB vB yB TB De ∆Hf

1.7 W 0 ms

100 µm 170 C 1 11 · 104 Jkg

Figure 7.1.: Temperature and phase field for exposure by a stationary beam forthree time steps with the parameters given in table 7.1

To compare the numerical results with the experimental findings the peaktemperature reached at the top of the substrate is compared with the data providedby L. Lanzl [90]. Figure 7.2 shows the results for different exposure times and thetwo limits for the enthalpy of fusion ∆Hf,min = 5 · 104 J

kg, which corresponds to

84


Figure 7.2.: Comparison of numerical results with the experimental findings byL. Lanzl [90] of the maximum reached temperature at the top of thesubstrate.

used powder and ∆Hf,max = 11 · 104 Jkg, which corresponds to virgin powder. The

calculated maximum temperatures, using ∆Hf,max indicated by the blue line, arewithin the scattering of the experimental data and follow a linear trend. It canbe assumed that virgin powder was used for the experiment. As expected thetemperatures reached for ∆Hf,min are higher, but follow the same trend.

Even though phase change and hydrodynamics were taken into account, nosignificant movement of the surface or powder fusion is observed within theseexposure times. This is in good agreement with the results of section 6.5, sincethe time scale of sintering is much larger. The melt pool size in figure 7.1 is ratherlarge due to the relatively long exposure time and thereby the high amount ofenergy absorbed.

7.2. Single line melting

Melting a single line is the most basic setup to investigate the consolidation char-acteristic of the powder by the irradiation of a moving heat source. The beam is

85


initialized at the starting position of y0 = 1000 µm and moves along a straightline perpendicular towards the simulation plane. As the beam crosses the simula-tion plane the energy is absorbed according to equation (5.16) and the absorptionmodel. The powder bed is generated as described in section 5.3 with an initialtemperature of TB = 172 C. If the absorbed energy is sufficient, the powdermelts and the simulation is continued until everything is solidified again. Duringthe fluid state the powder particles can fuse due to surface and gravity forces.

7.2.1. Temperature field


PB vB TB De ∆Hf

5 W 1 ms

172 C 1 5 · 104 Jkg

Figure 7.3.: Temperature and phase field with the parameters given in table 7.2for three different times.

Figure 7.3 shows the temperature and phase field with the parameters givenin table 7.2 for three different times. At the beginning of the interaction between

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Figure 7.4.: Time traces of the maximum Tmax and mean temperature Tmean withinthe melt pool and melt pool area Amelt for the parameters given intable 7.2 and vB = 0.5 m

s.

the beam and the powder bed at t = 0.002 s hot spots can be observed, dueto lens effects. The temperature field is the direct result of the absorbed energydistribution and the powder bed is divided into several small funnel shaped meltpools. At t = 0.02 s the temperature field is more homogenized towards a radialdistribution with its maximum at the crossing point of the center of the beam andthe simulation domain. The melt pool is already established over several powderparticles. However, the sintering process did not progress far and the particles arestill separated. After t = 0.6 s a fully homogenized temperature distribution isreached within the melt pool and the powder particles are consolidated, while thesolidification process did not start yet.

Detailed time traces of the maximum Tmax and mean temperature Tmean

within the melt pool for the parameter given in table 7.2 and a second param-eter set, changing only the scanning velocity to vB = 0.5 m

s, are shown in figure

7.4. Additionally, the evolution of the melt pool area Amelt is illustrated. The

87


peak temperature is reached during the irradiation of the laser. For vB = 1.0 ms

the peak is observed sooner since the scanning velocity is higher and, therefore,the beam crosses the simulation domain earlier. Furthermore, the absolute valueof the peak is significantly smaller because only half of the energy is absorbedcompared to vB = 0.5 m

s. After the irradiation the maximum temperature decays

due to thermal diffusion and convection within the melt pool until it overlaps withthe mean temperature and a fully homogenized distribution is reached.

The evolution of the melt pool size can be separated into four phases. Inthe first phase the melt pool extends rapidly during the interaction of the beamwith the powder bed. Here, new energy is absorbed and the energy distribution isdominated by radiation transport. During the second phase the melt pool growsmuch slower due to transport of the absorbed energy by diffusion and convection.

The more the powder particles sinter the faster the temperature equalizationuntil a fully homogenized energy distribution within the melt pool is reached inphase three. As can be seen in figure 7.3 for t = 0.6 s the melt pool has onlylimited contact with the surrounding powder bed due to wetting effects. In thatphase a constant size is observed, since the small contact areas reduce the heatflow and thereby isolate the melt pool. Eventually, in the last phase, the energycan dissipate into the powder bed and the melt solidifies.

7.2.2. Melt pool shape

Final melt pool shapes, indicated by the red line, of a parameter survey by varyingbeam power and scanning velocity are shown in figure 7.5. Table 7.3 summarizesthe process parameters used. Increasing the power with constant scanning velocityincreases the total absorbed energy and thereby the size of the melt pool. It isthe other way around for increasing speed while keeping the power constant. It isintuitive that the melt pool size scales with the energy input. For the parametersvB = 3.0 m

sand PB = 2.0 W the absorbed energy is so low that only a few cells can

overcome the melting temperature. Those cells are located at hot spots created bythe refraction of the incoming laser light by spherical particles. All particles keeptheir initial shape and no fusion is observed. Increasing the energy input, the firstsintering of a few particles can be seen with the melt pool being divided into several

88



PB vB TB De ∆Hf

2.0− 8.0 W 0.5− 3.0 ms

172 C 1 5 · 104 Jkg

Figure 7.5.: Melt pool shapes, indicated by the red line, for single line melting withthe process parameters summarized in table 7.3.

segments. Finally, for higher energies a single melt pool with full consolidation isestablished. With the increasing size of the melt pool also the surface becomesmore smooth. Larger size means longer life time in which the melt pool is liquidand, therefore, the surface and gravity forces can smooth out humps.

Figure 7.6 shows the final melt pool geometries with process parameters given

89



PB vB TB De ∆Hf

6 W 0.5ms

172 C 0.01− 100.0 5 · 104 − 11 · 104 Jkg

Figure 7.6.: Final melt pool geometries with process parameters shown in table7.4.

in table 7.4 for the upper and lower limit of the enthalpy of fusion and threeDeborah numbers.

The size of the melt pool clearly depends on ∆Hf , as discussed in section 5.1.2.A considerable portion of the energy is converted into enthalpy in this temperaturerange. Furthermore, the melt pool shape is dependent on the viscoelastic regime.For De = 0.01 most deformations are frozen into the solidified track. This isconsistent with the results of section 6.5, which showes the slowest sintering ratein this case. On the other hand, the surface is very smooth for De = 100.0 and themelt pool is even divided by wetting forces. The internal forces of the fluid couldnot stabilize the melt pool because of the low Newtonian viscosity. Additionally,the viscoelastic stress did not build up yet due to the long relaxation time. Theshape for De = 1.0 is at intermediate state between the other two.

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7.2.3. Melt pool depth

One of the most important features of the melt pool is its depth, since it cangive an indication if there will be a connection to the underlying layer or notwhen compared to the thickness of the applied powder. It is also a quantityeasy to measure and to compare with experimental results. Riedlbauer et al.[64] published measured melt pool depths for single tracks in a powder bed andcompared them with the thermal field calculated by a finite element method. Theirresults, even though not considering single powder particles but bulk material, werein good agreement with the experimental findings. Still, the simulated depthsoverestimated the measured ones in some cases by a factor of two.

Figure 7.7 shows the experimental results of Riedlbauer et al. [64] togetherwith simulated melt pool depths dm calculated with the presented scheme in thiswork over the energy per unit length El defined as

El = PBvB

(7.1)

The numerical process parameters used are shown in table 7.5, which also includesthe experimental process parameters. Additionally, for each parameter set eightdifferent bulk powder were generated to increase the statistical relevance, since themelt pool depth is in the same order of magnitude as the powder diameter. Thescattering of the numerical and experimental results is significant compared to theabsolute value and shows the statistical nature of this process. Still, almost allexperimental results lie within the range of the simulation results using ∆Hf,min.

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PB vB TB De ∆Hf

1.0− 6.0 W 0.5− 4.0 ms

172 C 1 5 · 104 − 11 · 104 Jkg

Figure 7.7.: Experimental results of Riedlbauer et al. [64] together with simu-lated melt pool depths dm with parameters shown in table 7.5. Foreach parameter set eight different bulk powder are generated. Eachdot represents one simulation. The dots for ∆Hf,max are omitted forclarity.

7.3. Single layer melting

Before simulating the melting of several layers it is important to identify the keyfeatures and parameters in a single layer. Here, the hatching scanning strategyis used, which is described in section 5.4. It consists of several superposed singlescanning vectors with an alternating direction. The parameters length and widthare set to l = w = 6 mm and the hatching spacing is hs = 250 µm, whichis a typical value for selective laser sintering of polymers. All simulations are

92


initialized with the temperature TB = 172C. The width of the simulation domainis set to 8 mm with periodic boundary conditions at the left and right boundary.The distance from the top of the powder bed to the substrate is 0.7 mm and thesubstrate thickness is 0.75 mm with fixed temperature TB at the bottom boundary.

7.3.1. Temperature field

The evolution of the temperature field within a layer is more complex than for asingle line due the non linear interaction of several neighboring scanning vectors.Figure 7.8 shows, as an example, the temperature distribution for the parameterssummarized in table 7.6.

At t = 0.004 s the temperature field is identical to the findings for a singleline, since the beam only crossed the simulation domain once at that time step.After twelve passings of the beam at t = 0.08 s, the temperature distribution ismore homogenized at the first intersection but the surface temperature is still highand increases towards the last crossing. The beginning of the sinter process canbe seen at the first passing. The fusion of the particles progresses as the beamcrosses a last time at t = 0.16 s and thereby illuminates with the last scan vectorthe full width of 6 mm. During the exposure time the energy did not diffuse veryfar into the powder, but was rather spread within the melt pool. The maximumsize of the melt pool, indicated by the black line, can be seen at t = 1.0 s as thetemperature is fully homogenized.

The corresponding time traces of the maximum and mean temperature andthe melt pool area are illustrated in figure 7.9. Analogous to the single line results,the melt pool life time of a layer can be structured into four phases. During thefirst phase the melt pool grows rapidly due the interaction of the beam with thepowder bed. Each time the beam passes, more energy is absorbed. For eachcrossing a peak in the maximum temperature occurs and the mean temperaturerises slowly. Furthermore, Amelt increases stepwise with every interaction. Therapid expansion is slowed down in the second phase because the absorbed energyis distributed through diffusion and convection. A long steady state phase followsfor almost 2 s, where the maximum and mean temperature overlap and the meltpool area is constant. In the fourth and final stage after t = 3 s the melt pool

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Table 7.6.: Parameters used for the calculations shown in figures 7.8 and 7.9.

PB vB hs TB De ∆Hf

10 W 1.0ms

250 µm 172 C 1 5 · 104 Jkg

Figure 7.8.: Temperature distribution within a single layer for the parameters sum-marized in table 7.6 at four different time steps. The black line in thelower plot indicates the final melt pool.

94


Figure 7.9.: Time traces of the maximum and mean temperature and the melt poolarea within a single layer using the parameters shown in table 7.6.

starts to solidify.

95


7.3.2. Melt pool depth

The melt pool depth dl is defined as the distance between the melt pool surfaceand its bottom along the z-axis. The mean of dl is the mean depth dmean. Figure7.10 shows the final shape of the melt pool, indicated by the red line. To calculateits mean depth, only the inner 4 mm between the dashed red vertical lines weretaken into account, to avoid boundary effects by the first and last scanning vectors.

Figure 7.10.: Final shape of the melt pool, indicated by the red line. dmean iscalculated from the inner 4 mm between the dashed red verticallines.

Figure 7.11 shows the mean depth for different beam powers, scanning veloc-ities and Deborah numbers for ∆Hf = 5 · 104 J

kg. The values of the parameters

used are given in table 7.7.

The computed depths range from 80 µm to 400 µm. No significant impacton the depths by the change in viscoelastic regime is observed. The contour linesof the depth distribution within the parameter field of beam power and scanningvelocity show a linear characteristic, which indicates that the melt pool depth of asingle layer is mainly dependent on the energy per unit length El and not on eachparameter combination individually, at least within the investigated range.

Analogous calculations are shown in figure 7.12 with ∆Hf = 11 · 104 Jkg. The

range of the depths is lower than that for ∆Hf = 11·104 Jkg, which is expected since

more energy is needed to overcome the enthalpy of fusion. However, the trendsare very similar, no impact of the Deborah number is found and the contour linesare linear within the parameter field.

For the comparison of the energy input on a surface a more convenient quan-tity is needed than the energy per unit length. Therefore, the energy per unit area

96



PB vB hs TB De

5− 30 W 0.5− 4.0 ms

250 µm 172 C 0.01− 100

Figure 7.11.: Mean depth of a single layer for different beam powers, scanning ve-locities and Deborah numbers with ∆Hf = 5·104 J

kg. The parameters

are given in table 7.7.

Ea is introduced as

Ea = PB · texpA

(7.2)

where texp is the exposure time of the beam onto a given area A. For the hatchingstrategy used with the width w, length l and vector line spacing hs, Ea can be

97


Figure 7.12.: Mean depth of a single layer for different beam powers, scanningvelocities and Deborah numbers with ∆Hf = 11 · 104 J

kg. The pa-

rameters are given in table 7.7.

simplified to

Ea =PB · whs

lvB

w l= PBhs vB

(7.3)

Figure 7.13 (left) shows the mean melt pool depth over the energy per unit areaEa with De = 1. Additionally, linear fits, with the parameters a0 and a1, weremade for the lower and upper limit of the enthalpy of fusion.

dmean = a0 + a1 · Ea (7.4)

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The minimal energy per unit area Ea,min can than be calculated as

Ea,min = −a0

a1(7.5)

The resulting fit parameters and Ea,min are summarized in table 7.8.

Table 7.8.: Fit parameters of equation (7.4) for the two enthalpy limits shown infigure 7.13 and the corresponding minimal energy per unit area.

∆Hf a0 a1 Ea,min

5 · 104 Jkg

−1.44 · 10−5 m 9.5 · 10−6 m3

kJ1.5 kJ

m2

11 · 104 Jkg−2.15 · 10−5 m 6.2 · 10−6 m3

kJ3.5 kJ

m2

The good agreement of the simulation results with the fit confirms the previousfindings of a linear correlation between Ea and dmean. Difference between theminimal energy per unit area Ea,min needed to overcome the melting temperaturefor ∆Hf,min and ∆Hf,max is roughly a factor of two, which is in good agreementwith the estimation given in section 5.1.2.

Figure 7.13 (right) shows the melt pool depth over beam power with constantEa = 40 kJ

m2 . The simulation results show no significant dependency on the powervariation. This means increasing the scanning velocity and thereby decreasingexposure time while keeping the total amount of energy constant, which equalto increasing the heating rate, has no effect on the melt pool depth within theconsidered parameters. This is in contradiction with the findings presented byDrummer et al. [91], illustrated by the red crosses in figure 7.13. In the publishedwork it was concluded, that the melt pool depth decreases with higher heatingrates. This effect can not be reproduced considering thermo- and hydrodynamicsat the powder scale with a ray trace model for the absorption. Therefore, it isprobably necessary to consider further aging or hysteresis effects due to heatingrate, polymer degradation or cooling effects due to radiation or the inert gas.

A simple upper limit for the radiation losses Prad at the surface of the pow-der bed can be estimated by using the Stefan–Boltzmann law and difference be-tween the ambient temperature of TB = 172C and the maximal temperature Tmaxreached at the surface

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Figure 7.13.: Left: Mean melt pool depth over Ea with De = 1 and linear fits forthe lower and upper limit of the enthalpy of fusion. The resulting fitparameters are presented in table 7.8. Right: Melt pool depth overbeam power with constant Ea = 40 kJ

m2 . The experimental results byDrummer et al. [91] are indicated by the red crosses.

Prad = Arad σSB (Tmax − TB)4 (7.6)

where σSB is Stefan–Boltzmann constant and Arad is the radiation area. Thenumber of particles npar within a square of l = 6 mm can be estimated using themean particle diameter d3,50 = 60 µm and assuming equal distance between theparticles. It is assumed that only the half surface of each particle points towardsthe build chamber and therefore radiates energy away from the powder bed. Themaximal temperature during exposure can be estimated to Tmax = 500C (see fig.7.9). When assuming that the whole surface is at Tmax, which is an overestimation,than Prad can be calculated to

Prad = 12npar π d

23,50 σSB (Tmax − TB)4 < 0.1 W (7.7)

100


which is considerably smaller than the used beam power. Therefore this is notsignificant enough to cause relevant cooling rates. However, if the powder bed isnot optically thick for the radiation wavelength, than the surface contribution toradiation losses would increase drastically. Further experimental investigations arenecessary to determine the transmission characteristic of the relevant wavelengthrange.

7.4. Melting of several layers

To create complex three dimensional parts, it is necessary to melt several lay-ers. The most important quantity, which defines the resulting part quality, isthe relative density ρrel of the molten bulk material. The first layer is initial-ized and scanned analogous to the previous section. After the layer is solidifieda new powder layer is applied on top of the previous one, adding a layer heightof hl = 200 µm, and scanned with the same beam parameters. The simulationis terminated after five layers. The relative density is then evaluated in the firstthree layers. Figure 7.14 shows the results after five layers and the area where therelative density is evaluated, indicated by the red dashed line. The parametersused are presented in table 7.9.

By increasing the scanning velocity while keeping the beam power constantthe total energy input is reduced. This can cause a transition from fully densebulk material to porous structures due to lack of fusion between the layers. In thetransition region the binding faults are isolated within the bulk material, whilefor higher porosities the layers form band structures perpendicular to the builddirection. These layers are connected within one layer but not with the layersabove or below. For even lower energy input the connection is lost within thelayer. These cases are not discussed in this work. Similar results are found forlowering the beam power while keeping the scanning velocity constant. The limitsin this work are defined to be at ρrel > 0.98 for good parts and for porous partsρrel < 0.95, between is the transition zone.

The process map in terms of relative density in the parameter field of beampowder and scanning velocity is shown in figure 7.15 with parameters used sum-

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PB vB hs hl TB De

20 W 2.0− 6.0 ms

250 µm 200 µm 172 C 1

Figure 7.14.: Final mass distribution after five layers and the area where the rela-tive density is evaluated, indicated by the red dashed line. Parame-ters used are presented in table 7.9.

102



PB vB hs hl TB De ∆Hf

2.0− 40.0 W 0.5− 10.0ms

250 µm 200 µm 172 C 1 5 · 104 Jkg

Figure 7.15.: Process map in terms of relative density with the parameter usedshown in table 7.10. Each dot represents a calculated parameterset. The dashed line is a linear fit to the interpolated contour lineslimiting the region of transition.

marized in table 7.10.

A linear regression is fitted to both interpolated contour lines limiting theregion of transition and the resulting fit parameters are shown in table 7.11. Thedeviation of the contour lines from the linear fit is small and thereby indicatesthat the relative density, within the investigated parameter field, is independentof the heating rate. The slope a1 is the minimum energy per unit length neededto produce the corresponding relative density.

The influence of the viscoelastic regime on the process map is presented in fig-ure 7.16 with ∆Hf,min. The limits of the transition zone for De = 1 are illustrated

103


Table 7.11.: Fit parameters for equation (7.4) shown in figure 7.15.

ρrel a0 a1

0.98 1.84 W 5.65 Jm

0.95 0.53 W 4.2 Jm

by the dashed line for better orientation. With De = 100.0 no significant changeof the limits is observed, which means that lower Newtonian viscosity does notimprove the processing ability compared to De = 1. This is not surprising sincethe lifetime of the melt pool is several seconds, hence enough time to flatten thesurface of each layer. A slide shift of the positions of the limits and their slope isobserved for De = 0.01. In the rubber like regime some humps and valleys remainat the surface. Therefore, the melting depth needed to melt a connection to theunderling layer is increased, which increases the portability for lack of fusion.

It is expected that the transition zone shifts significantly towards higher en-ergies with ∆Hf = 11 · 104 J

kgfor all viscoelastic regimes, as can be seen in figure

7.17. With De = 0.01 the limits almost move entirely out of the considered pro-cess map. The strong influence of the enthalpy of fusion on the resulting relativedensity and thereby on the part quality is in good agreement with the findings ofK. Wudy [67]. It was found that virgin powder with ∆Hf = 11 · 104 J

kghad a

higher porosity than aged powder using the same process parameters and, there-fore, parts build from aged powder have better mechanical properties. However,after 20 hours of processing time the negative effects of aging start to dominateand the relative density is decreased.

The scaling of the relative density with the energy per unit area Ea is summa-rized in figure 7.18 for all presented process maps, including the upper and lowerlimit of the enthalpy of fusion and all Deborah numbers, summarized in table7.12. The relative density for all parameter sets is lower for ∆Hf,max comparedto ∆Hf,min until a threshold is reached and the energy input is sufficient to meltfully dense parts for both cases. At this value the necessary melt pool depth of themolten layer is high enough to generate a consistent connection to the underlyinglayer.

104


Figure 7.16.: Process map in terms of relative density with the parameters usedshown in table 7.10, while changing De. Each dot represents a cal-culated parameter set. The dashed lines indicate the limits of thetransition zone for De = 1.

105


Figure 7.17.: Process map in terms of relative density with the parameters usedshown in table 7.10, while changing De and ∆Hf = 11 · 104 J

kg. Each

dot represents a calculated parameter set. The dashed lines indicatethe limits of the transition zone for De = 1 and ∆Hf = 5 · 104 J

kg.

106


Table 7.12.: Parameters used for the calculations shown in figure 7.19 and 7.18.

PB vB hs hl De ∆Hf

2.0− 30.0 W 0.5− 8.0ms

250 µm 200 µm 0.01− 100.0 5 · 104 − 11 · 104 Jkg

Figure 7.18.: Relative density of several layers over the energy per unit area Eawith the parameters given in table 7.12. Marker: () De = 0.01, (4)De = 0.1, () De = 100

Each point in figure 7.19 represents the relative density of several layers andthe mean depth of a single layer for the same parameter set, covering the parameterfield given in table 7.12.

The trends of ∆Hf,min and ∆Hf,max overlay within the scattering of the datapoints. The results indicate a clear relation between melt depths and relativedensities. It is not sufficient for dmean to be in the same range as the layer heighthl to melt fully dense parts, due to an uneven substrate and a statistical powderbed. As a rule of thumb it can be estimated that it is necessary to melt half of theunderling layer, or to reach a single layer melt pool depth of 1.5 hl, to generate asound material connection. Using this relation can be beneficial to estimate theprocess map boundaries from experiments or simulations of a single layer.

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Figure 7.19.: Relative density of several layers over the mean depth of a single layerfor the same parameter set. The parameter range is given in table7.12.

7.5. Scan strategy

In the previous sections the influence of the beam power and scanning velocity fordifferent material property limits were discussed. The parameters for the scanningstrategy scan line length and scan line spacing were kept constant. The effect ofthose two parameters on the thermal field and thereby on the melt pool depth andrelative density will be investigated in the following section.

7.5.1. Scan length

The presented process map and scaling in the previous sections are all calculatedwith the same vector scan length of w = l = 6 mm. In general, it is desired to meltarbitrary geometries, which results in different scan vector lengths l. Dependingon the scanning velocity vB and l, the turning back time tb, which is the timeneeded for the beam to scan one line, varies for a constant Ea.

tb = l

vB(7.8)

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Ea vB l hs De ∆Hf

16− 64.0 kJm2 0.5− 8.0 m

s12− 96 mm 250 µm 1 5 · 104 J

kg

Figure 7.20.: Melt pool depth of a single layer for four scanning vector lengths andfour different energies per unit area, with the parameters shown intable 7.13.

With given tb the diffusion length lD in bulk material can be calculated,

lD = 2√√√√ λ

cpρtb (7.9)

which ranges from 28 µm to 113 µm for l = 12 mm and from 80 µm to 320 µm forl = 96 mm and the considered scanning velocity window. This is in range of thespacing between two scan vectors hs = 250 µm and an interaction of thermal fields,in terms of a preheated substrate, generated by the previous scan lines is expected.Therefore, the dependency of the melt pool depth of a single layer is investigatedby doubling the scan length stepwise from 6 mm to 96 mm. Figure 7.20 shows theresults of the mean melt pool depth for these scanning vector lengths and differentenergies per unit area, with the parameters used summarized in table 7.13.

109


In agreement with previous results, a linear scaling of dmean with Ea is ob-served and the points lie, within a small scattering, on top of each other for allscan lengths. No significant dependency of the melt pool depth for a single layeron the scan vector length is found, which makes the process parameters indepen-dent of the geometry within the considered length range. However, this result israther surprising considering the ratio of the diffusion length scale and hatchingspacing. Due to the time needed for the particles to fuse tc, which is much longerthan tb, the molten powder has still the same shape as the initial powder bed whenthe beam returns. Therefore, the thermal conductivity is significantly lower thanwithin bulk material, which was assumed for the calculation of the thermal length.The advection of the energy can also be neglected at this time scale. Therefore,it can be concluded that the melt pool depth is independent of the vector scanlength if the turning back time is smaller than the particle fuse time.

tb tc (7.10)

7.5.2. Hatching spacing

The second important parameter of the hatching scan strategy is the hatchingspacing hs. This parameter can change the absorbed energy distribution whilekeeping the energy per unit area and exposure time constant. Lowering hs andincreasing the scanning velocity vB by the same factor will homogenize the energyinput and should thereby reduce the maximum temperature. This is desired sincePA12 can experience severe degradation at a temperature of 420 C and above[91]. Additionally, avoiding high peak temperatures can reduce or eliminate steepgradients and thereby reduce thermal stresses.

Time traces of the maximum temperature reached in the melt pool and themelt pool size for the hatching of a plane surface without powder are shown infigure 7.21, using the parameters given in table 7.14. The exposure of a plainsurface has the advantage to study the impact on the temperature and melt poolsize without the overlaying effects of thermal isolation and advection due to powderfusion. The peaks of the maximum temperature are considerably lower for smallerhatch spacings and the growth of the melt pool is smoother. However, small

110



Ea vB l hs De ∆Hf

16 kJm2 0.5− 2.5 m

s6 mm 50− 250 µm 1 5 · 104 J

kg

Figure 7.21.: Time traces of the maximum temperature and the melt pool size forthe hatching of a plane surface without powder, using the parametersgiven in table 7.14.

steps can still be observed for hs = 50 µm. To keep the energy input and theexposure time constant, the scanning velocity needs to be increased from 0.5 m

s

for hs = 250 µm to 2.5 msfor hs = 50 µm.

Analogous calculations for the melting and solidification of a single layer areshown in figure 7.22 with the parameters from table 7.15. The energy per unit areaused melts fully dense parts for hs = 250 µm, as shown in section 7.4. The timerange of the highlighted zoom insert ranges from the beginning of the simulationto 0.04 s.

The difference in the peak temperature is roughly 100 K for this parameterset and the overall maximum temperature with hs = 50 µm is lower than thedegradation temperature. With hs = 250 µm the temperature is exceeded, whichcan cause significant mass loss, even though it is only for a short period of time

111



Ea vB l hs De ∆Hf

40 kJm2 1.0− 5.0 m

s6 mm 50− 250 µm 1 5 · 104 J

kg

Figure 7.22.: Time traces of the maximum temperature and the melt pool size forthe hatching of a single layer, using the parameters given in table7.15. Zoom insert from 0 s to 0.04 s.

[91]. On the other hand, the mean temperature within the melt pool is basicallyidentical for both cases. In agreement with the results for the plane surface, themelt pool growth rate is smoother for lower scan line distances, while the final sizeand life time are equal. Some minor differences are observed in the solidificationphase, which are based on a slightly different interface between powder bed andmelt pool. Therefore, the solidification does not start at the same point whichresults in changes in solidification rate during that phase.

The mean melt pool depth of single layer over Ea for two scan hatchingspacings is presented in figure 7.23 with the parameters shown in table 7.16. Alldata points overlay for both scan line spacings and show also the same linear

112


Table 7.16.: Parameters used for the calculations shown in figure 7.23 and 7.24.

Ea vB l hs De ∆Hf

0− 40 kJm2 0.5− 20.0 m

s6 mm 50− 250 µm 1 5 · 104 J

kg

Figure 7.23.: Mean melt pool depth of a single layer over Ea for two scan linespacings with the parameters shown in table 7.16.

scaling as in section 7.3.2, thereby indicating that there is no negative influenceon the melt pool depth when lowering the hatch distance.

Figure 7.24.: Relative density of several layers over Ea for three scan line spacingswith the parameters shown in table 7.16.

It is also interesting to investigate the influence on relative density by melting

113


several layers. In section 7.4, a direct correlation between the melt pool depthof a single layer and the relative density of several layers was found. Hence, itis not expected to find any influence of lowering the scan line spacing onto therelative density, since no influence is found on the melt pool depth of a single layer.Figure 7.24 shows the relative density of several layer over Ea. The layers weresimulated and evaluated as described in section 7.4, while changing the hatchingdistance in the range shown in table 7.16. No significant influence of lowering thehatching spacing on the relative density for a wide range of Ea could be found.Scanning strategies with small hatch spacing should be preferred to avoid polymerdegradation due to their lower peak temperature at constant energy per unit area.One limit needs to be considered though, to keep the build rate constant thescanning velocity needs to be increased. The maximum velocity is determinedby the working distance and the mirror positioning speed. The inertia of themirror due to its mass sets a technical limit for deflection speed. On the otherhand, increasing the working distance will also lead to higher demands on theaccuracy of the beam optical and deflection system, since the projection error alsoscales with the working distance. Those two facts limit the maximum achievablescanning velocity and thereby also limit the feasibility of scanning strategies withsmall hatch spacing.

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8. Conclusion

The development and first application of a simulation framework for modeling se-lective laser sintering of viscoelastic polymers at powder scale was demonstrated.First, the derivation of the governing equations for the hydrodynamics of viscoelas-tic fluids was presented by means of kinetic theory, consisting of the continuity,Navier-Stokes and Oldroyd-B constitutive equation. Together with the convection-diffusion equation it was possible to formulate a complete system of equations forthe description of selective laser sintering. Due to the complexity of the SLS pro-cess a numerical approach is required. The Lattice-Boltzmann method (LBM)offers the possibility to solve the equations with the given boundary conditions.

While the LBM for the thermo- and hydrodynamics of Newtonian fluids is wellcovered in the literature, the description of the numerical scheme for the Oldroyd-Bmodel and its coupling with the other equations is rather rudimentary. Therefore,Onishi’s scheme [32] was adapted to the requirements needed to model the SLSprocess which includes the development of a new transport model. Analytical testscould show that the presented solver is second order in space and time, while thetransport model is only first order accurate. However, the absolute error is smallcompared to other state of the art methods [37] and it is rather stable due to itsnon-oscillating characteristic, so that the overall accuracy is not affected. Thiscould also be demonstrated in complex flow benchmark problems, namely the 4to 1 contraction and the four-roller mill. The developed scheme reproduces allrelevant features and can compete with other numerical methods [25].

A second important part of modeling the SLS process is the absorption ofthe laser energy. The developed model is based on a ray tracer approach and isable to describe the absorption, reflection and refraction of coherent laser light ina powder bed. As an analytical validation test, the irradiation of a plane surface

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with a laser beam was examined. The calculated results correspond very well tothe analytical values for a wide range of resolutions.

With the numerical scheme for the thermodynamics and hydrodynamics andthe absorption model it was possible to create a simulation tool for the selectivelaser sintering of viscoelastic polymers. The material properties of polyamide 12,which are necessary as an input for the model, were collected through a thoroughliterature research. Essential properties such as enthalpy of fusion can differ bya factor of two between virgin and aged powder. Unfortunately, no data for theviscoelastic relaxation time could be found. Both parameters were varied wherenecessary to check their influence on the SLS process. The required optical prop-erties have been measured experimentally.

Using the simulation tool it was found that the laser absorption in a powderbed notably differs from bulk in material. Although less mass is passed by a rayin the powder bed than in solid material, the effective absorption depth can beshorter, due to reflection and refraction. This results in a higher energy density inthe powder particles. In addition, the round powder particles act as lenses and canthus create local hot spots. This inhomogeneous absorption behavior is difficultto represent by an simple absorption model, especially if the absorption length isof a similar order of magnitude as the particle diameter. During the initial stateof melting funnel like melt pools arise along the focused beam path. If the energyinput is sufficient a melt pool develops which extends over the full beam width. Nosignificant consolidation was observed during the short interaction time betweenthe laser and the powder bed.

The time scale of the sintering process of polyamide 12 powder, used in SLS,ranges in the order of seconds depending on the viscoelastic regime. For a re-laxation time shorter than the characteristic time, the fluid shows a rubber likecharacteristic and the sintering process is long. The viscoelastic stress builds upimmediately and contributes to the viscous drag. Thereby, the sintering rate drivenby the surface tension is reduced. This changes with very long relaxation times.Here, the viscoelastic tension cannot build up and thus also cannot contribute tothe viscous resistance. An intermediate state is observed for a relaxation time inthe order of the sintering time scale. At the beginning, the sinter rate is fast until

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the viscoelastic stress is fully developed and slows down the sintering process. Nosignificant influence on the melt pool depth of a single line or a single layer was ob-served, even though the melt pool surface is affected and ranges from flat for longto uneven for short relaxation times. Since most of the consolidation takes placein the beginning of the sintering process and due to the long melt pool life time inthe SLS process, long and intermediate relaxation time results in similar relativedensities when melting several layers. In the rubber like regime some humps andvalleys remain frozen during solidification at the surface. When the next layeris applied these valleys are filled with powder and the depth needed to create amaterial connection is increased compared to the powder layer thickness, whichincreases the probability for lack of fusion. Therefore, the processing window interms of scanning speed and beam power gets smaller and the limit for dense partsis shifted towards higher power.

The aging behavior of the powder has a far greater impact on the processingwindow than the viscoelastic regime. The temperature of the build chamber is justbelow the melting point of polyamide 12, therefore, the energy deposited by thebeam is mostly needed to overcome the enthalpy of fusion. As a result the limitfor dense parts moves significantly towards higher beam powers. To minimize thisinfluence and to guarantee the repeatability of the material properties, a powderrecycling system is necessary. One could argue that using process parametersfor virgin powder will melt dense parts for any combination of virgin and usedpowder. But what needs to be considered is that introducing more energy thanneeded will increase the peak temperature and lead to polymer degradation. Oneway to compensate is to decrease the hatching spacing and thereby homogenizethe energy input. This has lower peak temperatures as a consequence, creatinga broader processing window and thereby being more robust to changes in theenthalpy of fusion. Since there is no negative influence on the relative density,when keeping the energy per unit area constant, this strategy should be consideredin future experimental and theoretical studies. Always taking into account thatthe scanning velocity must be increased in order to keep the build rate constantand that there are physical limits for the maximum velocity given by the workingdistance and the mirror positioning speed. Nevertheless, these limits seem to beworth exploring.

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The developed simulation tool was validated with experimental data for theabsorptance, reflectance and transmittance of PA12 powder layers, peak temper-atures in stationary beam exposure, measured by high speed DSC, melt pool di-mensions of single line melting and melt pool depth of single layers. Most of thesimulated results were in good agreement with the experimental findings. How-ever, the dependence of the mean melt pool depth of a single layer on the heatingrate could not be found. Here, the potential for further improvements of the modelcould be considered, e.g. aging or hysteresis effects due to heating rate, polymerdegradation or cooling effects due to radiation or the inert gas. It was still possibleto validate the presented framework in a wide range of process parameters and itcan, therefore, offer valuable insights into the SLS process at powder scale.

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[90] L. Lanzl et al. “Laser-high-speed-DSC: Process-oriented thermal analysis ofPA 12 in selective laser sintering”. English. In: 83 (2016), pp. 981–990.

[91] Dietmar Drummer, Maximilian Drexler, and Katrin Wudy. “Impact of Heat-ing Rate During Exposure of Laser Molten Parts on the Processing Windowof PA12 Powder”. In: Physics Procedia 56 (2014). 8th International Con-ference on Laser Assisted Net Shape Engineering LANE 2014, pp. 184–192.issn: 1875-3892. doi: 10.1016/j.phpro.2014.08.162.

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Appendices

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A. List of Symbols

A.1. Latin

Symbol Description

a0 Fit parameter 1a1 Fit parameter 2A Medium AAmelt Melt areaArad Radiation areaB Medium BBo Bond numbercf Friction coefficientcp Specific heat capacityc∗p Modified heat capacityCa Capillary numberd Particle diameterd Dimensionless particle diameterd3,50 Median particle diameterdl Distance between the melt pool surface and its bottomdm Melt pool depthdmean Mean melt pool depthDe Deborah numberd~x Volume elementd~ξ Volume in velocity spaceD Numerical diffusion coefficient~ei Discrete velocity vector

129

APPENDIX A. LIST OF SYMBOLS

Ea Energy per unit areaEa,min Minimal energy per unit areaEi Discrete energy density distribution functionsEI Energy of the incoming rayER Energy of the reflected rayET Energy of the transmitted rayEl Energy per unit lengthEm Remaining EnergyEmin Minimal energy thresholdEn Total absorbed energy with refractive index nErr Error~en Unit vector in Cartesian coordinatesf Particle distribution functionf eq Equilibrium particle distribution functionfi Discrete particle distribution functionf eqi Equilibrium discrete particle distribution functionfl Focal length~f Force vector~fc Connecting force between dumbbells~fext External forceF magnitude of the forceg Gravity acceleration constanth Half width of downstreamhl Layer heighth∗l Effective layer heighths Hatch spacingH Hook constanti Discrete lattice directioni Inverse direction of iI Intensity distributionIm Mean intensityj Shape parameter of inverse ellipsekB Boltzmann constant

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~k Normalized direction vector~kR Normalized direction of the reflected ray~kT Normalized direction of the refracted rayl Length of scan vectorlc Characteristic lengthlD Diffusion lengthL Domain lengthm MassM1 Mesh 1M2 Mesh 2M3 Mesh 3M4 Mesh 4nP Density number of dumbbellsnA Refractive index of the medium A

nB Refractive index of the medium B

nh Number of scan linesnPA12 Refractive index of the medium PA12nmax Maximum number of samplesnmed Refractive index of the mediumN Number of lattice cells evaluatedN0 Domain size in cellsNx Number of cells within one period L~n Normal vectorp Pressurepatm Atmosphere pressureP0 Total powerPB Beam PowerPB,max Maximum beam powerPrad Power of radiation lossPe Péclet numberq Particle diameter distribution densityq Particle diameter probability distribution~q Dumbbell connection vector

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~qi Discrete dumbbell connection vectorQ Heat sourceQ Dumbbell connection vector tensorr Particle radiusrB Build rateRe Reynolds numbers0 Distance in x-directiont Timetb turning back timetc Characteristic timetexp Exposure timets Simulation timeti,min Minimum beam interaction timetα Diffusive time scaleT TemperatureTB Build chamber temperatureTc Crystallization temperatureTm Melting temperatureTmax Maximum temperatureTmean Mean temperatureTs Solidification temperatureTi Discrete temperature distribution functionsT Stress tensorTN Newtonian stress tensorTP Viscoelastic stress tensorv Mean velocityvB Scanning speedvB,max Maximum scanning speedvc Characteristic velocityvx x-component of velocityvy y-component of velocity~v Velocityw Width of scan area

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ws Sinter neck half widthWi Weissenberg numberxB x beam coordinatexr Ray at positionxR Characteristic silent vortex length~x Space coordinateyB y beam coordinate

A.2. Greek

Symbol Description

α Thermal diffusivityγ Shear rate∆H Specific enthalpy∆Hf Enthalpy of fusion∆Hf,max Lower limit for the enthalpy of fusion∆Hf,min Upper limit for the enthalpy of fusion∆r Casting distance∆x spatial resolutionε Effective Weissenberg numberΘ Simulation domainλ thermal conductivityλ1 Viscoelastic relaxation timeλ2 Viscoelastic retardation timeλeff Effective penetration depthλs Thermal conductivity solidλl Thermal conductivity liquidλα Thermal conductivityµ0 Total dynamic viscosityµa Absorption coefficient

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µN Dynamic viscosity of the Newtonian fluidµP Dynamic viscosity of the viscoelastic fluidνN Newtonian kinematic viscosityµα Attenuation coefficient~ξ Velocityρ Densityρrel Relative densityσB Standard deviation of beam distributionσSB Stefan–Boltzmann constantτ Molecular collision timeτN Relaxation time of the Newtonian fluidτT Relaxation time for scalar distribution functionsϕ Volume fractionΨ Configurational distribution functionΨi Discrete configurational distribution functionΨeqi Discrete equillibrium configurational distribution function

ω Vorticityωi lattice weightsΩ Collision operator

134

B. Notation and operators

The definitions of the notation and operators used in this work. For all examplesa two dimensional Cartesian coordinate system is considered. The notation waschosen with the aim of getting as close as possible to the literature presented andbeing consistent at the same time.

Scalars: x, X, ψ, ΨBig and small Latin and Greek letters.

Vectors: ~xSmall Latin and Greek letter with an arrow. The components are given by:

~x =(xxxy

)(B.1)

Second order Tensor: XBold big Latin letter. The components are given by:

X =Xxx Xxy

Xyx Xyy

(B.2)

Dot product: ~x · ~yThe dot product of two vectors is defined as:

~x · ~y = xxyx + xyyy (B.3)

The dot product of a tensor with a vector is defined as:

X · ~y =(Xxxyx +XxyyyXyxyx +Xyyyy

)(B.4)

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APPENDIX B. NOTATION AND OPERATORS

The dot product of two tensors is defined as:

X ·Y =XxxYxx +XxyYyx XxxYxy +XxyYyy

XyxYxx +XyyYyx XyxYxy +XyyYyy

(B.5)

Dyadic product: ~x~yThe outer product of two vectors is defined as:

~x~y =xxyx xxyy

xyyx xyyy

(B.6)

Transpose: XT

The transpose of a tensor is defined as:

XT =Xxx Xyx

Xxy Xyy

(B.7)

Frobenius inner product: X : YThe Frobenius inner product of two tensors is defined as:

X : Y = XxxYxx +XxyYxy +XyxYyx +XyyYyy (B.8)

Ensemble average: 〈X〉The ensemble average over a distribution fi of a tensor is defined componentwise:

〈X〉α,β =∑j ~xj,α~xj,βfj∑

k fk(B.9)

Total derivative: ddxf

Total derivative of f with respect to x.

Partial derivative: ∂xfShort notation for the partial derivative of f with respect to x.

∂

∂xf = ∂xf (B.10)

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Nabla operator: ∇The nabla operator with respect to space. The vector derivative of a scalarfield f , also called gradient, is given by:

∇f =(∂xf

∂yf

)(B.11)

The result of the nabla operator applied as a dot product to a vector field isa scalar function, also called divergence of the vector, and is given by:

∇ · ~f = ∂xfx + ∂yfy (B.12)

The divergence of two dimensional tensor is given by:

∇ ·X =(∂xXxx + ∂yXyx

∂xXxy + ∂yXyy

)(B.13)

The inner product with a vector is defined by:

∇~f =∂xfx ∂xfy

∂yfx ∂yfy

(B.14)

Laplace operator: ∆The Laplace operator of a scalar f in two dimensions is given by:

∆f = (∇ · ∇) f = ∂2xf + ∂2

yf (B.15)

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Upper convected time derivative:∇X

The upper convected time derivative is defined as:∇X= ∂tX + (~v · ∇) X− (L ·X + X · LT ) (B.16)

with

L = (∇~v)T (B.17)

the components can be written as:

∇Xαβ=∂tXαβ + vx∂xXαβ + vy∂yXαβ

− (∂xvαXxβ + ∂yvαXyβ)− (∂xvβXαx + ∂yvβXαx)(B.18)

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Modeling of Selective Laser Sintering of Viscoelastic Polymers

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