A Fully-Coupled Finite Element Analysis of Field Assisted Sintering Techniques

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Università di Pisa

Dipartimento di Ingegneria Aerospaziale

“Lucio Lazzarino”

A Fully-Coupled

Finite Element Analysis

of Field Assisted Sintering Techniques

Relatori:

Prof. Ing. Massimo De Sanctis

Prof. Ing. Luigi Lazzeri

Prof. Eugene A. Olevsky

Candidati:

Diletta Giuntini

Corso di Laurea in Ingegneria Aerospaziale A.A. 2013-2014

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Preface

The present work of thesis has been realized in the Powder Technology Laboratory at San Diego State University, under the supervision of Prof. Eugene A. Olevsky.

Prof. Olevsky is the founder and the director of the laboratory, which is nowa-days considered one of the world main centers of academic research in the powder technologies field.

The PTL is endowed with a spark plasma sintering machine, with which the in-novative Field-Assisted Sintering Techniques can be experimented.

Field-assisted sintering is a relatively new and still not fully understood technology, allowing powders from a variety of materials to be densified at elevated temperatures, with high heating rates and short processing times.

Among the several collaborations developed by PTL with domestic and foreign institutions, the joint work with a US laboratory and with the Dresden Fraunhofer Institute for Ceramic Technologies and Systems IKTS has led to a thorough investi-gation on the role of the tooling setup during field-assisted sintering procedures, here presented.

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List of Figures

2.1 Publications on FAST per year . . . 22

2.2 Countries percentages of FAST publications . . . 23

2.3 Patents on FAST per year . . . 23

2.4 Hot Pressing and FAST setups . . . 25

2.5 FAST machine . . . 26

2.6 Typical FAST tooling setup . . . 27

2.7 FAST setup in which graphite foil and felt are marked . . . 28

2.8 Mass transport mechanisms in sintering . . . 30

2.9 Necks formated during the FAST of vanadium carbide . . . 31

2.10 Evidence of discharge presence in FAST . . . 34

2.11 Evidence of microscopic thermal gradients in FAST . . . 36

2.12 Evidence of localized overheating in FAST . . . 37

3.1 FAST geometry imported in the COMSOL Multiphysics environment . . . 39

3.2 Porosity definitions . . . 42

3.3 Models of bulk modulus and sintering stress . . . 44

3.4 Imposition of electric current input at the tooling upper surface . . . 46

3.5 COMSOL interface ready to start the simulation . . . 47

4.1 Configurations utilized for the horizontal contact resistance individuation . 53 4.2 Configurations utilized for the vertical contact resistance individuation . . 54

4.3 System horizontal contact resistance . . . 56

4.4 Horizontal contact resistance results . . . 57

4.5 Imperfect contact among tooling components . . . 57

4.6 System vertical contact resistance . . . 59

4.7 Vertical contact resistance results . . . 60

4.8 FEM verification - Temperature profiles with and without ECR implemen-tation . . . 63

4.9 FEM verification - Axial temperature distributions with and without ECR implementation . . . 64

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List of Figures 7 4.10 FEM verification - Axial temperature distributions with and without ECR

implementation . . . 65

4.11 FEM verification - Temperature profiles with and without TCR implemen-tation . . . 66

4.12 FEM verification - Axial temperature distributions with and without TCR implementation . . . 66

4.13 FEM verification - Axial temperature distributions with and without TCR implementation . . . 67

5.1 Tooling setup presenting the overheating problem . . . 70

5.2 Geometries of (a) two transitional disks; (b) three transitional disks, and (c) four transitional disks configurations . . . 73

5.3 Final axial temperatures (in K) distribution for the conical configuration . 75 5.4 Peak axial temperatures (in K) distribution for the conical configuration . 76 5.5 Final axial temperatures distribution for the various 2 disks configurations 81 5.6 Final axial temperatures distribution for the various 3 disks configurations 82 5.7 Final axial temperatures distribution for the various 4 disks configurations 83 5.8 Peak axial temperatures distribution for the various 2 disks configurations 84 5.9 Peak axial temperatures distribution for the various 3 disks configurations 85 5.10 Peak axial temperatures distribution for the various 4 disks configurations 86 5.11 Temperature evolution with time for the two, three and four disks config-urations . . . 87

5.12 Temperature evolution—Effect of height variation . . . 88

5.13 Peak temperature disparity between hot spot and specimen - two and four disks . . . 89

5.14 Peak temperature disparity between hot spot and specimen - three disks . 90 5.15 Final temperature disparity between hot spot and specimen - two and four disks . . . 91

5.16 Final temperature disparity between hot spot and specimen - three disks . 92 6.1 Radial temperatures uniformization experimental setup . . . 96

6.2 Radial temperatures disparity in a conventional FAST setup . . . 98

6.3 Experimental fitting of the two pyrometers’ temperature profiles . . . 99

6.4 New punch geometries: Holes and Rings . . . 100

6.5 Experimental implementation of one Holes (left) and one Rings (right) configuration . . . 102

6.6 Comparison of the temperature disparity evolution with time for Full, Holes and Rings . . . 103

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List of Figures 8

6.7 Alpha phase content in the sintered silicon nitride specimens . . . 104

6.8 Density distribution of the sintered silicon nitride specimens . . . 105

6.9 FESEM micrograph with alpha and beta phases . . . 106

6.10 FESEM micrographs of the Full configuration - Edge (top) and center (bottom) . . . 107

6.11 FESEM micrographs of the Holes configuration - Edge (top) and center (bottom) . . . 108

6.12 FESEM micrographs of the Rings configuration - Edge (top) and center (bottom) . . . 109

6.13 Temperature disparity in the Holes configurations . . . 110

6.14 Temperature disparity in the Rings configurations . . . 111

6.15 Summary of the temperature disparities in the Holes and Rings setups . . 112

6.16 Temperature disparity in the three selected Rings configurations . . . 113

6.17 Optimized 3 Rings setup . . . 114

6.18 Radial temperature distribution comparison among Full and optimized Rings cases . . . 115

6.19 Experimental temperatures in the optimized 3 Rings configuration . . . 116

6.20 Densification of the specimen in the optimized 3 Rings case . . . 117

6.21 Schematics of the punches’ further geometric optimization . . . 120

6.22 Further geometric optimization results . . . 121

6.23 Rings configurations coated with boron nitride . . . 123

6.24 Temperature disparity for the hBN-coated rings configurations . . . 124

6.25 Optimized hBN-coated full punch configuration . . . 126

6.26 Central radial cross-section temperature distribution for the conventional full punch and the hBN optimized case . . . 127

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List of Tables

4.1 Isocarb Graphite I-85 properties . . . 52

4.2 Dimensions of graphite components . . . 52

4.3 Pressure factors - horizontal contact resistance . . . 58

4.4 Pressure factors - vertical contact resistance . . . 60

5.1 Dimensions of the overheated tooling setup . . . 70

5.2 Isocarb Graphite I-85 properties . . . 71

5.3 Alumina properties . . . 71

6.1 R7710 Graphite properties . . . 96

6.2 Silicon nitride properties . . . 97

6.3 Punch cross-section surfaces and sample final densities of the experimen-tally implemented configuration . . . 103

6.4 Boron nitride properties . . . 123

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Summary

Sintering is nowadays a well-assessed and thoroughly investigated technology, con-sisting in the densification of a powder material by means of the application of high temperatures.

The peculiarity of such technology is the sequence of processing steps involved: at-omization followed by densification. Essentially, the material needs to be “destroyed”, reduced to a powder state, before undergoing the sintering process, which will ulti-mately lead to the production of a dense component.

Another fundamental aspect characteristic of the sintering processes is the absence of melting. The temperatures necessary for powder densification to happen are located in a range around 70% of the treated material’s melting point.

The science of sintering was started more than a century ago in order to explain the main aspects of this innovative powder technology. Scientists wanted to find out why and how sintering happens. More than fifty years of research eventually led to the discovery of the answer to the “why” question, namely that the driving force for sintering consists in surface tension, i.e. the tendency of the powder material to minimize its free surface energy. In the meanwhile, answers to the “how” interrogative were found, by individuating a variety of mass transport mechanisms that play a role during powder materials processing at high temperatures, mainly related to diffusion and viscous flow mechanisms.

Interest kept growing in this field and a large number of novel technologies based on the principles of sintering were developed. Researchers and academics started referring to the original technique as “conventional sintering”, in order to distinguish the surface tension-driven phenomenon from the newly developed processes, which were mainly the result of the combination of pure sintering with the application of external loads (hot pressing) or with the employment of alternative heat sources with respect to the traditional furnaces (microwave, electric currents).

Among all these innovations, the most exciting development lies in the Field-Assisted Sintering Techniques (FAST, also known as Spark Plasma Sintering, SPS). These techniques consist in attaining densification of powder materials by means of

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List of Tables 11 the application of an electric field. The basic sintering principle is not altered, since the application of a voltage leads to a flow of current through or around the powder material, depending on its electrical conductivity properties, which generates heat by means of Joule effect. Frequently an external pressure is also applied in order to enhance densification while retaining the undesired grain growth effects typical of materials exposed to elevated temperatures.

While a conventional sintering machine essentially consists of a furnace and a push-rod, aimed at maintaining the powder specimen in place and measuring the change of volume occurring during densification, a FAST machine is significantly more complex. The core is constituted by the setup in which the powder material is located.

When excluding the more complex industrial FAST applications, a sintered speci-men is typically cylindrical. Such shape is imposed by the surrounding tooling, which consists of an assembly of components made of a highly conductive material, typically graphite. The powder is located in an apposite cavity inside a “die”, a cylindrical block with an axial cylindrical through hole, and is axially contained by two more cylinders, called “punches”. The die-punches assembly therefore consitutes the housing of the powder material, with the die creating the radial confinement and the punches taking care of the axial one. A slight cold pressing is usually applied, so that the powder compact reaches a relative density of 64-70%. The assembly is therefore located in the FAST machine, in a chamber provided with the two stainless steel electrodes that, once in touch with the upper and lower surfaces of the tooling setup, will apply the voltage and allow the current to flow through tooling and specimen, if conductive. The FAST procedure generally happens in vacuum. Often the contact between electrodes and tooling does not happen at the bases of the punches, but a series of disks with increasing dimensions and made of the same material as the tooling are interposed, in order to create a smoother transition. Such components are called spacers and are available in a variety of shapes and sizes.

This thorough description of the FAST tooling setup is motivated by the extremely significant influence that it exterts on the FAST procedure itself.

The main advantages of field-assisted sintering lie in a fast volumetric heating and in a consistent shortening of the densification time, which is reduced from several hours to a few minutes thanks to high heating rates (up to several degrees K per minute), allowing the retention of the initial grain size. Such efficiency leads to the consolidation of materials otherwise non processable, the absence of additives that are necessary in conventional sintering to densify certain materials, and the tailored production of composites, among other advantages. Many theories are also present in the literature concerning the actual role played by the electric current, which is

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List of Tables 12 sometimes thought to have a broader influence than the mere Joule heating. This usually holds for certain specific materials.

Nevertheless, even in the simple Joule heating case, a variety of results can be obtained in FAST procedures by simply altering the tooling setup. The tooling is the mean through which current and heating are transported to the specimen, and therefore its shape, dimensions and arrangement need to be carefully tailored.

One of the main issues that have hampered the diffusion of FAST technologies to the industrial field is the development of elevated thermal gradients within tooling and specimen. Thermal gradients endanger the tooling, by means of localized over-heating peaks that create permanent deformations or even changes of state. They also compromise the final outcomes in terms of sample densification, whose uniformity, as well as the microstructural one, is strictly dependent on the temperature distributions in the axial and radial directions. Thermal non-uniformities become more important as the sample’s size is increased.

An important limitation when attempting to monitor the FAST process is the lack of access to the specimen during the procedure. The FAST machine provides data on the voltage applied at the electrodes, the current input, the load applied at the upper ram, the axial displacements of the specimen’s top surface (therefore the densification curve), and the temperature at one, or two at most, specific locations. These specific locations are points belonging to the tooling setup, since only extremely rare setups allow the temperature measurement probes (optical pyrometers or thermocouples) to access the specimen itself, and even in those situations, they are limited to a single point.

A broader mapping of thermal and electrical phenomena is necessary, and in order to conduct this type of analysis finite element methods (FEM) have been utilized.

Up to now, there are no FEM softwares provided with a sintering interface, since discussion is still ongoing on the physical phenomena underlying most of the sintering processes usually conducted. In order to develop our own FAST models, we selected a commercial software, COMSOL Multiphysics®, which allows the combination of different interfaces.

A complete and fully coupled FAST simulation environment was created, by com-bining electric currents, heat transfer, solid mechanics and mathematics moduli, such that both the tooling and the specimen conditions were reproduced, starting from the application of the voltage, to the Joule heating effect, to the sample’s densification and consequent shrinkage.

The most critical part is naturally the densification step. In order to reconstruct its evolution, we employed the continuum theory of sintering, which treats the powder

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List of Tables 13 compact as a porous compressible continuum, therefore neglecting the single powder particles behavior. Such approach is well established for conventional sintering, and it has been shown, in the present and other works, that it applies to FAST processes too. The continuum theory of sintering provides a constitutive equation relating stresses and strain rates for the porous specimen and a densification kinetics equation, which connects the porosity change with the volume shrinkage. The whole specimen consolidation can therefore be reproduced.

By combining this sintering module, appropriately implemented in the FEM soft-ware, with the heating and electricity modules, every FAST procedure can be repro-duced.

While developing the FEM FAST models, an important issue arose. Since the tooling setup is constitued by an assembly of separated components, at the interface among them an electro-thermal contact resistance will be present, whose effect cannot be considered negligible. Differently from the bulk materials properties, this contact resistance cannot be measured with conventional techniques. An investigation was consequently carried out in order to individuate the entity of this contact resistance and to formulate an analytical expression to be used in any FAST context.

The study consisted in a main experimental section, in which a variety of FAST routines were applied to the tooling setup in absence of powder. The machine readings of current and voltage provided an immediate estimation of the electrical resistance associated with a contact interface, which was then analyzed as a function of tempera-ture and pressure, considered to be the two most influential factors on this parameter. It is interesting to point out that electrical resistance changes with its direction with respect to the current flow, therefore we distinguished between horizontal (perpen-dicular to the current flow) and vertical (axial, along the current flow). The former depends on the material’s creep and the interface’s quality, the latter on Poisson deformation and thermal expansion.

The thermal contact resistance was proven to be negligible instead, and FEM modeling of a simplified tooling was implemented in order to verify the experimentally assessed results.

Once the contact resistance had been modeled, our FEM simulations of FAST processes were complete and ready to be applied to specific investigations and the solving of actual issues.

A first study (Chapter 5) was conducted as a consequence of an overheating prob-lem localized at the top punch of a macroscale setup, where macroscale imples that the specimen’s was several centimeters in diameter. Experiments showed that the top punch was quickly becoming red and then white hot, leading to permanent damages

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List of Tables 14 to the entire tooling. In certain extreme cases the punch was even reported to have sublimated.

The cause for such behavior was individuated in the conically-shaped spacer, which did not allow for the heat to be released efficiently enough at the areas where current density was the highest, and was therefore replaced in the FEM environment by a stack of disks. A broad simulations campaign was conducted, which allowed for the individuation of a variety of alternative solutions, by modifying the number of spacers and their dimensions. The importance of surface radiation and current concentration phenomena was also pointed out.

While this section of the study was analyzing the temperature distributions in an axial cross-section, the second part (Chapter 6) concerned the radial direction. Also in this case the investigation arose from an experimentally encountered problem in a macroscale FAST procedure. The processing of an insulating material forces the current not to cross the powder compact, leading to a Joule effect enhancement inside the die wall and therefore to an overheating of the sample’s edge with respect to its center. A strong radial thermal inhomogeneity creates uneven density distributions and non-uniform grain growth, therefore hampering the final mechanical properties of the sintered specimen.

In this case we selected the punch as the component to be improved, since it is the one whose cross-section corresponds to the powder compact’s one. Instead of a conventional full cylinder component, we decided to drill a number of holes, in order to remove unneeded material and therefore mitigate the heat accumulation in the punch itself, enhance surface radiation by adding free boundary surfaces and direct the current flow along a specific path. A variety of configurations were numerically attempted until a final solution was individuated in the drilling of three concentric rings-shaped holes according to an optimized pattern. Another possible strategy was explored, which, instead of requiring the removal of material, was creating a preferred current path by applying a boron nitride insulating coating that was partially cover-ing the punch upper surface. Some of the optimized setups were also implemented experimentally, and experiments confirmed the beneficial results.

Both analyses, axial and radial temperature distributions and homozeneization, uncovered many aspects of electrothermal phenomena during FAST procedures. The role of current density and distribution, together with heat capacity and surface radi-ation, were better understood. The factors hampering experimental procedures were significantly mitigated by simple geometry modifications, confirming the fundamental role played by the tooling setup during field-assisted sintering. Notice that in both cases the material to be sintered was an insulating powder, meaning that the main

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List of Tables 15 consolidation mechanism in the FAST process was exactly the Joule heating effect.

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Chapter 1 Introduction

Field-Assisted Sintering Technology (FAST), a particularly efficient process for the densification of powder materials with the aid of electric currents, has gained growing attention in the last decade.

With respect to the conventional sintering techniques, FAST enhances the densifi-cation of powder materials, which are compacted in significantly shorter times thanks to extremely high heating rates (several hundreds degrees K per minute) and conse-quently shorter holding periods at the high temperatures. A limitation of the length of the timestep in which the material is exposed to the high sintering temperatures, usually around the 70% of the material melting point, has the beneficial effect of limiting the undesired grain growth once the powder compact has reached densifica-tion. Another important contribution to the development of highly dense structures is the rapid and volumetric heating simultaneous to the application of pressure. The pulsed DC current renders FAST suitable for a variety of materials, so that, in many cases, the need for sintering aids is eliminated [1–5]. As a direct consequence, costs are lowered and the production of dense nano-materials, i.e., endowed with improved mechanical properties, is possible.

The main characteristic that distinguishes FAST from convensional consolidation processes is the participation of an electric field. The powder material to be densified is therefore located in an electrically conductive tooling setup, at which ends a low voltage is applied, leading to the flow of a current throughout the conductive tooling and the internal powder specimen, which can be either conductive or insulating.

The heating necessary for the sintering process to happen is therefore generated by means of Joule effect, even though a more direct contribution of electric currents to the powder densification has been hypothesized and is still under investigation for a variety of materials.

When dealing with thermal and electrical studies is FAST procedures, interest has

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CHAPTER 1. INTRODUCTION 17 frequently shifted from the specimen to the entire tooling setup, usually constituted by graphite components, since experimental and numerical research has shown how significantly the overall tooling conditions affect the powder-specimen final properties, in terms of densification and microstructure.

Many efforts have consequently been dedicated to FAST processes optimization. In this area, most of the conducted studies concern a progressive refinement of the sintering regime parameters - heating rate, holding time-step, preset temperatures, externally applied pressure, utilization of sintering aids - in order to gradually improve the densified specimen properties [6-8], enhance certain specific characteristics of the final product [9], or realize opportunely tuned functionally-graded materials [10].

In certain cases, optimization resulted from the combination of FAST with other techniques in a tailored sequence [11], while other studies concerned more fundamental aspects, leading to the formulation or the review of apposite theoretical models [12-14]. A few coupled modeling with experimental verification [15], sometimes consider-ing even more innovative current-assisted sinterconsider-ing techniques, such as high-voltage electric discharge consolidation, in whose context Grigoryev and Olevsky analyzed the behavior of inter-particle contact zones [16].

Among the approximately 700 yearly publications addressing current-assisted sin-tering techniques, only a few works investigated on the role played by the tooling setup characteristics [17,18] and on the relative possibility of optimization, treated with experimental approaches, numerical modeling or a combination of both.

Such studies analyzed the consequences of modifying the constituent material [19-22], the geometrical design of the components [23,24], the manufacturing machine’s combination with characterization technologies [25],[26], or the scalability of the pro-cess [27].

This last aspect revealed to be of fundamental importance in the understanding of how thermo-electrical phenomena distribute within the powder specimen and the entire setup, since an increase in the characteristic dimensions unfailingly causes the development of augmenting internal temperature gradients, non-uniformities that so far have been assessed only for small sample’s diameters [28].

Some more specific analysis were operated in order to customize the tooling shape for the production of complexly shaped axial-symmetrical parts [26] and of functionally-graded materials [29], evidencing how extensive and versatile FAST-tooling optimiza-tion procedures can be.

More efforts are needed in order to attain satisfactory results in terms of repro-ducible production of bulky parts with homogeneous property distributions, suitable for industrial production.

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CHAPTER 1. INTRODUCTION 18 When considering Field-Assisted Sintering Technique or Spark Plasma Sintering (SPS) equipment, certain distinctions have been individuated and defined, in the context of comprehensive reviews of the various methodologies and patents released since the introduction of these electrically-assisted densification technologies [30,33].

A setup optimization process needs to follow a thorough analysis of the conven-tional configurations’ conditions, in which the influencing factors can be pointed out, classified and investigated.

The present work aimed at mitigating thermal gradients in the powder sample, therefore a preliminary study of the internal temperature distributions was performed. Because of the FAST device intrinsic difficulties in measuring the sample actual tem-peratures, after a few experimental approaches to the temperature distributions’ re-construction [34-37], numerical tools have revealed to be particularly effective in com-pensating for this lack of experimental data [38,39].

Finite-element models have been developed, capable to simulate the electrical and thermal evolutions of the specimen and tooling components during the FAST proce-dures [40-44].

In order to faithfully recreate the experimental conditions, fully-coupled models, combining electrical currents, heat transfer, mechanics and densification kinetics, need to be implemented. A variety of studies have addressed this issue, providing indis-pensable information on the inherent mechanisms of such innovative densification techniques [45]-[59].

Our study arose from actual issues of raising temperature disparities when dealing with sintering on the macro-scale (specimen diameter of several cm, as an order of magnitude). Specifically, non-negligible thermal gradients were appearing, sometimes in the axial direction or in other cases in the specimen cross-section’s radial direction. Such definition arise from the characteristic axial symmetry of the conventional FAST tooling setup.

In all the cases investigated, the powder specimen was constituted by a non elec-trically conductive material, which is typically the most critical situation in terms of severe temperature gradients and localized overheating. When an insulating powder material is located in the conductive tooling, the current is prevented from flowing through the specimen itself, and is therefore forced to addensate in the adjacent tool-ing sections, leadtool-ing to strong non-homogeneities in the temperature distributions. Generally the specimen’s edge is overheated while the center undergoes temperatures that can be even hundreds of degrees lower. Such non-uniformities significantly impact the final outcome in terms of microstructure homogeneity and mechanical properties. Similarly, axial thermal gradients, usually even more severe, can endanger the tooling

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CHAPTER 1. INTRODUCTION 19 setup and therefore damage the whole FAST machine.

In order to address such issues finite element simulations were developed and cali-brated on the experimental results, leading to a reconstruction of full FAST processes and subsequently to a numerical campaign aimed at modifying, improving and op-timizing the classic FAST tooling setup, all the way to the proposal of some novel configurations.

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Chapter 2 Field-Assisted Sintering Technique

Field-Assisted Sintering Technique (FAST), also known as Spark Plasma Sintering (SPS), is a recently emerged powder consolidating technique, which provides poten-tially revolutionary capabilities to the processing of materials into previously unattain-able configurations.

The coexistance of high heating rates with an electric current flow, combined with the possibility of applying high pressures, leads to extremely good results in terms of densification of powder materials, while hampering undesired grain growth phenomena.

It essentially consists of the conjoint application of high temperature, high axial pressure and electric current assisted sintering.

A careful review of the develoment of FAST technologies was conducted by Grasso [4].

According to the open literature, the FAST technology was pioneered by Duval d’Adrian in 1922. However, a more careful review attributed to Bloxam [58,59] in 1906 the first patent on pure direct current (DC) resistance sintering (RS). Thereafter, Tay-lor developed the first resistive sintering process combining a capacitor bank, trans-formers and special switching devices. This originated the so-called electric discharge compaction (EDC), a first rough version of the modern FAST techniques.

The progress of FAST technology, however, was rather discontinuous. Its main driving force was initially motivated by the current industrial needs. Prior discour-aging results were attributed to both the lack of fundamental understanding of basic principles and to poor devices (i.e. energy sources, capacitor switches, controlling systems, etc). These factors initially hindered the industrial development although the inherent benefits and potential of FAST were readily recognized.

Indeed, very few successful applications were reported before 1989. The patent published by Inoue [60] in 1966 added a solid backbone to the existing FAST

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CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 21 nology. It introduced unprecedented technological innovations with new basic sin-tering principles using different electric current waveforms, (i.e. low-frequency ac, high-frequency unidirectional ac and/or pulsed dc). These methods were combined in one sintering process named ‘Electric-discharge sintering’ (EDS), also known as spark sintering (SS). Spark sintering employs a unidirectional pulsed direct current or a unidirectional alternate current, eventually superimposed to a direct one.

This process inspired the development of the subsequent pulsed electric current sintering (PECS) methods, e.g. plasma assisted sintering (PAS), spark plasma sinter-ing (SPS) and plasma pressure compaction (P2C).

The most recent Inoue’s patents have been focused on the design of more effi-cient current sources as well as on the introduction of auxiliary/peripheral devices to increase the reproducibility and controllability of the processes.

FAST/SPS is today considered as a mature and viable manufacturing technology. This is shown by the increasing rate of published journal papers and comprehensive reviews, represented in Figure 2.1. About 4000 papers were published from 1970 to 2013 on FAST. Figure 2.2 shows the contributions of the most involved countries. The number of published papers started to grow exponentially from the early 1990s, the so-called exploitation stage. Similarly, the number of annual international conferences, workshops and specialized symposia increased worldwide—US-ARO ’89, US ARO ’99, Pacrim, Sintering, SPS Forum and NEDO 2000, to mention a few. The same trend is individuated in the patents context, as Figure 2.3 shows. Notice that in this same figure FAST is addressed as ECAS (Elecric Current Assisted Sintering), another denomination for this family of technologies.

Depending on the discharge time, a basic FAST process is classified into two classes, i.e. fast and ultrafast. In our study we are considering the fist case.

Basic FAST apparatuses can be primarily classified with respect to the discharge time, repetition frequency of the pulse trains, electric current density and waveform. Classification based on discharge time is more practical. Conventionally, 0.1 s dis-charge time can be assumed as the threshold between fast and ultrafast technologies. According to several reviews, such technologies may take more than 50 different names. The inherent classification, made with reference to the electric current wave-form and apparatuses, focused on scientific papers rather than on original patents. This number is too large and inappropriate for capturing the essential features of the methods and apparatuses.

The distinction between fast and ultrafast points out the differences in the hard-ware employed in basic field-assisted sintering (e.g. current sources) as well as its features (e.g. current density and waveform, response time, control unit and loading

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CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 22

Figure 2.1: Publications on FAST per year

method). In fast processes, usual discharge time, current density and voltage are on the order of minutes, up to 1 kA cm2 and tens of volts, respectively. The sintering

procedure performed using with either an electrically conductive or insulating powder. As stated before, FAST is a class of consolidation methods in which mechanical pressure is combined with electric and thermal fields to enhance interparticle bonding and densification. The starting materials can be in the form of either powders or green compacts. The primary purpose of imposed electric currents is to provide the required amount of resistive heat. Moreover, electric currents may additionally enhance powder sintering by activating one or more concurring mechanisms, which we will describe later. The overall resistive heat consists of a localized and massive heat. The former is concentrated at particle interfaces and serves to bond particles with each other. The latter promotes plastic deformation upon sintering.

The established advantages of FAST are:

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Figure 2.2: Countries percentages of FAST publications

Figure 2.3: Patents on FAST per year

• The absence of sintering aids;

• Control of the thermal gradient (for functional graded materials (FGMs); • Selective control of the density in specified regions;

• Accurate control of the porosity; • Single step sintering-bonding;

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CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 24 • Particle surface cleaning;

• High heating rate;

• Near-net-shape capability.

The short sintering time is particularly suitable for:

• Preserving initial powder grain size or nanostructure; • Consolidating amorphous materials;

• Improving bonding strength between particles;

• Controlling phase reactions or decomposition (in the case of composites). Materials produced by FAST therefore exhibit improved physical and mechanical properties compared with those obtained by conventional methods.

Furher noteworthy features resulting from FAST include superplastic behaviour of ultrafine ceramic, increased permittivity of ferroelectric materials, improved mag-netic properties of magmag-netic materials, enhanced product-scale bonding, augmented thermoelectric properties, superior mechanical properties and reduced impurities seg-regation at grain boundaries.

Essentially, basic current-assisted sintering exploits the same punch/die system concept as the more familiar hot pressing (HP) process. The two technologies setups are compared in Figure 2.4.

A powder or green compact (which we will generally refer to as “specimen” or “sample” is placed in the die and subsequently pressed between two counter-sliding punches. Mechanical loading is normally uniaxial.

It is well documented that in the HP process the powder mainly densifies ow-ing to a combination of thermal and pressure effects. However, FAST and HP differ significantly in the heating mode. Specifically, in HP an array of heating elements indirectly heats the punch-powder-die assembly by radiation and eventually by con-vection and/or conduction. Conversely, in FAST the punches transfer the electricity and Joule heat directly to the powder. As the supplied current density can be very large, the heating rate in the powder can approach 102 _{K/s (ultrafast process). This}

heating rate is much higher than 80 °C/min for the HP technique. Thus, the sintering time can be lowered and the production rate increased.

A FAST machine is represented in Figure 2.5. The section on which the present study and most of the publications present in the literature are focused is the setup depicted in Figure 2.6, namely the specimen, the tooling surrounding it and the

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Figure 2.4: Hot Pressing and FAST setups

interface with the machine, in a scheme in part (a) and in a photograph in part (b).

The tooling is usually constituted of graphite, but sperimentation has been con-ducted to investigate on the effect of the employment of more resistant materials with analogous thermal and electrical conductivity properties, and it serves as housing to the powder compact.

The single components forming a conventional tooling setup are:

Die a hollow cylinder, which surrounds the powder in the radial direction; Punches coming in pair, two cylinders with the same diameter as the specimen,

impeding the axial sliding of the compact;

Spacers coming in one or more pairs and sometimes denominated “upper punch” and “lower punch”, disks that consent a gradual transition between the

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Figure 2.5: FAST machine

specimen and the electrodes, where the voltage is applied.

The electrodes are made of stainless steel and they act as the interface between the graphite tooling and the main machine. They are cooled though a recirculating water system in order to avoid the high temperatures reached in correspondence of the specimen’s location to cause overheating of the remaining sections of the machine.

A graphite foil is generally located at the interfaces between the sample, the die and the punches, with the aim of preventing adhesion between the parts during the sintering process. A graphite felt is also located around the die in order to contain the heat losses by means of surface radiation. Figure 2.7 shows a setup in which these last two components are clearly represented. The two red arrows indicate the locations on which the pyrometers, responsible for temperature monitoring, are located. Notice that this specific setup was employed in a large section of our study, and will be more carefully decsribed in Chapter 6.

A typical pre-sintering procedure consists in:

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Figure 2.6: Typical FAST tooling setup

2. locating the powder in the tooling;

3. cold pressing the powder woth a few MPa load; 4. locating the tooling in the apposite machine chamber; 5. closing the chamber and create vacuum conditions;

6. setting the temperature and pressure regime on the machine; 7. starting the FAST cycle.

More precisely, FAST can be operated under various atmospheres, such as vacuum, inert gas or air, but the most common condition is under vacuum, since inadequate atmospheres or operating conditions may facilitate sparking.

Many processing parameters playing an important role in such routine can be distinguished.

The heating rate acts as a new important sintering parameter that further extends the potentials of FAST, as it will be clarified in the following paragraphs.

Moreover, the chamber atmosphere, along with the parameters of heating rate and mechanical loading can be adjusted to optimize the procedure or to tailor the material microstructure.

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Figure 2.7: FAST setup in which graphite foil and felt are marked

The electric supply is the primary source of heating of the punches, die and sinter-ing powder. The current density and waveform determine the type of current source. A strong nonlinear interaction exists between heat flow and current through the ma-terial properties (e.g. electrical resistivity, thermal conductivity, density and specific heat), which is also an object of our study, crucial for the finite-element modeling of FAST processes.

The mode of mechanical loading is also a crucial factor. It can be either static, periodic or impulsive. Depending on the loading mode, very different stress states, such as shear, uniaxial or pseudo-isostatic can be induced in the powder.

The system geometry is another fundamental factor affecting the heating method, and the core of the present thesis. In turn, the temperature distribution affects the homogeneity of the final microstructure. In the absence of suitable auxiliary or heating control devices, the product shape is restricted to simple geometries and the product diameter is typically below 2 or 3 cm.

The FAST machine can be operated in temperature-control mode, in which a specific temperature profile is pre-set and the machine regulates the current input accordingly, or current-control mode, in which the electrical input is what the operator

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CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 29 directly regulates.

In any of these modalities, once the sintering cycle has started, a voltage is applied in correspondence of the electrodes and the current flows through the setup in the axial direction. In case of electrically-conductive powders. the current will flow though the specimen itself, while, in case of insulating materials, it will be forced to avoid the powder compact and pass through the adjacent die.

Such effect is the reason lying behind the many theories and studies focused on the individuation of the main phenomena causing densification in a FAST setup.

Discussion is still ongoing on such matter because of the relative novelty of these techniques and because of the high variability of the processes involved when dealing with different materials.

As pointed out by Figure 2.2, Even though the first patents ascribable to FAST/SPS prototypes can be individuated already in the beginning of the 20th century, research on the matter didn’t begin fully until the late ’70s and ’80s, mainly in the former USSR and in Japan, and the real “explosion” of scientific studies in the field-assisted and spark sintering field happened only in the years 2000s, after the first machines started being produced on a regular basis by the japanese Sumitomo Coal Mining Company and later by the German FCT.

A whole variety of approaches and modifications are present in both the academic and the industrial world: resistance sintering electric-discharge, electroconsolidation, discharge powder compaction, electric spark sintering, plasma activated sintering, to mention a few.

Such variety gives a clear and immediate indication of the many theories that have been developed to explain the enhancement of densification processes during FAST.

While for the conventional sintering processes, happening by providing heat to a powder compact by means of a dilatometer, the mechanisms of mass transfer respon-sible for densification are considered well-assessed, the same cannot be stated with regards to field assisted sintering.

It’s worth having here a brief description of these conventional sintering phenom-ena, since several, if not all, of them can be also occurring in the most advanced FAST technologies.

Sintering requires mass transfer through diffusion, which can happen along many distinct paths (Figure 2.8), among which the principal ones are:

Surface-diffusion - typically responsible for the pheroidization of the pores inside the compact;

Volume-diffusion - or lattice-diffusion, effective in densification but the slowest among all the involved diffusion processes;

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CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 30 Boundary-diffusion - happening at the grain boundaries, more energetically-active than the interior of the grains and therefore the mechanisms responsible for most of the consolidation process;

Evaporation-condensation - from the surface of powder particles, generally negligible, unless specific dopants are added to the powder mixture.

Figure 2.8: Mass transport mechanisms in sintering

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con-CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 31 nection between particles growing during the process (Figure 2.9, a-d) and responsible for the strengthtening of the material. It must not be forgotten that the driving force for sintering is the decrease in surface energy to which powder materials spontaneously tend when heat is provided, and that the neck growth corresponds to a reduction of the overall free surface of the specimen.

Figure 2.9: Necks formated during the FAST of vanadium carbide

The four above-mentioned mechanisms are the ones relative to sintering in its purest definition, reduction of surface tension-driven, what is often referred to as free sintering. “Free” implies without the application of a load. Nevertheless, in reality a force, even only a few MPa, is applied to the specimen in order to accelerate and favor densification and avoid grain growth, therefore another mechanism that needs

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CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 32 to be taken into account is viscous flow, Newtonian or non-Newtonian depending on the entity of the loads and on the nature of the material involved.

FAST sintering involves experimental conditions more complex than in traditional sintering, so a higher number of mass transfer mechanisms might be involved, which would explain the charactiristic enhancement of matter transport and consolidation attained thanks to this technique. The number of involved mechanisms and their relative relevance is still being debated, mainly because of the high dependence on experimental conditions and on the material physical and chemical properties.

A first classification of such mechanisms can be based on the distinction between thermal and field effects.

Among the thermal effects one can list high heating rates, high local non-uniformities of temperature distribution, local melting and sublimation, macroscopic temperature gradients, thermal diffusion and thermal stresses.

The influence of high heating rates has been broadly assessed through experimen-tal investigations. It has been shown that an increase in such parameter considerably augments the consolidation rate of both conductive and non-conductive powders dur-ing FAST. A well-known example is the case of alumina, a material with which most of the initial FAST studies have been conducted. It was shown that the increase of heating rate from 50 to 300 °C/min, with the same maximum temperature and the corresponding six time decrease of sintering time, allowed obtaining the same final density. Physically, this was attempted to be explained as a result of the existence of additional defects in the material, directly related to the high heating rates and short processing times. A series of fundamental studies in this same field has been conducted by Gillia and Bouvard, who carried out a number of comparative experi-ments on sintering of WC-Co powder systems and varying the heating cycles. They employed cycles with the same with the same average heating rate but with distinct temperature histories, namely by utilizing sequences of steady ramps and isothermal periods. Their results showed evidence of the dependence oh the densification rate on the average heating rate, but an absence of dependence on the temperature history.

Thermal diffusion is another aspect of great importance in the field of these in-novative sintering techniques, which we will only briefly summarize here. The main phenomenon laying behind the enhancement of thermal diffusion is the Ludwig-Soret effect, predominant especially in two-components systems subject to temperature gra-dients. More generically, it is currently agreed that the intensity of this type of dif-fusion increases if the pulse frequencies are higher. Thermal difdif-fusion also promotes components’ separation, at the level of atoms and vacancies scales. At the early stages of sintering, this leads to the growth of inter-particle necks, and therefore to

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CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 33 the enhancement of the consolidation process. Nevertheless, at the final stages the pores may serve as vacancy sinks under thermal diffusion conditions, a phenomenon that hampers sintering. It is therefore possible that higher pulse frequencies and their effects on thermal diffusion phenomena have contrasting effects on the densification od the powder compacts, depending on the stage of sintering considered.

In the field effects category the most effective cases seem to be arc and spark discharge, electromigration and its consequent diffusion enhancement, electroplasticity (namely electron wind and magnetic depinning of dislocations), dielectric breakdown of oxide films at grain boundaries, surface plasmons.

To such list one needs to add the role played by pressure and stress gradients. The combination of a number of the above-mentioned mechanisms lead to a series of effects, in particular surface activation, high-speed diffusion and material transfer, efficient heating, plastic deformation promotion, high-density energy supply and quick cooling of the intergranular bondings. Futher consequences of the cooperation of a number of these effects end up constituting the main advantages of FAST methods:

• Short-time sintering;

• Increase in the sinterability of traditionally non-sinterable materials; • Low-temperature consolidation;

• Uniformity of the final densified product; • Bonding of dissimilar materials;

• Sintering of metastable phases; • Sintering of amorphous materials.

An important phenomenon in order to provide some orders of magnitude for the voltages and the amperages involved in FAST processes is the arc, glow and spark discharge.

Arc-discharge - high current (> 10 A) and low voltage (< 20 V);

Glow-discharge - low current (< 1 A) and relatively high voltage (300 V;

Spark-discharge - a transient condition with very high current (order of kA) and voltage (kV/cm).

The most common mechanism is the arc-discharge, since FAST machines usually operates with a voltage in the 5-15 V range, which is also the case of the present

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CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 34 study. Such low imposed fields are sufficient to cause polarization effects, and therefore localized fields orders of magnitude higher. The electric current delivered by the machine is always pulsed.

Evidence supporting the presence of discharge can be seen in Figure 2.10, where electric discharge patterns are shown on a CeSiNO2 specimen (top) and on

polyethy-lene fibers (bottom).

Figure 2.10: Evidence of discharge presence in FAST

A number of studies has been dedicated to the electromigration and electric field-induced diffusion effects, since they appear to be responsible for a significant enhance-ment of mass transfer through diffusion. Such investigations are beyond the purposes of the present study.

The main subject of this investigation lies, instead, in another important phe-nomenon, listed in the thermal effects category: temperature gradients.

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CHAPTER 2. FIELD-ASSISTED SINTERING TECHNIQUE 35 Microscopic thermal gradients happen within the single powder particle, or grain, dependending on the stage of the process that is being considered. The main causes have been individuated in the cross-section restriction and the additional grain bound-ary’s resistance, as depicted in Figure 2.11. A smaller cross-section available for the current to flow creates an abrupt increase in current density and consequently leads to a peak in Joule heating effects, causing localized overheating at the particle con-tacts and at the necks. Higher temperatures at the bridges among particles allow localized enhanced diffusivity, melting and vaporization, as one can infer from Figure 2.12, showing a micrograph of broken necks in a spherical powder compact. A grain boundary is, instead, an area of atomic disorder, through which the electric current encounters a higher resistivity, therefore leading, again, to more consistent Joule heat-ing. Together with what mentioned with regard to the cross-section restriction case, it’s worth noticing here that such higher resistivity can be due to an surface film of oxide, often covering a conductive core, and that localized overheating can alter such thin layer and subsequently the flow of current inside the powder material. Such microscopic temperature non-uniformities have been addressed as related to retarded grain growth and flash sintering, an interesting version of FAST processes that allows almost complete densification of samples in a timeframe of a few seconds.

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Chapter 3 Finite-element modeling of FAST

As described in the previous chapter, Field-Assisted Sintering Techniques procedures involve a wide variety of physics, spanning from electric currents, to heat transfer, to mechanics and densification kinetics. Such complexity renders the modeling of FAST techniques extremely non-trivial.

On the other hand, these sintering processes are both expensive to be conducted and non immediate to manipulate, analyze and control. Temperature distributions themselves, the principal aim of the present work, are one of the most coumbersome issues that one needs to address in order to fully understand and characterize pro-cedures and outcomes. Indeed, while an experimental setup can only monitor, and usually pre-set, the temperatures reached at one or two specific points of the tool-ing, the actual thermal conditions experienced by the specimen cannot be accessed at the current state of the art. This is the reason lying behind the broad interest lately developed in virtually reconstructing what occurs during a variety of FAST experiments.

An appropriate tool for the purpose of implementing this kind of simulations has been individuated in the finite element methods (FEM). Specifically, a commercial software was selected, COMSOL Multiphysics®, thanks to its distinguishing capa-bilities of simultaneously combining a variety of physical processes.

The models we built and that will be described in the following chapters are all three dimensional, macro-scale and fully coupled. The choice of three-dimensionality, instead of a less computationally heavy two-dimensional axy-symmetric geometry, was forces by the absence of axy-symmetry for certain punch geometries, as will be shown in Chapter 6.

By specifying that we are operating on the macro-scale we intend to distinguish our modeling procedure, including the entire FAST graphite tooling setup and the powder specimen, from the studies that focus on the sole evolution of the sample

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CHAPTER 3. FINITE-ELEMENT MODELING OF FAST 39 itself, by investigating on the microstructural evolution during the sintering process.

Fully coupled means that all the physics involved in the FAST processes have been included, therefore creating an electrical-thermal-mechanical set of simulations, in or-deer to closely replicate the experimental procedures previously conducted. Boundary, contact and initial conditions served at completing the reconstruction in every detail. The entire FAST tooling setup was reconstructed, comprehensive of powder com-pact specimen, punches, die and spacers. COMSOL offer the possibility of building or importing the CAD geometry, which in our case was realized in the SolidWorks® environment and subsequently transferred in the FEM context. When a geometry is constituted by several components, like in our case, after the import step two options are available: union or assembly. “Union” sees the whole geometry as a single part, while “assembly” distinguishes the various components as they were created by the CAD. This second approach was the one selected in our work, since it enabled us to imbed contact conditions among the various graphite tools and consequently im-plement contact conditions in between them, which play a crucial role in the FAST processes and therefore in the fitting of the experimental data, as will be demonstrated in the following chapter.

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CHAPTER 3. FINITE-ELEMENT MODELING OF FAST 40 Such contact conditions were also utilized to simulate the presence of the graphite foil, located at the specimen-punch, specimen-die and punch-die interfaces in order to prevent sticking and contamination among different materials, was simulated through an appropriate distributed-impedance contact condition.

The graphite felt thermal insulation was replicated with the corresponding bound-ary condition.

As anticipated, the FEM simulation of spark plasma sintering procedures requires the coupled implementation of an electrical module, describing the current flowing through the FAST setup, together with a thermal one, responsible for the consequent heat transfer by Joule heating phenomena, a mechanical section, allowing the applica-tion of external loads, the descripapplica-tion of the interacapplica-tions between components and the introduction of the powder compact constitutive behavior, and finally a mathematical section, reconstructing the densification kinetics.

The required equations are presented here. For the DC current distribution we use

J = σ (−∇V ) (3.1)

which is a result of the respective conservation equation, by expressing the electric field as the gradient of the electrostatic potential. Here, J (A/m2_{) is the current}

density, σ (S/m) the electrical conductivity and V (V) the voltage.

Coupled to equation 3.1, the heat transfer due to Joule heating is given by the equation for heat transfer by conduction

ρef fCp

∂T

∂t = ∇ · (k∇T ) + ∇ · J (3.2)

in which ρef f (kg/m3) is the effective density of the material, Cp (J/kg/K) is the

heat capacity, T (K) the temperature and k (W/m/K) the thermal conductivity. The effective density is simply the bulk material density when we consider the graphite tooling, but once we shift to the description of the powder compact which will undergo the sintering process, the presence of porosity must be taken into account.

A few definitions are here necessary:

ρth theoretical density of the bulk fully dense material, kg/m3;

ρef f effective density, taking into account the presence of pores inside the

ma-terial, kg/m3_;

ρ relative density, the volume fraction of fully dense material in the domain considered, dimensionless;

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CHAPTER 3. FINITE-ELEMENT MODELING OF FAST 41

θ porosity, the volume fraction of voids in the domain voids, complement to 1 of the relative density, dimensionless.

The three concepts are therefore related as follows

ρef f = ρthρ = ρth(1 − θ) (3.3)

Figure 3.2 summarizes these concepts. In the bottom part it also reports some mi-crographs in which porosity can be clearly identified, and a plot that distinguishes how the amounts of closed and open porosity evolve with time during a typical sintering process. It is worth reminding here that open porosity indicates a net of intercon-nected voids that ultimately can reach the outer boundary of the domain, while closed porosity is constituted by isolated pores. The plot shows how usually open porosity disappears when roughly 8% of global porosity is left in the specimen. Such conclusion is nevertheless under discussion in the more modern FAST technologies, which seem not to respect this rule of thumb.

The porosity evolution is defined through the continuity equation, from which the progressive densification of the powder compact can be derived, and which is expressed as

˙ θ

1 − θ = ˙εkk (3.4)

As for the mechanics model, we assume graphite not to undergo any creep phe-nomena, a hypothesis that was experimentally verified, while a specific constitutive equation is needed for the specimen.

The constitutive equation for the powder specimen has the form taken from the continuum theory of sintering [90], which treats a powder compact to be sintered as a continuum media, by defining bulk and shear moduli as function of porosity.

It is expressed in the following manner

σij = σ(W ) W h ϕ ˙εij + ψ − ϕ 3 ˙ εkkδij i + PLδij (3.5)

An explanation of every variable is given: σij stress tensor components, Pa;

˙

εij strain rate tensor components, 1/s;

σ(W ) equivalent stress, Pa, the term which describes the constitutive behavior of the bulk material of the powder particles. It can assume different ex-pressions depending on the type of sintering process, either free sintering,

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Figure 3.2: Porosity definitions

sintering of amorphous materials, hot or cold pressing. In our case we are dealing with high temperatures and pressures applied to a crystalline material, therefore a power law creep-like behavior is selected, described as

AmWm (3.6)

with

Am power law creep coefficient, related to the material’s activation

energy (Q, J/kg/K) through an Arrhenius-type relationship, Am = Am0exp _RTQ

;

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W equivalent strain rate, 1/s, talking into account the presence of porosity and aiming at switching from a tensor to a scalar quantity W = s ϕ ˙γ2 _{+ ψ ˙}_ε2 kk 1 − θ (3.7)

where ˙γ is the strain rate tensor deviator (1/s) and the remaining quanti-ties are defined at the next point;

m power law-creep exponent, a number comprised between 0 and 1, or, in some more advanced theories, a function of temper-ature. For our purposes, mainly concentrated on the graphite tooling intsead of on the sintered specimen itself, this approx-imation of the creep exponent as a constant does not cause significant alterations of the results. Nevertheless, a deeper investigation of finite element methods intended to implement the power-law creep mechanisms could lead to considerable improvements in FAST simulations.

σ(W )

W therefore becomes AmW m−1_.

It is important to notice that the power law creep relation is here inverted with respect to the conventional case, in which strain rate is expressed as a function of stress. This means that the exponent m is the reciprocal of the more commonly used n, always greater than 1 because of the intrinsic non-linearity of creep phenomena.

ϕ normalized shear viscosity, a dimensionless function of porosity defined as

ϕ = (1 − θ)2 (3.8)

ψ normalized bulk viscosity, a dimensionless function of porosity defined as ψ = 2 (1 − θ)

3

3θ (3.9)

PL sintering stress (or Laplace pressure), Pa, taking into account the actual

driving forse of sintering, namely the tendency to decrease the sample’s free surface energy, given as a function of powder particles size (r0, m),

surface tension (α, J/m3) and porosity

PL=

3α r0

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δij Kronecker delta.

It is worth mentioning that several models exist for the porosity-dependent material properties listed above. The graph given in Figure 3.3 shows a comprehensive list of bulk modulus and sintering stress expressions, developed by a variety of authors, together with a plot that shows how well they approximate the experimental results. Notice that the expressions we selected, theorized by Skorohod, generally constitute a reasonable choice, thanks to their “intermediate” collocation among all the pro-posed models. Research is anyways ongoing in order to develop more precise models, especially for the innovative FAST densification processes.

Figure 3.3: Models of bulk modulus and sintering stress

The material properties were considered as function of temperature and, for the specimen, porosity, and will be given for every part of the study in the respective chapters. Such choice is based on the fact that two different setups were employed as

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CHAPTER 3. FINITE-ELEMENT MODELING OF FAST 45 source of experimental data, depending of where the temperature gradients issue was happening.

The axial temperature inhomogeneities were encountered in a 4 cm-diameter setup while sintering alumina (Al2O3).

The radial thermal gradients were a substantial issue in a 6 cm-diameter setup during the processing of silicon nitride (Si3N4) powders.

Notice that in both cases the material involved was electrically insulating with respect to graphite. Therefore, as mentioned in the introduction, current could not flow through the sample itself, and was forced to follow a path inside the die, leading to a concentration of the Joule effect in the external area and a consequent overheating of the sample’s edge with respect to its center. Such gradients will be the subject of Chapter 6.

The imposed boundary and initial conditions reflected the actual experimental framework employed during the FAST procedures, and they are listed according to the section in which they were defined.

In the electrical currents module, the initial voltage was set to zero, insulation of the outer tooling surfaces was included, the bottom surface of the lower spacer was grounded, the current input (extracted from the FAST machine readings during the experimental process, Figure 3.4) was applied to the top surface of the upper spacer, and an electrical contact resistance was imbedded at the interfaces between the tooling components as a function of temperature and pressure. This last point required itself a separate study and will be introduced in the next chapter.

As for the heat transfer module, the initial temperature was set to be 25 °C, the same constant temperature of 25 °C was imposed at the bottom surface of the lower spacer and at the top surface of the upper one, corresponding to the cooling effect of the circulating water in the FAST machine, the outer surface of the die was thermally insulated, as the presence of the graphite felt imposes, ideal thermal contact between layers were implemented, since the role of thermal contact resistance was proven to be negligible (Chapter 4), and the external surfaces were endowed with a heat radiation boundary condition, according to the Stefan-Boltzmann formulation

−n · (−k∇T ) = ν T4 0 − T 4 w (3.11) where the left-hand side is the heat flux per unit area, ν is surface emissivity, set to be 0.8, is the Stefan-Boltzmann constant, 5.6704 × 10−8 W/m2/K4, T

w and T0

are the temperature of the external die surface and the ambient temperature (25°C), respectively.

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Figure 3.4: Imposition of electric current input at the tooling upper surface

was introduced, the bottom surface of the lower spacer was fixed, the top surface of the upper spacer was subjected to an externally applied pressure equal to the load utilized in the experiments, and the contact between the different tooling components was ensured.

The calculations correspond to an assumed conventional for FAST experimental setup when the temperature measurement was operated by the pyrometer, focused on the hole on the die surface. The processing is therefore supposed to be conducted in temperature-control mode (not current-control), which means that the temperature profile is imposed by setting it on the machine control panel.

The macro-scale model of FAST eventually enables the calculation of the evolution of voltage (V ), temperature (T ), stress (σ) and porosity (θ).

Figure 3.5 provides a picture of the COMSOL interface ready to simulate the procedure.

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Chapter 4 Electrical Contact Resistance

In the previous chapter, the fundamental role played by the electrical resistance among the various graphite tooling components has been pointed out.

The general lack of a uniform and complete description of such phenomenon im-posed a preliminary step to our main study, namely the derivation of a formulation for the contact resistance to be imbedded in our FEM simulations. We will show here how essential this preliminary investigation turned out to be in order for the numerical modeling to successfully reproduce experimental conditions.

The present chapter follows the steps of our investigation. After a literature re-view on the topic of contact resistance descriptions and implementations for FAST processes, a report of how the formulation was derived is presented.

The main part of the study was conducted by means of experiments in the Powder Technology Laboratory at San Diego State University, USA.

A simple numerical set of simulations was finally imbedded, in order to assess what experimentally obtained and compare the roles of electrical and thermal contact resistances.

4.1 The role of electrical contact resistance in

FAST procedures

A general agreement has been reached in the FAST research field about the funda-mental role played by contact resistance in a typical graphite (or other materials) tooling setup, but a uniform approach on how to treat and model such parameter was still absent.

The contact resistance of the FAST system can be classified according to many criteria:

A Fully-Coupled Finite Element Analysis of Field Assisted Sintering Techniques

Università di Pisa

Dipartimento di Ingegneria Aerospaziale

“Lucio Lazzarino”

A Fully-Coupled

Finite Element Analysis

of Field Assisted Sintering Techniques

Prof. Ing. Massimo De Sanctis

Prof. Ing. Luigi Lazzeri

Prof. Eugene A. Olevsky

Diletta Giuntini

Preface

Contents

List of Figures

List of Tables

Summary

Chapter 1

Introduction

Chapter 2

Field-Assisted Sintering Technique

Chapter 3

Finite-element modeling of FAST

Chapter 4

Electrical Contact Resistance

4.1

The role of electrical contact resistance in

FAST procedures