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Article / Book Information
題目(和文)

微構造の3次元可視化による焼結の熱力学的駆動力の解明

Title(English)

Clarification of thermodynamic driving force for sinteringfrom 3-Dvisualization of the microstructure

著者(和文)

大熊学

Author(English)

Gaku Okuma

出典(和文)

学位:博士(工学),学位授与機関:東京工業大学,
報告番号:甲第10999号,
授与年月日:2018年9月20日,
学位の種別:課程博士,
審査員:若井 史博,東 正樹,尾中 晋,安田 公一,吉田 克己,西山 宣正,田中諭

Citation(English)

Degree:Doctor (Engineering),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第10999号,
Conferred date:2018/9/20,
Degree Type:Course doctor,
Examiner:,,,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Science Tokyo Research Repository)

Clarification of thermodynamic driving force for sinteringfrom 3-D visualization of the microstructure

微構造の3次元可視化による焼結の熱力学的駆動力の解明

Tokyo Institute of Technology
School of Material and Chemical Technology

Gaku OKUMA

Contents

Contents
Chapter 1 Introduction
1.1
Mechanical reliability of ceramics
1.2 What is sintering?
1.3 Continuum mechanics of sintering
1.4 Three-dimensional visualization of sintering
1.5 Objective of this research
1.6 Abstract of doctoral dissertation
References

1
1
2
5
10
12
13
15

Chapter 2 Interface topology for distinguishing stages of sintering
2.1
Introduction
2.2
Experimental procedure
2.3
Results
2.3.1
Microstructural evolution in sintering
2.3.2
Euler characteristic for describing interface topology
2.3.3
Transition between different stages
2.4
Discussion
2.4.1
Comparison with other sintering mechanisms
2.4.2
Euler characteristic per unit volume
2.4.3
Simulation of topological evolution
2.4.4
Topology and sintering mechanics
2.5
Conclusions
References

30
30
32
34
34
36
38
39
39
40
41
43
45
46

Chapter 3 Computation of sintering stress and bulk viscosity foreach stage of sintering
3.1
Introduction
3.2
Experimental procedure
3.3
Results
3.3.1
Evolution of pore morphology observed by 3D visualization
3.3.2
Relative density and specific surface area
3.3.3
Curvature of pore surface
3.3.4
Sintering stress derived from three methods
3.3.5
Bulk viscosity

59

i

59
61
62
62
62
63
64
64

Contents

3.4
Discussion
3.4.1
Sintering stress in the initial stage of sintering
3.4.2
Sintering stress in constrained sintering
3.4.3
Pore coarsening in constrained sintering
3.5
Conclusions
References
Chapter 4 Experimental verification of sintering stress andbulk viscosity estimated from X-ray microtomography
4.1
Introduction
4.2
Mechanics of sinter forging
4.3
Experimental procedure
4.4
Results
4.5
Discussion
4.5.1
Methods to estimate sintering stress from the microstructure
4.5.2
Comparison with sinter forging results
4.5.3
Difference between the viscous sintering and the sintering of
crystalline materials
4.6
Conclusions
References

65
65
67
68
70
71
82
82
83
85
86
88
88
89
90
92
93

Chapter 5 Determination of the size of representative volume
element for viscous sintering
5.1
Introduction
5.2
Experimental procedure
5.3
Results and discussion
5.3.1
Micro structural evolution in viscous sintering
5.3.2
Relative density, specific surface area and sintering stress
5.4
Conclusions
References

103

Chapter 6

120

Summary

Acknowledgements

103
104
106
106
107
110
111

128

ii

Chapter 1

Introduction

Chapter 1

Introduction

1.1 Mechanical reliability of ceramics
Ceramics has been widely used in modern industry due to its excellent
characteristics. Ceramic materials are produced by powder processing, followed bysintering, that is, the thermal process of transforming powder compacts into complex
shaped components (Fig. 1.1).1 Powder processing involves four steps as follows; thepowder manufacture, the powder preparation for consolidation, the consolidation to anengineering shape, and the densification development that eliminates void space. 2
However, each step in powder processing originate the microstructural heterogeneities.
This heterogeneous density distribution may lead to inhomogeneous shrinkage, andgenerate internal stress during sintering. Since ceramics is a brittle material, strength
depends on small flaws generated by internal stresses during sintering. Therefore, themechanical reliability of ceramics depends on the control of heterogeneity inprocessing. Evans3 described fracture occurs from void and cracks around inclusions
inside specimens, as observed in Figure. 1.2. The difference in the strain rate produces
stresses around rigid inclusions in sintering of composites, in sintering of thin films on
substrates, and in sintering of multilayered systems such as low temperature co-fired
ceramics (LTCC) and solid oxide fuel cells (SOFC). These stresses modify theshrinkage rate. They can generate cavities, cracks, delamination, and warping of thestructure, even though there is no large void or cracks in the initial powder packings.
Therefore, it is most important to understand what is the micro mechanics behind thedefect formation during sintering.
1

Chapter 1

Introduction

1.2 What is sintering?
Sintering is one of the oldest technologies dating back to around 10,000years ago,
nevertheless, the scientific approach to this technology has started only after 1940s.
Sintering is known as a common technique of consolidating powder compacts bythermal energy. This technique provides an engineering process for the production of
not only ceramics, but also metals, glasses, polymers and composites. A huge numberof particles undergoes a change in shape at elevated temperatures by matter transport
driven by surface tension, the kinetics of which is controlled by viscous flow for
amorphous particles4,5,6 or diffusion mechanisms for crystalline particles7,8,9. Since
sintering occurs so as to decrease the total interface energy, surface area per unit
volume, which is called specific surface area SV , decreases with sintering time.
Concurrently, the total pore volume decreases, thereby, the sintering process is
described as densification using the relative density ρ as a state parameter. Then, ρ
increases with time and approaches towards the maximum value of 1.
In 1961, Coble8 illustrated in Figure 1.3 the evolution of particle-pore structure
schematically, and identified three stages of sintering, (i) initial stage, (ii) intermediatestage, and (iii) final stage. Coble described about three stages of sintering, as follows;(i) Initial stage
Initial stage is characterized as the formation and growth of contact between
neighboring particles, where ρ increases from 0.5 to 0.6. In the initial stage, themechanism of material transport during sintering of crystalline particle is presented asfollows; lattice diffusion, grain boundary diffusion, surface diffusion, and gas phasetransport. On the other hand, the viscous flow mechanism, which was first proposed by
Frenkel9, can be applied in the sintering of viscous materials like glass.
2

Chapter 1

Introduction

Fig. 1.4 illustrates the material transport paths during sintering10. The material transport
due to the difference in interface curvature occurs under the parallel actions ofvarious mechanisms, but the dominant mechanism depends on particle size r0 , neckradius (contact radius between particles c), temperature and time11. Some of thesematerial transport mechanisms contribute to densification and shrinkage while others
do not. The interparticle distance can be reduce only by bulk material flow via
viscous flow or by grain boundary diffusion from the grain boundary to the surface.
The materials transport from the particle surface to the neck increases the neck size,
but do not contribute to the shrinkage of the interparticle distance. Therefore, thegrain boundary is the source of material transport for densification and shrinkage incrystalline powder compacts. The lattice diffusion of atoms from the grain boundary
to the neck allows the boundary to act as a site for vacancy annihilation. The
shrinkage kinetics is expressed as follows12;

Dl  Dl γ sVm  1 2=tl  RTa 3 
12

(1.1)

where l is the sample size, γ s is the solid surface energy, Dl is the lattice diffusion
coefficient, Vm is the molar volume of the solid, and a is the particle radius.
In some respects material transport from the grain boundary to the neck by grainboundary diffusion is similar to diffusional creep by grain boundary diffusion.13
For the grain boundary diffusion from grain boundaries, the shrinkage kinetics is
expressed as follows;Dl  3Dbδ bγ sVm  1 3t=l  4 RTa 4 
13

(1.2)

3

Chapter 1

Introduction

where Db is the grain boundary diffusion coefficient and δ b is the diffusion thickness ofgrain boundary diffusion. In this case the material transported to the neck surface bygrain boundary diffusion should be redistributed via another mechanism. If theredistribution of material is not fast enough compared to the material transport by grainboundary diffusion, this secondary redistribution may control the neck growth.
For the viscous flow mechanism, if the material follows the behavior of a Newtonian
fluid, the shrinkage kinetics is expressed as follows4;∆l γ s
≈
t
l
µa

(1.3)

where µ is the viscosity of material, a is the particle radius, and t is the sintering time.
(ii) Intermediate stage
In the intermediate stage, the pore structure evolves into an interconnected channel withcylindrical pore lying along three grain edges (see Fig 1.3(b)). Here, for the atom flux,
two mechanisms are available: lattice diffusion and grain boundary diffusion14.
For the lattice diffusion, the densification rate is expressed as follows;d ρ 336 Dl γ sVm
=
dt
RTG 3

(1.4)

where d ρ dt is the densification rate, Dl is the lattice diffusion coefficient, Vm is themolar volume of the solid, G is the grain diameter. In reality, grain growth during
sintering of crystalline particle, and hence, G is not constant parameter.
(iii) Final stage
The final stage begins when closed pores are formed by pinch-off of interconnected
pore channels at ρ = 0.9. The final stage of sintering of crystalline particles is usuallyaccompanied by coarsening and grain growth15, 16. Classical sintering theories, which
predict the rate of density change, have been proposed with the assumption of4

Chapter 1

Introduction

simplified geometrical model for each stage8, 16. For this model, the densification rateis expressed as follows10;d ρ 288 Dl γ sVm
=
dt
RTG 3

(1.5)

This equation indicates that the densification rate is inversely proportion to the cube ofgrain size. This result is the same as that found for the dependence of neck growth andshrinkage on particle size in the initial stage model.

1.3 Continuum mechanics of sintering
Continuum mechanics is a powerful method to predict the macroscopic
dimensional changes, and also to analyze internal stresses generated during sintering ofheterogeneous porous materials. When the microstructure is isotropic, the macroscopic
deformation during sintering is expressed by the following constitutive equation17-19:
E ij =

Σ ′ij
2G

+ δ ij(Σ − Σ )
s

m

(1.6)

3K

where E ij is the macroscopic strain rate, Σ'ij and Σ m are deviatoric and hydrostatic
components of macroscopic stress, Σ s is the sintering stress, and G and K are the shear
viscosity and the bulk viscosity, respectively. Note that the sign convention used forsintering stress, or sintering pressure,20 is opposite to that used for stress, since pressure
is positive if it is compressive. The densification rate is given by the trace of themacroscopic strain rate tensor ρ / ρ = −3E . In the absence of external and internal
stresses, the true strain rate E in free sintering is proportional to the sintering stress:−ΣE f =
3K

s

(1.7)

5

Chapter 1

Introduction

Here, Σ , G, and K, are not constant, and depend on the relative density, ρ. Theysdepend on different factors, such as the nature of powders, powder processing, particle
packing, sintering temperature, and grain growth.
Practically, the continuum model is useful in analyzing the effect of applied stressin stress-assisted sintering, such as hot pressing, spark plasma sintering, and hot
isostatic pressing. It is also useful to predict the internal stress, which is generated fromthe difference in shrinkage rate in constrained sintering. In microscopic scale, void andcracks are generated by the microscopic stresses during constrained sintering. 17,21
For example, in the sintering of thin film on substrate (Fig. 1.5(a)), tensile stress isgenerated inside the film. This stress is proportional to the free sintering strain rate E ,which is caused by the sintering stress Σ s . Here, the sintering stress is driving force forshrinkage, but it generates internal stress Σ film in the film which suppresses shrinkage inconstrained sintering. The internal stress is given by17:EΣ film = E f
1 −ν

(1.8)

where E f is the free sintering strain rate (Eq. (1.7)), E is the uniaxial viscosity, and ν
is the viscous Poisson’s ratio. The internal stress suppresses shrinkage in constrained
sintering, and induces cracks, debonding, or delamination in co-sintering of low
temperature co-fired ceramics and solid oxide fuel cell.
The continuum mechanical parameters for sintering, Σs, G, and K, are notconstant since they depend on the relative density of the compact. These sinteringparameters are physical quantities that can be determined experimentally by sinterforging, in which strain rates are typically measured at a fixed applied stress for a
specific microstructure. Figure 1.6 shows the uniaxial viscosity and sintering stress of6

Chapter 1

Introduction

alumina powder during sintering determined by sinter forging.22 This experimental
result shows the uniaxial viscosity increases with increasing the relative density (Fig.
1.6 (a)), and the sintering stress varies with the relative density (Fig. 1.6 (b)). Sintering
stress of pressure-filtrated compact increases, and becomes very large, more than 30
MPa. This stress is almost the same with that in hot pressing and spark plasma sintering.
On the other hand, the sintering stress of dry pressed compact is small and does not varysignificantly with densification. Their result is very important, because it shows thatsintering stress depends on powder processing. The starting powder is most importantfor sintering. Furthermore, there are many powder processing methods; press forming,
tape forming, slip casting, injection molding, and extrusion. The powder packing, which
is either isotropic or anisotropic, depends on these processing methods. The
macroscopic sintering stress and viscosity depends on the powder packing. The
macroscopic shrinkage depends on sintering stress and viscosity. Therefore, the increaseof sintering stress by optimized powder processing is the core technology ofpressureless sintering. After all, clarification of the relationship between macroscopic
and microscopic sintering parameters, such as sintering stress and bulk viscosity, is very
important.
The sintering stress in macroscopic scale arises from the microscopic pore
structures. The sintering stress can be defined rigorously for pore structures inequilibrium where the mechanical force just balances the surface tension forces, so thatthe porous materials do not shrink. The sintering stress is calculated from curvature andenergy for periodic porous structures under the constraint of fixed volume, where thecurvature is constant at any point on pore surface.23 In the constant mean curvature(CMC) surface, the sintering stress Σs for viscous sintering of glass is calculated from7

Chapter 1

Introduction

the curvature κ:Σ s =γ sκ(1.9)

where γs is the surface energy. Alternatively, the sintering stress is given by24:=
Σ s rγ sκ +2γ s Apore(1.10)

3V

where Apore is the total surface area in a volume element V.
Substitution of Equation (1.9) and r= (V − V pore ) / V into Equation (1.10) gives25:
2γ s AporeΣs =3V pore(1.11)

where Vpore is the total pore volume in the volume element. The porous microstructure ischaracterized by the specific surface area SV ≡ Apore V , then, Equation (1.11) isexpressed as follows26:

2γ S
Ss = s V
3(1 − ρ )(1.12)

The curvature method (Equation (1.9)), the mixed method (Equation (1.10)), and thesurface energy method (Equations (1.11) and (1.12)) give the identical value for the
CMC surface of glass.
However, real porous structures are nonequilibrium, nonperiodic, and nonuniform,
then the curvature is not constant on pore surface. Here, we consider the final stage ofsintering where closed pores are dispersed in a viscous material as shown in Fig. 1.7(b)25. Sintering stress of a pore of arbitrary shape is defined as:

=
s ijs

1
γ s (d ij − ni n j ) dA
Vn A∫n(1.13)

8

Chapter 1

Introduction

where ni is the unit (outward) normal to the pore surface. The macroscopic sintering
stress in V is defined as the volume average of local sintering stress of pores:

1Σijs = ∑ Vns ijs
V pore n(1.14)

where V pore = ∑ n Vn is the total pore volume.
A non-spherical pore has deviatoric components of local sintering stress. However, themacroscopic sintering stress is hydrostatic in isotropic structures, because a random
distribution of arbitrary shaped pores does not confer any directional macroscopic
properties. And hence, the hydrostatic sintering stress Σ s is expressed as Eq. (1.11), andΣ s is simply calculated from the total surface area and the total pore volume. Thus, theenergy method (Equations (1.11) and (1.12)) can be applied to nonequilibrium process
of viscous sintering in the final stage, where closed pores with different sizes are
dispersed randomly in a viscous material.

9

Chapter 1

Introduction

1.4 Three-dimensional visualization of sintering
The microstructural evolution during sintering has been usually observed by 2D
imaging techniques such as SEM and optical microscopy. Although the quantitative
microscopy provides geometric quantities from the 2D images, the understanding ofcomplicate 3D microstructure was difficult.
Recent advances in X-ray microtomography enable us to observe the internal
structures of the actual sample by three-dimensional (3D) visualization. Here, X-ray
microtomography uses X-ray beam to create (2D) cross sectional image of the object
that can be used to reconstruct a 3D model (Fig.1.8). FIB-SEM tomography is also
known as a micrometer or nanometer scale imaging technique which combines afocused ion beam (FIB) with a scanning electron microscope to obtain information on
the internal structure of specimen.27 Although FIB-SEM tomography provides the highresolution image, we cannot observe the same positon during the microstructural
evolution, because it is destructive technique. By contrast, X-ray microtomography is
the non-destructive technique and it overcomes the limitations of 2D imaging
techniques, like SEM and optical microscopy. The direct measurement of a 3D
structure, which is now readily available from X-ray microtomography, provides abasis for the statistical analysis of microstructural characteristics, such as relative
density ρ, specific surface area SV , surface curvature κ , particle size r , neck radius c
28,29, coordination number Z29, heterogeneous particle displacement30,31, particle

rotation32, pore orientation33, pore coarsening34,35, grain growth36, and microstructuralanisotropy37. The knowledge of microstructure obtained from X-ray
microtomography is the first step to understand the realistic property-microstructure
relationship during sintering. Since 2000s, 3D microstructural evolution in sintering of
10

Chapter 1

Introduction

metal particles38 was observed by synchrotron X-ray microtomography. Figure 1.928 isthe example of the viscous sintering of glass particles observed by synchrotron X-ray
microtomography. Fig. 1.9(a) shows the particle packing, and the Fig. 1.5(b) shows thespaces between particles, that is pore or void space. Actually, the local particle
arrangements and inhomogeneity at the particle scale during sintering are revealed by
X-ray microtomography.28, 38 For example, Wakai and Guillon26 showed the pore space
evolution in viscous sintering of spherical glass particles in Figure 1.10. Here, we notethat the 3D pore space evolution during sintering is far more complicated than thesimplified model shown in Figure 1.3. This limits the applicability of classicalmodels in real situations. Therefore, it is expected that the analysis of 3D
microstructural evolution by using X-ray microtomography provides new insights
beyond the classical models.

11

Chapter 1

Introduction

1.5 Objective of this research
The purpose of this study is to construct the experimental and theoretical methods
to estimate sintering parameters, such as sintering stress and bulk viscosity, which are
fundamental quantities for predicting the macroscopic shrinkage behavior in sintering,from 3D visualization of microstructures observed by X-ray microtomography. Glass
was chosen as a model material, because the mechanism of material transport during
sintering of glass is much simpler than that of crystalline particle, and hence, we only
consider about viscous flow mechanism. In this research, I propose the speedy andreliable method of predicting the sintering stress and bulk viscosity. Finally, I believe
the application of the new research methods and theory developed through this work
will be related to the innovative materials developments in ceramic fields.

12Chapter 1

Introduction

1.6 Abstract of doctoral dissertation
This doctoral dissertation was divided into 6 chapters, and abstract of eachchapter was described as below;

Chapter 239: Sintering is a common process during which nanoparticles andmicroparticles are bonded, leading to the shrinkage of interstitial pore space.
Understanding morphological evolution during sintering is a challenge, because pore
structures are elusive and very complex. A topological model of sintering is presented
here, providing insight for understanding 3-D microstructures observed by X-ray
microtomography. I find that the topological evolution is described by Euler
characteristics as a function of relative density. The result is general, and applicable
not only to viscous sintering of glasses but also to sintering of crystalline particles. It
provides criteria to distinguish the stages of sintering, and the foundations to identify
the range of applicability of the methods for determining the thermodynamic driving
force of sintering.
Chapter 340: Sintering stress and bulk viscosity were derived as functions ofrelative density from microtomographic images in viscous sintering of glass particles.
Three methods were proposed to estimate the sintering stress from relative density,
specific surface area, and average of curvature on pore surface, which were directly
measured by X-ray microtomography. The surface energy method gave valid value inthe final stage of sintering, while the mixed method gave better estimation in theintermediate stage. For the initial stage of sintering, the sintering stress was calculated
from the average contact radius and the average coordination number observed by X‐
ray microtomography. The sintering stress at the final stage increased in free sintering,13

Chapter 1

Introduction

but it decreased in constrained sintering due to pore coarsening. The bulk viscosity wascalculated from the shrinkage rate and the sintering stress.
Chapter 441: The macroscopic sintering parameters, sintering stress, bulk
viscosity, and shear viscosity, were measured by the discontinuous sinter forging
experiment for viscous sintering of calcium aluminosilicate (CAS) glass. The calculated
results were compared with experimental values from the microstructural evolution
during viscous sintering of spherical soda lime glass particles by X-ray
microtomography. The sintering stress of CAS glass normalized by surface energy andthe initial particle size was in good agreement with values estimated from themicrotomography data of soda lime glass, despite the differences in particle shape andchemical composition. The bulk viscosity obtained by discontinuous sinter forging
agreed fairly well with that estimated by X-ray microtomography observation, when
they were normalized by glass viscosity.
Chapter 542: The representative volume element (RVE) is a basic concept in thecontinuum mechanics of sintering of random heterogeneous porous materials. A
quantitative determination of its size was performed by using synchrotron X-ray
microtomography data of constrained sintering of thin glass film on a rigid substrate. A
RVE size is associated with a property of interest; I determined it for relative density,
specific surface area, and hydrostatic component of sintering stress. The RVE size wasestimated to be from 11 to 17 times larger than the average initial particle size. The RVE
size was associated with a given precision of the property. It depended on the volume
fraction of porous structure, or, relative density, so that it varied with microstructural
evolution.
Chapter 6: The conclusion of each chapter was summarized.
14

Chapter 1

Introduction

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the sintering of heterogeneous powder systems by synchrotron microtomography anddiscrete element simulation. J. Am. Ceram. Soc. 92, 1492-1499 (2009).
31. S. A. McDonald, P. J. Withers, Combining X-ray microtomography andthree-dimensional digital volume correlation to track microstructure evolution during
sintering of copper powder. J. Strain Anal. Eng. 49, 257-269 (2014).
32. R. Grupp, M. Nöthe, B. Kieback, J. Banhart, Cooperative material transport
17

Chapter 1

Introduction

during the early stage of sintering. Nat. Commun. 2, 298 (2011).
33. O. Lame, D. Bellet, M. D. Michiel, D. Bouvard, Bulk observation of metal
powder sintering by X-ray synchrotron microtomography. Acta Mater. 52, 977-984(2004).
34. L. Olmos, D. Bouvard, L. Salvo, D. Bellet, M. D. Michiel, Characterization of theswelling during sintering of uniaxially pressed copper powders by in situ X-ray
microtomography. J. Mater. Sci. 49, 4225-4235 (2014).
35. T. Hondo, Z. Kato, K. Yasuda, F. Wakai, S. Tanaka, Coarse pore evolution in
dry-pressed alumina ceramics during sintering. Adv. Powder Tech. 27, 1006-1012(2016).

36. V. Tikare, M. Braginsky, D. Bouvard, A. Vagnon, Numerical simulation of
microstructural evolution during sintering at the mesoscale in a 3D powder
compact. Comp. Mater. Sci. 48, 317-325 (2010).
37. D. Bernard, O. Guillon, N. Combaret, E. Plougonve. Acta Mater. 59, 6228-6238(2011).
38. O. Lame, D. Ballet, M. Di Michiel, D. Bouvard. Bulk observation of metal powder
sintering by X-ray synchrotron microtomography. Acta. Mater. 52, 977-984 (2004).
39. G. Okuma, D. Kadowaki, T. Hondo, S. Tanaka, F. Wakai, Interface topology for
distinguishing stages of sintering, Sci Rep.7, 11106 (2017).
40. G. Okuma, D. Kadowaki, T. Hondo, A. Sato, S. Tanaka, F. Wakai, Computation ofSintering Stress and Bulk Viscosity from Microtomographic Images in Viscous
Sintering of Glass Particles, J. Am. Ceram Soc. 100, 867-875 (2017).
41. G. Okuma, J. Gonzalez-Julian, O. Guillon, F. Wakai, Comparison between sinterforging and X-ray microtomograhy methods for determining sintering stress and bulk
18

Chapter 1

Introduction

viscosity, J. Euro. Ceram. Soc. 38, 4, 2053-2058 (2018).
42. G. Okuma, D. Kadowaki, Y. Shinoda, T. Akatsu, O. Guillon and F. Wakai,
Determination of the size of representative volume element for viscous sintering, J.
Ceram. Soc. Japan. 124 [4], 421-425 (2016).

19

Chapter 1

Introduction

Fig.1.1. Powder processing and sintering. The continuum theory of sintering is of
considerable help in predicting the dimension and shape of products or in designingindustrial processes. 1

20Chapter 1

Introduction

Fig.1.2. Scanning electron micrograph of (a) large void fracture (b) cracklike cavity.3

21Chapter 1

Introduction

Fig.1.3. Coble’s simplified geometrical model for distinguishing stages of sintering.
(a) Initial stage of sintering; model structure represented by spheres in tangential
contact. (b) Intermediate stage; dark grains have adopted shape oftetrakaidecahedron, enclosing white pore channels at grain edges. (c) Final stage;pores are tetrahedral inclusions at corners where four tetrakaidecahedra meet. 822Chapter 1

Introduction

Fig.1.4. Material transport paths during initial stage of sintering.10

23

Chapter 1

Introduction

Fig.1.5. Schematic image of (a) constrained sintering, (b) co-sintering. Internal
stresses induce cracks, debonding, or delamination in co-sintering.

24

Chapter 1

Introduction

Fig.1.6. (a) Uniaxial viscosity, and (b) Sintering stress of alumina powder during
sintering determined by sinter forging. 22

25

Chapter 1

Introduction

Fig.1.7. Sintering models. (a) Intermediate stage, (b) Final stage.25

26

Chapter 1

Introduction

Fig.1.8. Schematic explanation of X-ray microtomography. X-ray microtomography
used x-ray beam to create (2D) cross sectional image of the object that can be used toreconstruct a 3D model.

27

Chapter 1

Introduction

Fig.1.9. Example of viscous sintering of glass particles observed by synchrotron X-ray
microtomography. (a) Solid phase, (b) Void space.28

28Chapter 1

Introduction

Fig.1.10. Evolution of pore space in sintering: (a) relative density, ρ = 63.5%, (b) ρ =87.8%, (c) ρ = 94.1%, (d) ρ = 98.4%.26

29

Chapter 2

Interface topology for distinguishing stages of sintering

Chapter 2 Interface topology for distinguishing stagesof sintering

2.1 Introduction
As mentioned in the introduction, Coble1 illustrated the evolution ofparticle-pore structure schematically in Fig. 1.3, and identified three stages ofsintering. The initial stage is characterized as the formation and growth of contact
between neighboring particles, where relative density increases from 0.5 to 0.6. In theintermediate stage, the pore structure evolves into an interconnected channel withcylindrical pores lying primarily along three grain edges. The final stage begins when
the pinch-off of interconnected pore channel forms closed pores at the relative densityof 0.9. The final stage of sintering of crystalline particles is usually accompanied bycoarsening and grain growth2,3. Classical sintering theories, which predict the rate ofdensity change, have been proposed with the assumption of simplified geometricalmodel for each stage1,4.
However, understanding actual morphological evolution during sintering is achallenge, because pore structures are elusive and very complex, as observed in X-ray
mocrotomography image (Fig. 1.10). And hence, no theoretical criterion exists for
distinguishing stages of sintering by using relative density ρ and specific surface area

SV , which are metric properties and vary monotonically during sintering.
Alternatively, topological properties are required to quantify the complex
microstructural changes in three stages of sintering. Rhines, DeHoff, andAigeltinger5,6,7 made a pioneering attempt to analyze the topological properties (e.g.,30

Chapter 2

Interface topology for distinguishing stages of sintering

the connectivity or genus per unit volume, and the number of pores per unit volume)by using a stereological method. But, their analysis has not been used widely in thesintering community due to the difficulty in the analysis of two dimensional crosssectional data. Here, the aim of this chapter is to obtain quantitative knowledges on
topological properties, which provide insights for distinguishing stages in sintering.
The microstructural evolution in sintering is an example of morphogenesis, defined as
the ensemble of mechanisms responsible for the formation of patterns and shapes8.
The observation of interface topology by X-ray microtomography affects our thinking
on sintering significantly. While most of sintering studies are concerned with
distinguishing matter transport mechanisms, we show that the evolution of interface
topology shows remarkable similarity between viscous sintering of glass anddiffusional sintering of crystalline particles.

31

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2.2 Experimental procedure
The soda lime glass powder used in this work consisted of spherical particles(Spheriglass GB-AD, Potters Industries). It had a composition of 72.0 wt% SiO2,
13.5 wt% Na2O/K2O, 9.0 wt% CaO, 3.4 wt% MgO, 2.0 wt% Al2O3, and 0.1 wt%
Fe2O3. The glass spheres were sieved to obtain a homogeneous particle size distribution
between 155 and 183 μm in diameter. Particle size is large because the spatial resolutionof X-ray microtomography used in this work is not high. This powder was mixed with
polyvinylalcohol and surfynol, and resulted in aqueous slurry (60 vol% solid content).
The slurry was casted on an alumina substrate using a doctor blade with the front andrear doctor blades adjusted to a gap height of 4 and 5 mm, respectively. Tape casting
speed was 10 cm/s. Drying was performed at room temperature for 72 hour. The dried
green sheets were removed from the substrate and cut to the desired sample size
(4.5 mm × 6.8 mm × 3.0 mm). Binder burnout and calcination were conducted byheating the sample at a rate of 3°C/min up to 450°C, and 5°C/min up to 690°C, then,held for 30 minute. The isothermal sintering treatment was performed in an external
furnace, and then, taken off for microtomography measurement. This step was repeatedfor one sample. In each sintering step, the sample was heated at 5°C/min and held at
690°C in air with a holding time of 30 minute. The final sintering step was conducted at
700°C in air for 240 minute.
The sample was analyzed by X-ray computed microtomography (SKYSCAN
1172; Bruker, Kontich, Belgium). The X-ray source was set at a voltage of 80 kV and acurrent of 100 μA. The sample was rotated by steps of 0.1° until 180°. The 3D
mappings with voxel size of 2.5 μm × 2.5 μm × 2.5 μm were reconstructed from theacquired data using the filtered back‐projection method. The 3D visualization and32

Chapter 2

Interface topology for distinguishing stages of sinteringgeometrical measurements were performed using Amira (VSG, Burlington, MA) in thisstudy. A Gaussian filtering was applied to reduce the noise in 2D images. Local
thresholding method was used to segment the gray value image into pore and material,so as to determine the pore volume Vpore. The pore surface was discretized using
triangular meshing, from which the pore area Apore was calculated. The curvature oneach triangle was calculated from the eigenvalues and eigenvectors of the quadratic
form.

33

Chapter 2

Interface topology for distinguishing stages of sintering

2.3 Results
2.3.1 Microstructural evolution in sintering
The viscous sintering of spherical glass particles was observed by X-ray
microtomography. The microstructural evolution is illustrated in Figure 2.1. In theloose packing of particles (relative density ρ = 64%, Fig. 2.1(a)), individual particles
have contacts with neighbor particles, but some neighbors are not touching yet. In thepore space view (Fig. 2.1(b)), particles, which intersect with the bounding box, areseen as concave surface, where circular apertures indicate contacts with neighborparticles. For example, in sintering of a cluster of four particles (A, B, C, and D),
three apertures (arrows) can be seen on the surface of particle A.
The arrangement of spherical particles reconstructed from a tomography imageat ρ = 74% is illustrated in Fig. 2.2(a), and the topology of solid phase isschematically shown in Fig. 2.2(b). A point (node, vertex) represents each particle. A
contact between two particles is represented by an arc (branch, edge) between the
representative points. A face is defined by a polygon composed of these vertices andedges. A cell is defined by a 3-D space partitioned by these faces9. There is a void ineach cell, from which a closed pore may be formed during sintering. The irregularbond network can be characterized by a combination of faces and partial polyhedral
cells. Some local structure models depicted in Fig. 2.2(c)-(e) consist of triangular,
rectangular, and pentagonal faces, respectively. There is a hole at the center of eachface, then, the porous solid phase structure is a continuous network with numerous
holes. In the pore space view, this structure is represented as voids inside cells which
are connected by pore channels passing through faces.

34

Chapter 2

Interface topology for distinguishing stages of sintering

The cluster of four particles (Fig. 2.2(c)) is indeed observed by X-ray
microtomography (particles A, B, C, and D in Fig. 2.1(b)). As the contact radius (i.e.,neck radius) grows with time, the size of a hole in the ring of three particles becomes
smaller, and can be seen as a pore channel or a ligament (red arrows in Fig. 2.1(c)).
The breakup of pore channel among three particles (A, B, and D) has alreadyoccurred in Fig. 2.1(c). When two remaining pore channels are pinched off, a single
small closed pore is formed at the center of the tetrahedral particle cluster. But, theformed pore disappears quickly. The shrinkage of a tetrahedral pore, which is thecharacteristics of the “final stage” in the classical sintering model, takes place in very
early stage actually.
The cross section of a pore channel is a polygon with rounded corners, where
the number of edges is the number of particles surrounding the channel. For example,
the pore channel outlined in red (Fig. 2.1(e)) is formed by five particles (A, E, F, G,
H). The size of the pore channel decreases with densification, and becomes zero at
the pinch-off finally. The initial size of a pore channel usually increases with thenumber of particles surrounding it. The pore channel formed by the ring of threespheres has a small initial size, and can be seen in the early stage (red arrows,
Fig. 2.1(c)). But, such pore channel along three-grain junction is pinched off quickly.
Large pore channels surrounded by many particles are important in the intermediatestage. An example of pore channel formed by the ring of four spheres (G, H, I, J) isseen in Fig. 2.1(e) (blue arrow). Such large pore channel remains up to the relative
density of 94% before the pinch-off (Fig. 2.1(g)). When a pore channel is pinched off,
one hole is closed, thereby decreasing the number of holes.

35

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Interface topology for distinguishing stages of sintering

Many tetrahedral cells form small closed pores by the pinch-off of porechannels, and these small tetrahedral pores shrink and disappear quickly. The size ofa cell, and its void space, usually increases with the number of particles surroundingit, or pore coordination number10. The closed pores larger than the particle size areformed from large voids preexisting in the random packing of particles. Such voids
are connected with open pore channels initially, and are separated later throughconsecutive pinch-offs. Closed pores formed by viscous sintering in air become
spherical (Fig. 2.1(i)), and shrink by gas diffusion in the glass. The number ofspherical pores decreases to zero ultimately in the final stage of sintering.

2.3.2 Euler characteristic for describing interface topology
The naturally evolving interface changes its topology during sintering. The
topological transitions are summarized as follows: (1) Formation of contacts among
particles leading to the increase of number of holes Gin the networks, (2) Pinch-off ofpore channels, i.e., the decrease of G by the closure of holes, (3) Formation of closed
pores, i.e., the increase of number of pores N, (4) Disappearance of closed pores, i.e.,the decrease of N. The term “holes” is used for pore channels between porous cells
throughout this paper, then, “hole closure” means the pinch-off of pore channel. The
topology of a surface is characterized by its genus g; roughly speaking it is thenumber of holes in the surface. A single sphere has g = 0, and a torus (doughnut
shape) has g = 1. The genus is mathematically related to the Euler characteristic
as χ = 2 − 2 g. For partially sintered particle clusters, tetrahedron (Fig. 2.2(c)), cube
(Fig. 2.2(d)), and dodecahedron (Fig. 2.2(e)), the Euler characteristic is −4, −8, and36

Chapter 2

Interface topology for distinguishing stages of sintering−20, respectively. Here, we consider the half of total Euler characteristic X/2 as a sumof Euler characteristics of all poresN∑ (1 − g )

X=
2

(2.1)

n =1

Since the sum of genus ∑g is approximately equal to the total number of holes G, Eq.
(2.1) becomes X/2 ≈ N − G. The evolution of interfacial topology is, then, describedby using the total Euler characteristic.
Using the Gauss-Bonnet theorem, the total Euler characteristic is calculated
from the integral of Gaussian curvature K = κ1κ2 over all pore surfaces
X 2=

1
KdA
4π ∫A

(2.2)

The normalized Euler characteristic per unit volume V is given as

X 2V = KSV 4π(2.3)

where K is the average Gaussian curvature (Figure 2.3(a)), SV = Apore V is the
specific surface area (Fig. 2.4), and Apore is the total surface area in the unit
volume. K , SV are derived from the tomography data, and plotted as functions ofrelative density in Fig. 2.3(a) and Fig. 2.4, respectively. X/2V is calculated from these
data, and is plotted as a function of relative density in Fig. 2.5(a). The unit
volume V is defined as a cube with edge length L = 20r0, where r0 is the initial
particle radius. Three stages of sintering can be distinguished by using Euler
characteristic. In the initial stage of sintering, Euler characteristic is negative, anddecreases slightly with relative density up to ρ = 74%. The intermediate stage is theregion where Euler characteristic increases with relative density until it has amaximum value (positive) at ρ = 96%. The final stage is characterized as a region
37

Chapter 2

Interface topology for distinguishing stages of sintering

where Euler characteristic decreases to zero ultimately. These quantitative criteria for
distinguishing stages of sintering agree to the classical qualitative definition based on
microstructures fairy well, as seen in Fig. 2.1).
The changes of Euler characteristic can be analyzed by comparing the numberof closed pores N/V, which is plotted in Fig. 2.5(b). The increase in the number ofclosed pores shows that they are formed continuously during both the initial and theintermediate stages, although formation rate is limited during the initial stage(<ρ = 74%). This result indicates that the slight decrease of Euler characteristic
occurs because the rate of formation of holes is higher than that of hole closure in theinitial stage. New holes are created as new contacts with neighbor particles areformed during densification. At the beginning of intermediate stage the increase ofEuler characteristic occurs due to hole closures. The rate of hole closure decreases
with densification, while the number of closed pores increases. The maximum Euler
characteristic at ρ = 96% is almost the same with the number of closed pores. The
ratio of open porosity to total porosity (1-ρ) is also plotted in Fig. 2.5(b). The ratio of
open porosity decreases significantly as large closed pores are formed. In the final
stage of sintering, the Euler characteristic decreases as the number of closed poresdecreases.

2.3.3 Transition between different stages
The transition from the initial to the intermediate stage is related to an
evolution towards a system of interconnected channels with more or less constantcurvature in the conventional approach. However, the areal distribution functions ofGaussian curvature K = κ 1 κ 2 and mean curvature H = (κ 1 + κ 2)/2 (Fig.2.6) show that38

Chapter 2

Interface topology for distinguishing stages of sintering

the heterogeneity in curvature does not seem to be a pertinent parameter for
distinguishing stages of sintering. Both mean curvature H (Fig. 2.3(b)) and Gaussian
curvature K (Fig. 2.3(a)) increase with relative density monotonously, so that they do
not distinguish stages. On the other hand, the transition from the initial to theintermediate stage can be distinguished clearly as an augment of the Euler
characteristic.

2.4 Discussion
2.4.1 Comparison with other sintering mechanisms
Aigeltinger and DeHoff6 analyzed the number of holes G and the number ofpores N in sintering of copper particles by using quantitative microscopy. The Euler
characteristic was calculated by using their data, and plotted in Fig. 2.5(a) forcomparison. The Euler characteristic curves are of similar shape, when they arenormalized by the average particle size. Three stages of sintering can be distinguishedby using Euler characteristic not only for viscous sintering of glass but also forsintering of copper particles by diffusion.
It should be noted that sintering of amorphous and sintering of crystalline
materials are rather different. The particle coarsening results from the surface motion
in sintering of crystalline materials by evaporation-condensation and surface
diffusion. The grain growth results from grain boundary motion by curvature.
Although both coarsening and grain growth affects microstructure evolution insintering of crystalline materials2, such effects are not involved in viscous sintering.
Nevertheless, as far as we focus on topological nature of microstructure, common
features are observed for sintering of both amorphous and crystalline materials.
39

Chapter 2

Interface topology for distinguishing stages of sintering

The effect of particle size on viscous sintering of glass particles can bepredicted by using scaling law of Herring11, thereby, we suppose the microstructure isself-similar for different particle size. In Fig. 2.4(a), the normalized X/2V in viscous
sintering of large glass particles (r0 = 80 μm, present work) was compared with that insintering of small glass particles (r0 = 4 μm), which was observed by synchrotron
X-ray microtomography12. Euler characteristic vs relative density curves wereself-similar despite the difference in particle size.
The sintering of crystalline materials is affected significantly with the decreaseof particle size into the nanometer range13. The particle rotation and sliding
contribute to densification in the initial stage of sintering of loosely packed
nanocrystalline powder14. Common features observed in sintering of coarse particles
may not be observed in sintering of such loosely packed powder. For example,
Schleef and co-workers15 reported that Euler characteristic increased monotonicallyin sintering of fresh snow with relative density of 0.1. However, at present, the voxel
resolution (0.28–2.5 μm) of X-ray microtomography is insufficient to study sinteringof nanocrystalline particles.

2.4.2 Euler characteristic per unit volume
Consider the initial stage of sintering of identical spheres periodically arranged
in a simple cubic lattice. Because there is one unique interconnected pore (N = 1 in Eq.
(2.1)), the half of total Euler characteristic is a sum of genus made on the porous
cells X 2 = 1 − ∑ g . The periodic cubic cell structure is topologically the same with
Schwartz P surface, which has the genus 3 for a unit cell (Fig. 2.7(a)). The average
number of particles P in the unit volume is given as P = 6000ρ/π where ρ is the40

Chapter 2

Interface topology for distinguishing stages of sintering

relative density. At ρ = 0.6, the normalized Euler characteristic per unit
volume X/2V is about −2800, which is in good agreement with experimental results in
Fig. 2.5(a). The Euler characteristic per unit volume depends on the particle shape,
the distribution function of particle size, and that of pore size. It is clearly shown thatthe number of small pores, where the pore size is normalized by the average particle
radius, is larger in sintering of large particles than in the sintering of small particles(Fig. 2.8). This is partly due to the difference in relative resolution (the ratio of voxel
size to the particle radius), which is 0.03 and 0.07 for the sintering of large particles(present work) and the sintering of small particles, respectively. It should be notedthat Euler characteristic is sensitive to the relative resolution, because small bodies
are weighted equally with large bodies.

2.4.3 Simulation of topological evolution
In order to visualize the topological evolution in sintering, we performed acomputer simulation using a mathematically simple model for the case the grainboundary energy is zero, so as to compare with the microstructure evolution inviscous sintering. The computer simulation of sintering was conducted by assuming acase where the bulk diffusion is so fast that the sintering rate is controlled by the rateof creation/annihilation of vacancies on the surface16,17.
As a mathematically simple sintering mechanism, we consider a model, where thebulk diffusion is so fast that the shrinkage rate is controlled by the interface reaction(creation and elimination of vacancies by the surface)16. We assume grain boundary is
not the source/sink of vacancy for simplicity. Closed pores can shrink by bulk diffusionfrom surface to pore surface. The chemical potential in the bulk will be uniform, if the41

Chapter 2

Interface topology for distinguishing stages of sinteringbulk diffusion is infinitely fast. It is supposed that the interface reaction rate is
proportional to the difference in chemical potential between the bulk and the surface.
The surface moves inward when a vacancy is created on the pore surface. The normal
velocity of surface is expressed asυ = Mγ s (κ − κ av )(2.4)

where M is the mobility, γ s is surface energy, κ = 2 H is curvature, and κ av is theaverage curvature. The detailed derivation of Eq. (2.4) is given in Appendix D of Ref.
38.
Brakke’s Surface Evolver program18 was used to simulate the microstructural
evolution in sintering by interface-reaction controlled bulk diffusion. The outline of the
program is described here briefly. Both the surface and the grain boundary of particles
are represented as a set of triangular finite elements, or facets. Each facet consists ofthree edges and three vertices. The surface and the grain boundary have energies
proportional to their area. The Surface Evolver program evolves the surface toward
minimal energy by a gradient descent method under any constraint. The gradient ofenergy at a vertex is a force, which must be converted to a velocity vector for the
motion. This conversion involves what may be called the mobility factor: how a vertex
responds to the force on it. In the interface-reaction controlled sintering, the resistanceof motion is actually due to the surface, not the vertex. In order to approximate this, theresistance to motion of a vertex is proportional to the area associated with vertex. The
actual motion is found by multiplying the velocity by a scale factor. The physicalinterpretation of the scale factor is the time step. The surface motion can beapproximated by enforcing the constraint on conservation of the total volume of42

Chapter 2

Interface topology for distinguishing stages of sintering

particles. The Surface Evolver program had been applied to simulate theinterface-reaction controlled sintering16,17, for example, evaporation-condensation,interface-reaction controlled bulk diffusion, and interface-reaction controlled surface
diffusion.
The sintering of a cluster of 128 spherical particles was simulated. The result was
analyzed by using the dimensionless time defined as t * = γ s Mt r02 , where r0 is theinitial radius.
Closed pores shrink by bulk diffusion from surface to pore surface4.
Figure 2.9(a) show the evolution of pore structures in sintering of a cluster of 128
spheres. The movie clearly shows that large closed pores are formed by consecutive
pinch-off of pore channels in a similar way to the microstructural evolution in viscous
sintering. Figures 2.10 illustrate how a void and pore channels evolve from the
topological cell of particles. Figure 2.9(b) shows the Euler characteristic varies with adimensionless time. The Euler characteristic increased after a plateau region, reached
to a maximum, and decreased toward zero. This topological feature agreed withexperimental observations qualitatively, although sintering mechanisms differed witheach other.

2.4.4 Topology and sintering mechanics
The sintering mechanics depends on the topology of microstructure. For the
initial stage, large scale simulations of sintering of many particles have beensuccessfully achieved by using discrete element method (DEM) recently19,20,21. The
mechanics underlying this method is a relationship between the relative velocity ofparticles and the sintering force acting among neighbor particles in both sintering by
43

Chapter 2

Interface topology for distinguishing stages of sintering

grain boundary diffusion22,23 and in viscous sintering24. The macroscopic sintering
stress can be estimated from the microscopic sintering force, which is a function of
the average contact radius and the average particle coordination number25. In the final
stage where closed pores are dispersed randomly, each pore has a local sintering
stress16. For viscous sintering, the macroscopic sintering stress is defined as a volume
average of local sintering stress of pores26. It is simply calculated from the relative
density and the specific surface area12. The present authors have shown that sintering
stress can be derived as functions of relative density directly from the knowledge of
microstructure observed by X-ray microtomography12, 25, and proposed a method to
calculate the sintering stress in the intermediate state by using the average curvatureof pore surface. The range in application of these three methods can be clearly
defined by distinguishing stages of sintering from Euler characteristic.

44

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Interface topology for distinguishing stages of sintering

2.5 Conclusions
Three stages of sintering are distinguished by using Euler characteristic,
which is given as the number of closed pores minus the genus. The random packingof particles is expressed as vertices, edges, faces, and cells topologically. The genus
is the number of holes, which is equivalent to the number of pore channels
connecting voids inside each cell. The elementary processes in morphologicaltransformation of pore structure is the creation and annihilation of pore channels andthose of closed pores. Although the Euler characteristic vs relative density curve wasstudied only for viscous sintering of silicate glass particles here, I believe the result isgeneral and may explain the sintering behavior of many ceramic and metallic
particles.
These results may open the way to control internal defects formed during
sintering, because it is crucial to understand the evolution of heterogeneous pore
structures for improving the mechanical reliability of products. The interfacial
topology provides a description of stages of sintering, and helps to recognize the rolesof forces behind the microstructural evolution.

45

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Interface topology for distinguishing stages of sintering

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3. I.-W. Chen, X.-H. Wang. Sintering dense nanocrystalline ceramics without
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Interface topology for distinguishing stages of sintering

4152-4165 (1982).
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13. R. Chaim, M. Levin, A. Shlayer, C. Estournes. Sintering and densification of
nanocrystalline ceramic oxide powders: a review. Adv. Appl. Ceram. 107, 159–169(2008).
14. R. Chaim. Densification mechanisms in spark plasma sintering of nanocrystalline
ceramics. Mater. Sci. Engn. A 443, 25–32 (2007).
15. S. Schleef, H. Löwe, M. Schneebeli. Influence of stress, temperature and crystal
morphology on isothermal densification and specific surface area decrease of new snow.
The Cryosphere 8, 1825–1838 (2014).
16. F. Wakai, T. Akatsu, Y. Shinoda. Shrinkage and disappearance of a closed pore in
the sintering of particle cluster. Acta Mater. 54, 793-805 (2006).
17. F. Wakai. Modeling and simulation of elementary processes in ideal sintering. J.
Am. Ceram. Soc. 89, 1471-1484 (2006).
18. K. A. Brakke. The Surface Evolver. Exp. Math. 1, 141-165 (1992).
19. F. Parhami, R. M. McMeeking. A network model for initial stage sintering. Mech.
Mater. 27, 111-124 (1998).
20. A. Wonisch, et al. Stress-induced anisotropy of sintering alumina: Discrete element
modelling and experiments. Acta Mater. 55, 5187-5199 (2007).
21. C. L. Martin, Z. Yan, D. Jauffres, D. Bouvard, R. K. Bordia. Sintered ceramics with
controlled microstructures: numerical investigations with the Discrete Element Method.
J. Ceram. Soc. Japan. 124, 340-345 (2016).
47

Chapter 2

Interface topology for distinguishing stages of sintering

22. H. Riedel, H. Zipse, J. Svoboda. Equilibrium pore surfaces, sintering stresses andconstitutive equations for the intermediate and late stages of sintering-II. Diffusion
densification and creep. Acta Metall. Mater. 42, 445-452 (1994).
23. F. Wakai, K. A. Brakke, Mechanics of sintering for coupled grain boundary andsurface diffusion. Acta Mater. 59, 5379-5387 (2011).
24. F. Wakai, et al. Sintering force behind the viscous sintering of two particles. Acta
Mater. 109, 292-299 (2016).
25. G. Okuma, et al. Computation of sintering stress and bulk viscosity frommicrotomographic images in viscous sintering of glass particles. J. Am. Ceram. Soc.
100, 867-875 (2017).
26. F. Wakai. Mechanics of viscous sintering on the micro- and macro scale. Acta Mater.
61, 239-247 (2013).

48

Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.1. Microstructural evolution in viscous sintering of spherical glass particles.
(a) solid phase at the initial relative density ρ of 64%, (b–i) pore space view as
seen in the direction of arrow in (a). Particle surfaces are shown in white, while
pore surfaces are shown in yellow. The size of the reconstructed subvolume is
500 × 500 × 500 µm.

49Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.2. Packing model of spherical particles. (a) Sphere packing at ρ = 74%. (b)The bond network model consisting of vertices, edges, faces, and cells. A
quadrilateral face and a pentagonal face are shown for example. Some polyhedral
cells ((c) tetrahedron, (d) cube, and (e) dodecahedron) illustrate that voids inside
cells are connected by pore channel (or hole) at the center of each face.

50Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.3. (a) Average Gaussian curvature K , (b) Mean curvature H on pore surface
as a function of relative density. The local curvature on pore surface varies widely,so that we choose large volume elements with L = 1.5 mm to define the average.
The error bar indicates the heterogeneity inside the specimen (4.5 mm × 6.8 mm ×
3.0 mm).

51

Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.4. Relationship between specific surface area SV and relative density ρ .

52

Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.5. Packing model of spherical particles. (a) Sphere packing at ρ = 74%. (b)The bond network model consisting of vertices, edges, faces, and cells. A
quadrilateral face and a pentagonal face are shown for example. Some polyhedral
cells ((c) tetrahedron, (d) cube, and (e) dodecahedron) illustrate that voids inside
cells are connected by pore channel (or hole) at the center of each face.

53

Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.6. Distribution function of curvatures on pore surface. (a) Mean
curvature H = (κ 1 + κ 2)/2, (b) Gaussian curvature K = κ 1 κ 2. The mean curvature
is defined as positive for a spherical pore and negative for a spherical particle.

54

Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.7. Genus of cubic structures. (a) Schwartz P surface with genus 3, (b) A cubic
cluster of particles. For an isolated cluster (left model), we can stretch the top holeso that it is large enough. Then, collapse the top portion to flatten the model (rightmodel). The number of through holes is actually five, and then the genus is 5
( χ = −8 in Fig. 2 d).

55

Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.8. Normalized pore size distribution in the final stage of sintering. (a) The
number of pores in the unit volume, (b) Cumulative number of pores, (c)Cumulative pore volume fraction.

56

Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.9. Computer simulation of sintering by bulk diffusion. (a) Snap shot at theinitial stage, (b) Euler characteristic.

57

Chapter 2

Interface topology for distinguishing stages of sintering

Fig.2.10. Computer simulation of sintering by bulk diffusion. (a) An example ofinitial particle packing, (b) A void and pore channels together with the topologicalcell model.

58

Chapter 3

Computation of sintering stress and bulk viscosity for each stage of sintering

Chapter 3 Computation of sintering stress and bulk
viscosity for each stage of sintering

3.1 Introduction
Although powder processing is the most efficient method to form ceramics, it isprone to heterogeneities and nonuniformity in both microscopic and macroscopic
scales.1 Inhomogeneous regions in powder compacts results in nonuniform densification
rates and internal stresses, which in turn, lead to inhomogeneous shrinkage and may
create defects during sintering.2 Therefore, the mechanical reliability of products
depends on microstructural evolution of heterogeneity during sintering.
The continuum mechanics fully described in section 1.3 is useful to predict themacroscopic dimensional changes, and also to analyze internal stresses generated during
sintering of heterogeneous porous materials. The shrinkage rate is proportional to thesum of sintering stress and the mechanical stress, and it is inversely proportional to thebulk viscosity.3-7 The continuum mechanics is normally concerned with the behavior ofmatter on a macroscopic scale that is large compared with particles. The macroscopic
properties in heterogeneous materials are defined as the average over the representative
volume element (RVE), which will be described in chapter chapter5.8 This separation ofscales is known as the Micro-Meso-Macro principle,9 so that the property of RVE
provides a basis for multiscale analysis.
As mentioned in section 1.4, X-ray microtomography allow us to observe
three-dimensional (3D) microstructural evolution in sintering, and provide us a basis forthe statistical analysis of microstructural characteristics, such as relative density ρ,59Chapter 3

Computation of sintering stress and bulk viscosity for each stage of sintering

specific surface area SV , surface curvature κ , particle size r , neck radius c , andcoordination number Z . While constitutive parameters, such as sintering stress andbulk viscosity, are physical quantities that can be determined by macroscopic sinterforging experiments, they depend on the details of local particle packing, which are
influenced by powder processing.10 In this chapter 3, I describe methods to estimate
constitutive parameters in the scale of RVE from 3D visualization of microstructures
observed by X-ray microtomography in viscous sintering of glass particles. I present
microstructural observations of 10 stages in the range of relative density from 0.6 to
0.98. This approach will be useful for understanding the relationship between
microstructure and microscopic densification behavior.

60

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Computation of sintering stress and bulk viscosity for each stage of sintering

3.2 Experimental procedure
Sintering of glass particles and X-ray microtomography have been fully describedin chapter 2 and will be outlined here. The soda lime glass powder used in this work
consisted of spherical particles (Spheriglass GB-AD, Potters Industries). It had acomposition of 72.0 wt% SiO2, 13.5 wt% Na2O/K2O, 9.0 wt% CaO, 3.4 wt% MgO,
2.0 wt% Al2O3, and 0.1 wt% Fe2O3. The viscosity of the glass was predicted using themulticomponent model of Fluegel11 developed for industrial glass-forming silicate melts.
The temperature dependence of viscosity μ (Pa·s) of the glass is given by the following
Vogel–Fulcher–Tammann equation:

log10 ( µ ) = −2.6879 +

4475.9T − 524.037(3.1)

where T stands for temperature in Kelvin. The viscosity was 3.2 × 107 Pa·s and1.9 × 107 Pa·s at 690°C and 700°C, respectively.
The glass particles were sieved to obtain a homogeneous particle size distribution
between 155 and 183 μm in diameter. We assumed the average radius of 80 μm for thepolydispersed particles. This powder was mixed with polyvinylalcohol (PVA) andsurfynol, and the resulted aqueous slurry (60 vol% solid content) was casted on analumina substrate using a doctor blade. The dried green sheets were removed from thesubstrate and cut to the desired sample size (4.5 × 6.8 × 3.0 mm). Binder burnout andcalcination were conducted by heating the sample at a rate of 3 °C/min up to 450 °C,
and 5 °C/min up to 690 °C, then held for 30 minutes. The isothermal sintering treatment
was performed in an external furnace, and then, taken off for microtomographymeasurement. This step was repeated for one sample. In each step, the sample washeated at 5 °C/min and held at 690 °C in air for 30 minutes.

61

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Computation of sintering stress and bulk viscosity for each stage of sintering

3.3 Results
3.3.1 Evolution of pore morphology observed by 3D visualization
Figure 3.1 shows the evolution of pore morphology during viscous sintering,where the size of the reconstructed subvolume is 750 μm × 750 μm × 750 μm. Pore
space is shown in Figure. 3.1. As mentioned in chapter 2,12 the topology of pore
structure changes drastically as the volume fraction of pore space decreases in sintering.
In the initial stage (Fig 3.1(a)), spherical voids represent particles, and neck i.e. contacts
between particles can be seen as circular holes. In the intermediate stages (Fig. 3.1(b)),
voids are connected by pore channels. In the final stage (Fig. 3.1(c)) closed pores shrink
and disappear.

3.3.2 Relative density and specific surface area
The relative density ρ and the specific surface area SV were defined for cubic
volume elements with edge length of 1000 μm, which is about 7-8 times larger than the
average particle size. The average value of the sample was defined from eight
nonoverlapping volume elements, then it ensures the relevant results. The averagerelative density is plotted as a function of sintering time in Figure 3.2, which is the sumof holding time at 690 °C. The final holding time at 700 °C, ∆t (700 °C), was corrected
for effective time at 690 °C using viscosity ∆t = ∆t (700 °C) µ (690 °C) / µ (700 °C).
The densification curve obtained from microtomography resembles that for isothermalsintering. The error bar in Fig.3.2 indicates the spatial heterogeneity in density.
Specific surface area, shown in Fig. 2.4, SV decreases with relative density. The
linear relationship between SV and relative density14-16 has been found using
stereologi-cal method, where SV is estimated from the mean pore intercept length.17
62

Chapter 3

Computation of sintering stress and bulk viscosity for each stage of sintering

Wakai18 also reported a linear relationship in viscous sintering based on 3D observationof four stages in viscous sintering using synchrotron X-ray microtomography. However,the data in this study were best fitted by a curved line suggesting a nonlinear
relationship.

3.3.3 Curvature of pore surface
The sintering of glass powder occurs by viscous flow driven by surface tension.
The traction on the pore surface is proportional to local curvature, κ=κ1 + κ2, whereκ1 and κ2 are two principal curvatures. In this work, the sign of curvature is defined as
positive for a spherical pore and negative for a spherical particle. The local curvature on
pore surface varies widely, so that we choose large volume elements with L=1500 μm todefine the mean curvature H. Figure 2.3(b) shows the relationship between mean
curvature H = (κ 1 + κ 2)/2 and relative density ρ. The error bar indicates that the local
heterogeneity of H is still larger than those of relative density (Fig.3.2) and specificsurface area (Fig.2.4). The average curvature is negative in the initial stage of sintering,and is close to the curvature of spherical particles. It increases with densification, as thesaddle-shaped interparticle contacts grow and closed pores with positive curvatures areformed in the intermediate and the final stages of sintering.

63

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Computation of sintering stress and bulk viscosity for each stage of sintering

3.3.4 Sintering stress derived from three methods
I estimated the sintering stress of nonequilibrium porous structure by substituting
the average curvature κ = 2H into Equations (1.9) and (1.10). The values obtained by
the curvature method and the mixed method are compared with that obtained by thesurface energy method (Equations (1.11) and (1.12)) in Figure 3.3. Here, Σs isnormalized by γ S / r0 where r0 is the average initial particle radius. Actually, the particles
are polydisperse, then, we assumed r0 of 80 μm for convenience. The three methods
gave different values of Σs in the initial stage; the surface energy method gave thelargest value, the curvature method gave the smallest value, and the mixed method gave
the intermediate value. The curvature method is not valid since it predicts negative Σs,which is unphysical, in the initial stage. As the relative density increases, these values
increase, and coincide with each other in the final stage. The sintering stress at ρ=0.98 isestimated to be 16.2 kPa by assuming the surface energy of 0.304 J/m2, which ismeasured from the surface tension at 1200°C.19 The sintering stress was very low,
simply because we used glass particles with very large size.

3.3.5 Bulk viscosity
The true strain rate E is calculated from the data of Figure 3.2, and plotted in
Figure 3.4. A smooth curve is fitted to the plotted data, and the true strain rate is given
as a function of relative density. The bulk viscosity was estimated using Equation (1.7)
from sintering stresses determined by the energy method and the mixed method, andplotted in Figure 3.5. The bulk viscosity starts from values about 3.3×107 Pa・s in theinitial stage and increases gradually up to relative density of 90%. The bulk viscosity
increases steeply around relative density of 95% and reaches to 5.4× 109 Pa・s.
64

Chapter 3

Computation of sintering stress and bulk viscosity for each stage of sintering

Mackenzie and Shuttleworth20 gave the relation between bulk viscosity and porosity inthe final stage of viscous sintering by assuming spherical pores of identical size. It is
written as a function of relative density:K=4µρ
3(1 − ρ )(3.2)

where µ is the viscosity of glass. Wakai21 showed that Equation (3.2) can be appliedalso to closed pores with wide size distribution. The solid line in Figure 3.5 shows thetheoretical prediction of bulk viscosity by assuming µ of 6.0×107 Pa・s. This value
agree fairly well with the viscosity of the glass (3.2×107 Pa・s at 690 °C), which ispredicted from the chemical composition.

3.4 Discussion
3.4.1 Sintering stress in the initial stage of sintering
Shrinkage in the initial stage is the result of the motion of many particles, whichinteract with their neighbors. The interaction between two spherical particles is the firststep for understanding the mechanics of viscous sintering. The shrinkage rate between
two identical particles is proportional to the sintering force, F S , which can beapproximated as a parabolic function of contact or neck radius, c, in the early stage(0.1 ≤ c / r0 ≤ 0.7) 22:[

F S = γ S r0 − 11.53(c / r0 ) 2 + 13.58(c / r0 ) + 0.40

]

(3.3)

The microscopic sintering stress can be calculated from the sum of sintering
forces acting on a single particle in the initial stage. For simplicity, consider thesintering of periodically packed monodispersed spheres with radius r0. The coordination
numbers are Z=6 for the simple cubic lattice, and Z=8 for the body-centered cubic

65

Chapter 3

Computation of sintering stress and bulk viscosity for each stage of sintering

lattice when only nearest neighbors are in contact as shown in Figure 3.6. For the simple
cubic lattice, a sphere is contained in the Wigner-Seitz cell, which is a cube bounded bysix square faces with edge length of 2r0. Circular contacts lies on each face of the cell,then, the sintering stress in the initial stage is defined as

SS = F S / A(3.4)

where A is the area of the face. For the body-centered cubic lattice, the sintering stress iscalculated by considering an inscribed sphere in an octahedron, which is bounded byeight triangular faces with an edge length of r0 6 . The average contact radius and theaverage particle coordination number were directly measured by X-ray
microtomography. The coordination number was measured by counting the number ofcontacts, which appeared as circular holes in Figure 3.1(a). The average coordination
number was defined using at least 10 particles. The average contact radius was definedusing all contacts on these particles. The average contact radius at ρ =0.63 and 0.68 was
c=20.3 and 27.8 μm, respectively. The coordination numbers Z ranged from 6 to 8 in theinitial stage. Vagnon14 reported the average coordination number of 7.8 at ρ =0.7, and Z
increased with densification. Therefore, we assumed that Z=6 is the lower bound and
Z=8 is the upper bound for the initial stage. The particles are polydisperse, but thesintering force in viscous sintering of two particles of different sizes is not known yet.
We estimated the range of sintering stress among polydispersed particles using Equation(3.3) as follows. For a given contact radius, the sintering force between two particles ofdifferent sizes is estimated to lie between that in sintering of two particles of minimum
size (77 μm) and that in sintering of two particles of maximum size (91μm). The
sintering stress is calculated by Equation (3.4) using face areas corresponding to either
particles of maximum size or those of minimum size. Furthermore, the coordination
66

Chapter 3

Computation of sintering stress and bulk viscosity for each stage of sintering

number is either Z=6 or 8. We calculated every combination of grain sizes andcoordination numbers. Rectangular bars in Figure 3.3 show the range of sintering stressderived by this procedure. The sintering stress estimated from Equation (3.4) agrees
with the value calculated by the mixed method fairly well. The result indicates that thesurface energy method, which is valid for the final stage of sintering, cannot be applied
to the initial stage. The authors conclude that the mixed method gives a better
estimation of sintering stress in the initial and the intermediate stages.

3.4.2 Sintering stress in constrained sintering
The present work analyzed the microstructure of a tape in free sintering, where
the sheet was sintered after it was removed from the substrate. On the other hand,
Wakai and Guillon18 studied the sintering of a sheet under constraint by a rigid substrate,and estimated deviatoric and hydrostatic components of sintering stress from themicrostructural observation by synchrotron X-ray microtomography. The hydrostatic
component of sintering stress in constrained sintering is calculated using the threemethods, and plotted as a function of relative density in Fig. 3.7. Since they used glass
particles with smaller radius (r0 = 4μm), the sintering stress is about 20 times higher
than the present work. However, the sintering stress normalized by γ S / r0 is comparable
to the present work. The three methods gave different values of sintering stress in theinitial stage of sintering (ρ = 0.635). As the relative density increased, the difference insintering stress decreased in the intermediate stage, and all three method gave theidentical value at ρ = 0.941. The sintering stress in the final stage of sintering iscorrectly predicted by the surface energy method from the distribution function of poresize. 21 In the final stage (ρ = 0.984), the value obtained by the surface energy method
67

Chapter 3

Computation of sintering stress and bulk viscosity for each stage of sintering

decreased, while values obtained by the curvature method and the mixed method, which
are valid only when all pores has the same size, increased. Figure 3.8 shows the
distribution of normalized pore size in constrained sintering (ρ= 0.984) and freesintering (ρ= 0.98). The pore size, r, is defined as the radius of equivalent sphere, andnormalized by the initial particle radius r0. Although the number of large pore ( r / r0 >1)
is small, they constitute a large volume fraction. It is clearly seen that large pores areformed much more in the constrained sintering than in the free sintering. The
macroscopic sintering stress of the surface energy method is defined as the volume
average of local sintering stress of pores, so that it is governed by the largest pore. 21 On
the other hand, the macroscopic sintering stress of the curvature method is defined as
the surface average of local sintering stress of the pores. The origin of the low sintering
stress by the energy method is the pore coarsening in the final stage during constrained
sintering.

3.4.3 Pore coarsening in constrained sintering
Scherer23 presented a model of pore coarsening of a viscous material underhydrostatic constraint. The smaller pores can shrink at the expense of the larger pores,
so that some regions densify while others dilate. Here, we formulate pore coarsening ina viscous sheet on a rigid substrate. From the boundary condition of constraint, Bordia
and Raj24 showed that the internal in-plane stress was generated in the sheet
due to the macroscopic sintering stress Σs:S11 = S 22 =6GSS
3K + 4G(3.5)

The hydrostatic component of the stress is given as follows:

68

Chapter 3

Sm =Computation of sintering stress and bulk viscosity for each stage of sintering

4GSS
3K + 4G(3.6)

Since Σs is positive, Σm is positive also, then so is the tension. Each closed pore has its
local sintering stress, σS, which is 2γ S / r for spherical pores. From Eq. (1.4), small poreswith high local sintering stress (σS > Σm ) shrink, and large pores with low sintering
stress (σS < Σm ) increase their volume. Therefore, the initial tensile stress is responsible
for pore coarsening in constrained sintering. Of course, every pore can shrink in freesintering. And pore coarsening is suppressed by the application of compressive stress( Σ m ≤ 0 ).

69

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Computation of sintering stress and bulk viscosity for each stage of sintering

3.5 Conclusions
I have shown that both sintering stress and bulk viscosity can be derived asfunctions of relative density directly from microtomographic images. In the initial stageof sintering, the sintering stress is estimated from the sintering force acting among
particles. The mixed method gives fairly good estimate of sintering stress in the initial
and the intermediate states. The surface energy method is valid in the final stage ofsintering. I have demonstrated also that the sintering stress obtained by the surface
energy method depends on the pore size distribution. The decrease of sintering stress inthe final stage results from pore coarsening. The sintering stress is the driving force forshrinkage, but can generate internal tensile stress when the shrinkage of the sample isconstrained. The presence of internal stress is the origin of microstructural instability, sothat pore coarsening occurs in the constrained sintering. Overall, we have demonstrated
the feasibility of combining microtomographic images with constitutive parameters to
predict the sintering behaviors of specific powder compacts. This approach can beapplied to study the relationship between powder processing and sintering mechanics
from the point of view of the nature of initial particle packing.

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Computation of sintering stress and bulk viscosity for each stage of sintering

References
1. F. F. Lange. Powder processing science and technology for increased reliability. J.
Am. Ceram. Soc. 72, 3-15 (1989).
2. A. G. Evans, Consideration of inhomogeneity effects in sintering. J. Am. Ceram. Soc.
65, 497-501 (1982).
3. R. K. Bordia, G. W. Scherer. On constrained sintering – I. Constitutive model for asintering body. Acta. Metall. 36, 2393-2397 (1988).
4. R. M. McMeeking, L. T. Kuhn. A diffusional creep law for powder compacts. Acta.
Metall. Mater. 40, 961-969 (1992).
5. H. Riedel, H. Zipse, Svoboda J. Equilibrium pore surfaces, sintering stresses andconstitutive equations for the intermediate and late stages of sintering – II. Diffusional
densification and creep. Acta. Metall. Mater. 42, 445-452 (1994).
6. E. A. Olevsky. Theory of sintering: from discrete to continuum. Mater. Sci. Engn. R.
23, 41-100 (1998).
7. T. Kraft, H. Riedel. Numerical simulation of solid state sintering; model andapplication. J. Eur. Ceram. Soc. 24, 345-361 (2004).
8. R. Hill. Elastic properties of reinforced solids: some theoretical principles. J. Mech.
Phys. Solids. 11, 357-372 (1963).
9. Z. Hashin. Analysis of composite materials. J. Appl. Mech. 50, 481-505 (1983).
10. O. Guillon, J. Rodel, R. K. Bordia. Effect of green-state processing on the sintering
stress and viscosity of alumina compacts. J. Am. Ceram. Soc. 90, 1637-1640 (2007).
11. A. Flugel. Glass viscosity calculation based on a global statistical modelling
approach. Eur. J. Glass. Sci. Technol. A. 48, 13-30 (2007).
12. G. Okuma et al. Interface topology for distinguishing stages of sintering. Sci Rep, 7,
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Computation of sintering stress and bulk viscosity for each stage of sintering

11106 (2017).
13. G. Okuma, et al. Determination of the size of representative volume element for
viscous sintering. J. Ceram. Soc. Japan. 124 [4], 421-425 (2016).
14. A. Vagon, J. P. Riviere, J. M. Missiaen et al. 3D statistical analysis of a copper
powder sintering observed in situ by synchrotron microtomography. Acta. Mater. 56,
1084-1093 (2008).
15. R. A. Gregg, F. N. Rhines. Surface tension and the sintering force in copper. Metall
Trans. 4, 1365-1374 (1973).
16. E. H. Aigeltinger, R. T. DeHoff. Quantitative determination of topological andmetric properties during sintering of copper. Metall. Trans. A. 6, 1853-1862 (1975).
17. R. T. DeHoff, F. N. Rhines. Quantitative Microscopy. New York City, New York:
McGraw-Hill (1968).
18. F. Wakai, O. Guillon. Evaluation of sintering stress from 3-D visualization of
microstructure: case study of glass films sintered by viscous flow and imaged by X-ray
microtomography. Acta. Mater. 66, 54-62 (2014).
19. A. E. Badger, C. W. Parmelee, A. E. Williams. Surface tension of various molten
glasses. J. Am. Ceram. Soc. 20, 325-379 (1937).
20. J. K. Mackenzie, R. Shuttleworth. A phenomenological theory of sintering. Proc
Phys Soc London. 62, 833-852 (1949).
21. F. Wakai. Mechanics of viscous sintering on the micro-andmacro-scale. Acta. Mater.
61, 239-247 (2013).
22. F. Wakai et al. Sintering force behind the viscous sintering of two particles. Acta
Mater. 109. 292-299 (2016).
23. G. W. Scherer. Coarsening in a viscous matrix. J. Am. Ceram. Soc. 81, 49-54
72

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Computation of sintering stress and bulk viscosity for each stage of sintering(1998).
24. R. K. Bordia, R. Raj. Sintering behavior of ceramic films constrained by a rigid
substrate. J. Am. Ceram. Soc. 68, 287-292 (1985).

73

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Computation of sintering stress and bulk viscosity for each stage of sintering

Fig.3.1. Evolution of pore space in viscous sintering and the change in poremorphology: (a) Initial stage (relative density, ρ=64%); (b) Intermediate stage(ρ=83%); (c) Final stage (ρ=98%)

74

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Computation of sintering stress and bulk viscosity for each stage of sintering

Fig.3.2 Densification curve obtained from the analysis of microtomography. The
sintering time is defined as a sum of holding time at 690°C. The final holding timeat 700°C (▲) was corrected for effective time at 690°C.

75

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Computation of sintering stress and bulk viscosity for each stage of sintering

Fig.3.3. Relationship between sintering stress and relative density ρ. Sintering stressΣs by curvature method (▲); Σs by the mixed method (■); Σs by the surface energy
method (●). Rectangular bars show the range of Σs estimated from the averagecontact radius and the average coordination numbers.

76

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Computation of sintering stress and bulk viscosity for each stage of sintering

Fig.3.4. Relationship between strain rate and relative density ρ. Strain rate wasestimated from the analysis of microtomography.

77

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Computation of sintering stress and bulk viscosity for each stage of sintering

Fig.3.5. Relationship between bulk viscosity K and relative density ρ. Bulk
viscosities were calculated from Equation (3.2) using estimated sintering stress.

78

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Computation of sintering stress and bulk viscosity for each stage of sintering

Fig. 3.6. Local coordination structure in sintering of periodically packed
monodispersed particles. (a) the simple cubic lattice (Z=6), (b) the body centeredcubic lattice (Z=8).

79

Chapter 3

Computation of sintering stress and bulk viscosity for each stage of sintering

Fig.3.7. Relationship between sintering stress and relative density ρ in constrained
sintering. [18] Sintering stress Σs by the mixed method (■): Σs by the surface energy
method (●).

80

Chapter 3

Computation of sintering stress and bulk viscosity for each stage of sintering

Fig.3.8. Normalized pore size distribution in the final stage of sintering (Free sintering(● Present work); Constrained sintering (▲ Ref. [18]). (a) frequency, (b) volume
fraction, and (c) cumulative volume fraction. Pore radius is normalized by the initial81

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

Chapter 4 Experimental verification of sintering stress andbulk viscosity estimated from X-ray microtomography

4.1 Introduction
The macroscopic shrinkage in sintering is described by continuum mechanics,
where the strain rate is expressed as a response to the mechanical stress and thesintering stress Σ s as described in Eq. (1.6) in section 1.3. The continuum mechanicalparameters for sintering, i.e. Σ s , shear viscosity G, and bulk viscosity K, are not constantsince they depend on the relative density of the compact. Theoretical andsemi-empirical formulas have been proposed to calculate those parameters, and havebeen well reviewed by Olevsky1. However, these parameters depend on different factors,
such as the nature of starting powder, powder processing, particle packing, sintering
temperature, and grain growth. Thus, for accurate simulation, it is necessary to
determine these parameters experimentally under different conditions of powder
processing and sintering. Among several experimental methods, sinter forging 2,3 hasbeen widely used to determine these parameters, in which strain rates are typicallymeasured at a fixed applied stress for a specific microstructure. Zuo, Aulbach, andRödel4 have proposed an approach of discontinuous sinter forging (DSF) in order toaccurately determine the sintering parameters, eliminating the microstructuralanisotropy during forging induced by the application of uniaxial load. This method was
successfully applied to the sintering of alumina powder4,5 and a glass/ceramic composite
for the low-temperature co-fired ceramics (LTCC) 6,7. Guillon, Rödel and Bordia8
revealed that the continuum mechanical parameters were affected by powder processing
method, even when the same powder was used. Discontinuous sinter forging is an82

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

effective approach to obtain reliable data, but it is an expensive and tedious procedure
since many pre-sintered specimens are required.
The purpose of this chapter 4 is to measure Σ s and K in viscous sintering of glass
particles experimentally by macroscopic sinter forging test, and to compare with thoseestimated from microstructure observation by X-ray microtomography, which is
obtained in chapter 3. I performed discontinuous sinter forging of calcium
aluminosilicate (CAS) glass particles at different uniaxial pressures. The continuum
mechanical parameters were compared with those estimated from X-ray
microtomography in viscous sintering of aluminosilicate glass. I show that bothmethods give consistent values, when the difference in viscosity and particle size arecorrected. These parameters in viscous sintering of glass will provide a basic forunderstanding more complex processes, such as the liquid phase sintering9,10 and thesintering of LTCC6,7.

4.2 Mechanics of sinter forging
The constitutive equation of sintering (Eq. (1.6)) is written in cylindricalcoordinates by using uniaxial viscosity11:ε z = ε f +1[σ z −ν (σ r + σ θ )]
E

(4.1)

εr = ε f +1[σ r − ν (σ θ + σ z )]
E

(4.2)

where ε z is the axial strain rate, εr is the radial strain rate, E is the uniaxial viscosity,and ν is the viscous Poisson’s ratio. ε f is the free strain rate, i.e. the strain rate
without any external applied load, and related to the sintering stress by:83

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomographyε f = − Σ s 3K

(4.3)

During uniaxial sinter forging of a cylindrical sample under a uniaxial stress σ z , theradial and hoop stresses should be σ r = 0 and σ θ = 0 . Thus, Eq. (4.3) can be simplifiedto:ε z = ε f + σ z E(4.4)

Experimentally, E is determined as the reciprocal of the slope of ε z vs. σ z plot insinter forging tests using specimens previously pre-sintered to obtain a fixed relative
density. The volumetric strain rate εV = 2εr + ε z , is equivalent to the trace of E ij in Eq.
(1.6):εV = 3ε f + σ z 3K

(4.5)

The bulk viscosity K is determined from the reciprocal of the slope of εV vs. σ z plot7.
The shear viscosity G and the viscous Poisson’s ratio ν

are calculated from E and K:

G = 3KE (9 K − E )(4.6)

ν = (3K − E ) 6 K(4.7)

Uniaxial sintering stress s s is defined as the uniaxial tensile stress necessary to stop theshrinkage ( ε z = 0 in Eq. (4.4)):s s = − Eε f(4.8)

From Eq. (4.3), Eq. (4.8), and the relation K = E 3(1 − 2ν ) , the hydrostatic sintering
stress is given by:

Σ σ = 3Kσ σ E = σ σ (1 − 2ν )

(4.9)
84

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

4.3 Experimental procedure
A commercially available glass powder composed of 35 wt% CaO, 30 wt% Al2O3,
and 35 wt% SiO2 (calcium aluminosilicate glass designated as CAS, EG2705; Ferro
Corp., Cleveland, OH) was chosen in this study. The glass had a softening point of 925 °C,
annealing point of 825 °C, and a grass transition temperature of 800 °C. The density of
the glass was 2.8 kg/m3, and the mean particle size was 2.7 µm (Cumulative percentile
values D10 and D90 were 1.8 and 3.8 µm, respectively). The powder was uniaxiallypressed at 200 MPa to shape cylindrical green bodies with diameter of 8.06 ± 0.01 mm
and height of 10.66 ± 0.14 mm. The relative density of green bodies was 56.7 ± 0.4 %(calculated from the weight, height, and diameter of the cylinders). The specimens werepre-sintered in a conventional furnace in air to achieve defined relative densities of, 68%,
74%, 82%, 90% and 95%. Final densities were also measured by the Archimedes
method, and it keeps a good consistency with those calculated values.
Samples were sintered in air using a modified sinter-forging setup12, composed of avertical split furnace fixed on a mechanical testing machine (Model 5565; Instron Corp.
Norwood, MA). The pre-sintered bodies were located at the centre of the furnace andbetween two alumina discs, which served as reference markers for measuring thespecimen height. A minimal load of 5 N was applied during the whole thermal cycle toensure the contact between the samples and the discs. Height and diameter of the samples
were measured in-situ by a high-resolution laser scanner (162-100; BETA LaserMike,
Dayton, OH) with a resolution of 0.5 µm every second. As a result, axial and radial truestrains can be continuously calculated during the densification of the specimens. The
temperature was controlled by a type R thermocouple in contact with the lower
supporting alumina disc, and at only 4-5 mm from the sample. The specimens were
heated at 30 °C/min up to 850 °C. The sinter forging tests were conducted at 5, 10, 15 and
20 N at 850 °C. The load was kept constant during the isothermal holding time.

85

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

4.4 Results
Figure 4.1 (a) shows the microstructure of a sample calcined at 850 °C for 0 min(relative density of 58 %). The large particles maintained angular shapes typical of
crushed glass, meanwhile the smaller particles became spherical during pre-sintering.
Spherical closed pores were formed in the final stage of sintering (Fig. 4.1 (b), relative
density of 95 %). The X-ray diffraction results showed no crystalline peaks even after thesinter forging tests.
Figure 4.2 shows sintering strain curves of specimens (pre-sintered to relative
density of 67 %) as a function of sintering time. The axial strain was negative (shrink)and the radial strain was positive (expand). The strain rate was defined as the slope of thecurve at 250 sec after the application of load. For pre-sintered specimens with different
relative densities, axial strain rate ε z and volumetric strain rate εV were plotted asfunctions of uniaxial stress σ z in Fig. 4.3 (a) and (b), respectively. The absolute value offree strain rate, which was estimated by extrapolation to zero stress, decreased withdensification generally. However, the inspection of Fig. 4.3 (a) showed that the freestrain rate at the relative density of 95 % was higher than that at relative density of 74 %.
Since it is unphysical, we did not include the data at relative density of 95 % in further
analysis. Assuming the linear relationship between strain rates and uniaxial stress,
uniaxial viscosity E and bulk viscosity K were calculated from the slope of lines in Fig.
4.3 (a) and (b), respectively. In addition, the shear viscosity G was also calculated by Eq.
(4.6). Figure 4.4 shows E , K , and G increase with relative density. K is lower thanE at low density region, and becomes larger than E

86

at high density region. G

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

increased with relative density in the range from E 2 to E 3 . The uniaxial viscosity inviscous sintering of CAS glass at 850 °C was higher than that of LTCC at 820 °C, inwhich the chemical composition of the glass phase was 20.8 wt% CaO, 9.7 wt% Al2O3,
63.4 wt% SiO2, and 6 wt% B2O3.7 It is supposed that the low uniaxial viscosity of LTCC
material arises from the difference in chemical composition of glass phase. The viscous
Poisson’s ratio ν was calculated by Eq. (4.7), and plotted in Fig. 4.5. Generally viscous
Poisson’s ratio increases with increasing relative density. The Poisson’s ratio obtained inviscous sintering of glass showed the same tendency with those obtained in sintering ofalumina ceramics5. On the other hand, the viscous Poisson’s ratio of LTCC showed aquite different curve, probably due to the initial anisotropy of the LTCC tape12.
The hydrostatic sintering stress Σ s was calculated from the free strain rate byusing Eqs. (4.8) and (4.9). We plotted sintering stress of CAS glass and LTCC7
determined by discrete sinter forging test in Fig. 4.6 (a). The sintering stress increased
with densification. While the sintering stress can reach several tens of MPa in sinteringof alumina with the initial particle size of several hundred nanometers4, 5, 8, it was lessthan 1 MPa in viscous sintering of glass particles with the size of a few µm.

87

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

4.5 Discussion
4.5.1 Methods to estimate sintering stress from the microstructure
As described in chapter 3, the sintering stress in viscous sintering is estimatedfrom microstructures obtained by X-ray microtomography. The microstructure in theinitial stage is characterized as the formation and growth of contact between
neighboring particles, where relative density increases from 50 % to 60 %. The sintering
stress is given by:Σs = F s A(Sintering force method)(4.10)

where F s is the sintering force, and A is the area of the face of Wigner-Seitz cell,which depends on the coordination number. The sintering force, F s , acting between
two identical particles can be approximated as a parabolic function of contact radius, c ,in the early stage (0.1 ≤ c r0 ≤ 0.7) 14:
F s = γ S r0 [−11.53(c r0 ) 2 + 13.58(c r0 ) + 0.40](4.11)In the intermediate stage of sintering, the pore structure evolves into interconnectedchannels where surface curvature depends on position. The sintering stress is estimatedby:Σ s = rγ s κ +2γ s A pore
3V

(Mixed method)(4.12)

where ρ is the relative density, γ s is the surface energy, κ is the average ofcurvature on pore surface, and A pore is the total surface area in a volume element V.
The final stage begins when the pinch-off of interconnected pore channel forms closed
pores at relative density of 90 %. The sintering stress is simply expressed by 15:88Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomographyΣs =2γ s A pore

(Energy method)

3V pore(4.13)

where V pore is the total pore volume in the volume element.
The microstructure observation by X-ray microtomography provides the averagecoordination number, the average contact radius, ρ , V pore , A pore , and κ in realporous microstructure during sintering. Figure 4.6 (b) shows sintering stress estimatedfrom X-ray microtomography observation of viscous sintering of spherical soda lime
glass particles with the average particle radius of 80 µm. The glass had the compositionof 72.0 wt% SiO2, 13.5 wt% Na2O/K2O, 9.0 wt% CaO, 3.4 wt% MgO, and 2.0 wt%
Al2O3, 0.1 wt% Fe2O3. The sintering stress Σ s in Fig. 4.6 (b) is normalized by γ s r0
where r0 is the average initial particle radius. The estimated sintering stress was about

γ s r0 in the initial stage, and increased to 4 γ s r0 in the final stage.

4.5.2 Comparison with sinter forging results
The sintering stress of CAS glass measured by discrete sinter forging was
normalized by assuming r0 = 1.35 µm and γ s = 0.4 J m-2. Here, the surface energy γ s
at 850 ºC was estimated from the surface tension at 1600 ºC (See Fig. 8 in Ref. 16) andthe temperature coefficient, dγ s dT

16-18

. The normalized sintering stress of CAS glass

in Fig. 4.6 (b) was in fair agreement with values estimated by the mixed method fromthe microtomography data of soda lime glass, although the initial particle size, particle
shape, and chemical composition were different.
89Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

Mackenzie and Shuttleworth gave the relation between bulk viscosity andporosity in the final stage of viscous sintering by assuming spherical pores of identicalsize. It is written as a function of relative density:K=4µρ
3(1 − ρ )(4.14)

where µ is the viscosity of glass. The experimental data in Fig. 4.4 was fitted by Eq.
(4.14) with µ = 2.0 × 109 Pa·s for the CAS glass. The dashed line depicts the fitted
curve. The bulk viscosity of CAS glass was normalized by µ , and K µ was plotted asa function of relative density in Fig. 4.7. The bulk viscosity of soda lime glass was also
estimated by using Eq. (4.3) from free strain rates and sintering stresses obtained by the
X-ray microtomography. The bulk viscosity of soda lime glass was normalized by usingµ = 6.0 × 107 Pa·s, and plotted in Fig. 4.7 for comparison. In the initial stage, these
normalized bulk viscosity K µ obtained by discrete sinter forging agreed fairly wellwith that obtained by the mixed method (X-ray microtomography). In the final stage,
the difference of K µ between the mixed method and the energy method decreased,
and K µ obtained by the discrete sinter forging agreed with the values obtained by
X-ray microtomography. This experimental result supports the theoretical analysis thatthe bulk viscosity is proportional to the viscosity of glass, and is simply a function ofrelative density, independent of the distribution function of pore size 15.

4.5.3 Difference between the viscous sintering and the sintering of
crystalline materials
Zuo et al. 4 reported that the sintering stress of alumina increased slightly withdensity, and then, decreased at the final stage of sintering as significant grain growth90

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

took place. For a single closed pore in sintering of crystalline material, Raj19 expressed
the sintering stress as follows:Σ s = g s κ pore + α

g gb(4.15)dwhere κ pore is the curvature of pore, g gb is the grain boundary energy, and d is thegrain size. Therefore, both pore coarsening and grain growth decrease the sintering
stress in the final stage. On the other hand, the sintering stress increased with decreasing
pore size even at the final stage in Fig. 4.6 (a), because there is no grain growth inviscous sintering. However, it should be noted that pore coarsening can occur inconstrained sintering due to tensile stress in the film. In this case, sintering stressdecreases with increasing density in viscous sintering 13, 20.

91

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

4.6 Conclusions
The discontinuous sinter forging technique was applied for the experimental
determination of parameters in viscous sintering of glass particles. The bulk viscosity
and shear viscosity increased with increasing relative density. The sintering stressincreased with relative density. The decrease of sintering stress in the final stage, which
had been reported during sintering of crystalline particles, was not observed in viscous
sintering. The parameters obtained by the discontinuous sinter forging were comparedwith those estimated from the microstructural observation by using X-ray
microtomography. Even though the initial particle size and chemical composition ofglass particles were different, both techniques provided consistent and reliable values
when the bulk viscosity was normalized by the viscosity of glass and the sintering stress
was normalized by the surface tension and the initial particle size.
Recently the vertical sintering technique was developed to measure uniaxial
viscosity of thin ceramic layer21 and LTCC22, because sinter forging experiments are notapplicable to thin layers. The knowledge of sintering stress and uniaxial viscosity areuseful to analyze camber evolution and stress development in bilayer laminates23. The
authors believe the present technique by using X-ray microtomography will be also
applicable to analyze thin layers in laminates.

92

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

References
1.

E.A. Olevsky, Theory of sintering: from discrete to continuum, Mater. Sci. Eng.
R23, 41-100 (1998).

2.

K.R. Venkatachari, R. Raj, Shear deformation and densification of powder
compacts, J. Am. Ceram. Soc. 69, 499-506 (1986).

3.

M.N. Rahaman, L.C. de Jonghe, G.W. Scherer, R.J. Brook, Creep anddensification during sintering of glass powder compacts, J. Am. Ceram. Soc. 70,
766-774 (1987).

4.

R. Zuo, R., E. Aulbach, J. Rödel, Experimental determination of sintering stresses
and sintering viscosities, Acta Mater. 51 4563-4574 (2003).

5.

R. Zuo, R., E. Aulbach, J. Rödel, Viscous Poisson’s coefficient determined bydiscontinuous hot forging, J. Mater. Res. 18, 2170-2176 (2003).

6.

J.B. Ollagnier, O. Guillon, J. Rödel, Viscosity of LTCC determined bydiscontinuous sinter-forging, Int. J. Appl. Ceram. Tech. 3 437-441 (2006).

7.

J.B. Ollagnier, O. Guillon, J. Rödel, Effect of anisotropic microstructure on theviscous poperties of an LTCC Material, J. Am. Ceram. Soc. 90, 3846-3851 (2007).

8.

O. Guillon, J. Rödel, R.K. Bordia, Effect of green-state processing on the sintering
stress and viscosity of alumina compacts, J. Am. Ceram. Soc. 90, 1637-1640(2007).

9.

T. Ostrowski, A. Ziegler, R.K. Bordia, J. Rödel, Evolution of Young’s modulus,
strength, and microstructure during liquid-phase sintering, J. Am. Ceram. Soc. 81,
1852-1860 (1998).

10. J. Gonzalez-Julian, O. Guillion, Effect of electric field/current on liquid phasesintering, J. Am. Ceram. Soc. 98, 2018-2027 (2015).
93

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

11. R.K. Bordia, G.W. Scherer, On constrained sintering – II. Comparison of
constitutive models, Acta Mater. 36, 2399-2409 (1988).
12. E. Aulbach, R. Zuo, J. Rödel, Laser-assisted high-resolution loading dilatometerand applications, Exp. Mech. 44, 71-75 (2004).
13. G. Okuma, D. Kadowaki, T. Hondo, A. Sato, S. Tanaka, F. Wakai, Computation ofsintering stress and bulk viscosity from microtomographic images in viscous
sintering of glass particles, J. Am. Ceram. Soc. 100, 867-875 (2017).
14. F. Wakai, K. Katsura, S. Kanchika, Y. Shinoda, T. Akatsu, K. Shinagawa,
Sintering force behind the viscous sintering of two particles, Acta Mater. 109,292-299 (2016).
15. F. Wakai, Mechanics of viscous sintering on the micro- and macro scale, Acta
Mater. 61, 239-247 (2013).
16. D.A. Weirauch Jr, D.P. Ziegler, Surface tension of calcium aluminosilicate glass
using computerized drop shape analysis, J. Am. Ceram. Soc. 79, 920-926 (1996).
17. K. Mukai, T. Ishikawa, Surface tension measurements on liquid slags in CaO-SiO2,
CaO-Al2O3 and CaO-Al2O3-SiO2 system by a pendant drop method, J. Japan. Inst.
Met. 45, 147-154 (1981).
18. L. R. Barret, A. G. Thomas, Surface tension and density measurements on molten
glasses in the CaO-Al2O3-SiO2 system, J. Soc. Glass Technol. 43, 179T-190T(1959).
19. R. Raj, Analysis of the sintering pressure, J. Am. Ceram. Soc. 70, C210-C211(1987).
20. F. Wakai, O. Guillon, Evaluation of Sintering Stress from 3-D visualization of
microstructure: Case study of glass films sintered by viscous flow and imaged by94

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

X-ray microtomography, Acta Mater. 66, 54-62 (2014).
21. M. Cologna, V. M. Sglavo, Vertical sintering to measure the uniaxial viscosity ofthin ceramic laryers, Acta Mater. 58, 5558-5564 (2010).
22. L. Yu, S. Qi, Z. Ma, R. Zuo, Experimental determination of the uniaxial viscosityof low-temperature co-fired ceramic tapes by vertical sintering, Ceram int. 40,
9367-9375 (2014).
23. D. W. Ni, V. Esposito, C. G. Schmidt, T. T. Molla, K. B. Andersen, A. Kaiser, S.
Ramousse, N. Pryds, Camber evolution and stress development of porous ceramic
bilayers during co-firing. J. Am. Ceram. Soc. 96, 972-978 (2013).

95

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

Fig.4.1. Microstructure of CAS glass sample. (a) fracture surface of a calcined
specimen (relative density of 58%), (b) polished surface of a specimen pre-sintered
to relative density of 95%.

96

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

Fig.4.2. Sintering strain curves of CAS glass as a function of sintering time at
850 °C: εz is the axial strain and εr is the radial strain. A fixed uniaxial load (5, 10,15, and 20 N) was applied to each pre-sintered specimen with the relative density of
67%.

97

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

Fig.4.3. Linear relationship between strain rate and uniaxial stress of CAS glass. (a)Axial strain rate εz vs. uniaxial stress σz, (b) Volumetric strain rate εv vs. uniaxial
stress σz. The relative densities of pre-sintered specimens were 68% (●), 74% (▲),
82% (■), 90% (▼), and 95% (♦).

98

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

Fig.4.4. Viscosities as a function of relative density at 850 °C. Dashed line shows
Mackenzie-Shuttleworth’s model on bulk viscosity.

99Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

Fig.4.5. Viscous Poisson’s ratio as a function of relative density at 850 °C.

100

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

Fig.4.6. Sintering stress as a function of relative density. (a) Sintering stressdetermined by discrete sinter forging test, (b) Comparison with sintering stressestimated from X-ray microtomography observation.

101

Chapter 4 Experimental verification of sintering stress and bulk viscosity estimated from X-ray microtomography

Fig.4.7. Normalized bulk viscosities determined from DSF tests and estimated from
X-ray microtomography [13].

102Chapter 5

Determination of the size of representative volume element for viscous sintering

Chapter 5

Determination of the size of representative

volume element for viscous sintering

5.1 Introduction
The continuum mechanics is normally concerned with the behavior of matter ona macroscopic scale that is large compared with discrete particles. However, although
there is general agreement on the need for a unified treatment of continuous media,
there is no agreement as to the proper level at which this unification should take place.
In this chapter, I introduce the concept of the representative volume element (RVE),
which is a volume that is sufficiently large to contain enough information at the micro
scale in order to be representative, but it should be much smaller than the macroscopic
body.1 The macroscopic properties in heterogeneous materials are defined as theaverage over the RVE. This separation of scales is known as the Micro-Meso-Macroprinciple,2 so that the property of RVE provides a basis for multiscale analysis. Several
works have investigated the existence of an RVE and the possibility to determine its
size by using statistical-numerical analysis.3-7 These studies intended to predict theeffective mechanical properties of composites. Recent advances in X-ray
microtomography8 and focused ion beam-scanning electron microscopy9 have opened
the door for observation of detailed microstructural evolution during sintering, andprovides an opportunity to analyze the RVE from the knowledge of microstructure
experimentally. In the field of particle packing, for example, Razavi and coworkers10
reported that the concept of RVE provides an effective means of developing
macroscopic measures in the description of granular materials. The RVE size has been103

Chapter 5

Determination of the size of representative volume element for viscous sintering

determined for electrodes of a Li-ion battery11 and solid oxide fuel cell.12-14
In this chapter, the RVE size was determined from the X-ray microtomographyobservation for fundamental properties in sintering: relative density, specific surface
area, and hydrostatic component of sintering stress. I show that the minimum size of
RVE depends on the material property and level of precision required for the analysis.
In this chapter, the energy method was used to calculate sintering stress on each stage ofsintering. The hydrostatic component of sintering stress in a volume element V is simply
calculated from its relative density ρ and its specific surface area SV = Apore / V for
viscous sintering15,16

2γ S
Ss = s V
3(1 − ρ )(5.1)

where γ s is the surface energy, and Apore is the surface area of pores.

5.2 Experimental procedure
Processing of glass films from spherical particles and synchrotron X-ray
microtomography experiment has been performed by Bernard, Guillon, Combaret andPlougorven in Ref. 4. Experimental procedure is described as follows in detail;A model material was chosen in this study for viscous flow sintering. Spheriglass
5000 powder (Potters Industries, Valley Forge, PA, USA) is composed of glass
microspheres of mean size 3.5-8 μm. At least 75% of the particles have a diameter
under 12 μm (all particles being smaller than 45 μm). It is a soda lime A glass,composed of 72.5% SiO2, 13.7% Na2O, 9.8% CaO, 3.3% MgO, and impurities ofaluminium oxide, iron oxide and potassium oxide, at less than 0.4% each (density 2.50

104

Chapter 5

Determination of the size of representative volume element for viscous sinteringg cm−3, softening point 740 °C).
To obtain a more homogeneous particle size distribution around 8 μm, particles
were wet sieved twice (nickel sieves, Precision Eforming, Cortland, NY, USA), in orderto remove particles smaller than 7 μm and bigger than 15 μm. Particle size distribution
was then measured by means of laser scattering (Analysette 22, Fritsch, Germany). If
99% of all particles of the remainder were below 15 μm and the modal grain size was 7
μm, 25% were still under 3 μm after longer sieving. This will have consequences on the
microstructure of tapes and subsequent sintering.
Films ∼300 μm thick were tape cast on alumina substrates (Rubalit 710,
CeramTec, Germany) by means of a laboratory tape-casting apparatus. First,
polyvinylalcohol (PVA) (MW = 10,000-15,000, Erkol, Spain) was dissolved indeionized water to a concentration of 22 wt.% to act as a binder. Powder was gradually
added to the solution to obtain an aqueous slurry with 45 vol.% solid content.
Polyelectrolyte Dolapix CE64 (Zschimmer & Schwarz GmbH & Co. KG, Germany)
was used as a dispersant (0.7 mg m−2 of glass powder surface). 1-3 drops of Surfynol
DF110D (Air Products BV, the Netherlands) were added to avoid air bubbling. Slurry
was homogenized by ultrasonication for >10 min under cooling conditions. After
mixing in containers on the rolling bench at medium speed overnight the slurry washomogenized by ultrasonication for a second time and tape-cast on alumina substratewith a blade height of 0.5 mm and at a speed of 1.6 m s−1. After drying overnight inambient conditions the tapes were cut into 2 × 2 cm2 squares. The initial film density of
about 61.5% was measured by the Archimedes’ method performed in water on slightly
consolidated specimens, which were easily removed from the underlying substrate.
Sintering conditions were as follows: heating rate of 5 °C min−1 up to 600 °C to105

Chapter 5

Determination of the size of representative volume element for viscous sinteringobtain a complete binder burn-out and then 20 °C min−1 up to 700-750 °C. Isothermal
times between 0 and 10 min result in four different densities ranging from ∼64% to ∼98%. Small pieces with dimensions smaller than 600 μm were cut with a diamond wire
saw and glued onto a specimen holder for the synchrotron computed microtomography.
Due to the difficulty of specimen preparation, it was impossible to coincide its
coordinate system with the global coordinate system of the tape-casting process (except
for the thickness).
X-ray microtomography data were acquired by a monochromatic beam (20.5
keV) at the experimental station ID19 of the European Synchrotron Radiation Facility(ESRF, Grenoble, France). Radiographs were acquired by rotating the sample by stepsof 0.12° until 180°. The 3-D mapping with voxel size of 0.28 × 0.28 × 0.28 μm werereconstructed were reconstructed from the acquired data by the filtered backprojection
method. The 3-D visualization and geometrical measurements were performed using
Amira (VSG) in the present study. The watershed transform was used to segment thegray value image into pore and material. The pore surface was discretized using
triangular meshing, from which the surface area and the pore volume were calculated.

5.3 Results and discussion
5.3.1 Micro structural evolution in viscous sintering
The pore space evolution in viscous sintering of spherical glass particles
observed by synchrotron X-ray microtomography are shown in Figure 5.1. In the initialstage (Fig. 5.1(a), Stage 1), the pore structure is a continuous network with numerous
circular holes resulting from contacts between particles. As holes expand with the neckgrowth, ligaments are pinched-off, breaking the continuous network into fragments:
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Determination of the size of representative volume element for viscous sinteringclosed pores are formed one by one in the intermediate stage (Fig. 5.1(b) Stage 2, and(c) Stage 3). Complicated shaped pores become spherical in the final stage of sintering(Fig. 5.1(d) Stage 4).
Figure 5.2 shows a 2-D section extracted from the specimen in Stage 1, which iseasier to examine the microstructural heterogeneity. The smallest particles havesegregated at the bottom of the layer, then, I analyze the average properties of the layer
in the middle region marked in Fig. 5.2. Sub-volumes extracted from a total material
domain are restricted to cubic geometries. As many as possible independent cubes withedge length of 160, 80, 40, 20, and 10 μm were extracted from each sample for fourstages. The RVE size is given by using the edge length L of cubic volume element.

5.3.2 Relative density, specific surface area and sintering stress
The relative density of sub-volume in each stage is plotted as a function of theedge length of cubic volume element in Fig.5.3. I measure the relative density of
sub-volume by the local threshold method using Amira (VSG). Figures in brackets referto the number of sub-volumes. The mean relative density is almost independent of edge
length when sufficiently large number of sub-volumes is analyzed; the number ofvoxels per particle diameter was 29 in the present experiments. The scatter of data at
small edge length indicates the scale of microstructural inhomogeneity. Figures 5.4 and5.5 show the specific surface area and the sintering stress as functions of edge length.
For all three quantities the scatter of data increases with decreasing the edge length L,
while their mean values were almost independent of the edge length.
Both mean value of specific surface area and that of sintering stress depends on
sintering stage or mean value of relative density. The linear relation was observed107Chapter 5

Determination of the size of representative volume element for viscous sintering

between mean specific surface and mean relative density.15 The mean sintering stress
was almost constant from stage 1 to stage 3, and decreased slightly at the final stage. In
the final stage of sintering, where closed pores are dispersed sparsely, the macroscopic
sintering stress is dominated by the largest residual pores.16 The mean sintering stressdecreased in stage 4, because the coarsening in pore structure resulted in residual pores
more than two times larger than the initial particle size as shown in Fig. 5.1(d).
Standard deviation for given edge length is shown in Fig. 2.6, as a function of L.
It can be seen that the standard deviation is quite large for a small L; it decreases as L
increases, and converges to a constant value. The variation of standard deviation on L
reflects the microstructural heterogeneity. I may assign the edge length Lr , above which
the heterogeneity no longer affects the standard deviation, as the minimum size of theRVE. From this definition the RVE size is estimated to be from 90 μm to 135 μm for
relative density, specific surface area, and sintering stress. Here, the non-dimensional
RVE size L * is defined as
L*= L/d(2.2)

where d is the average initial particle size (8 μm). The non-dimensional RVE size wasfrom 11 to 17 in the present experiments. This result agrees with the values reported for
granular materials.10 However, this normalization is done with respect to the initial
microstructure, which completely disappears during sintering.
Alternatively the RVE size can be defined from standard deviation-edge length
curves. Since both mean value and standard deviation of each quantity varies withrelative density, we normalized standard deviation by the mean value at stage 1(ρ=63.5%).
The RVE size was defined with the normalized standard deviation; either 5 % or108

Chapter 5

Determination of the size of representative volume element for viscous sintering

2 % for the RVE size of relative density Lr,d , and specific surface area Lr,a , and either
8 % or 5 % for that of sintering stress Lr,s .The RVE size was plotted as a function ofrelative density in Figure 5.7. The RVE size must be larger when the desired accuracy ishigher. The RVE size of relative density (Fig. 5.7(a)) is relatively small in stage 1 where
inhomogeneities can be envisioned as local fluctuation caused by differences in random
particle packing. However, larger RVE size in the later stages suggests that thecharacteristic length to describe microstructural heterogeneity increases from theparticle size to the distance between coarse spherical pores during the microstructural
evolution. The increase in the characteristic length can be called coarsening in a broad
sense. The RVE size of specific surface area (Fig. 5.7(b)) decreased at stage 4. We
suppose it is an artifact due to the definition of Lr,a , from the normalized standard
deviation, because the mean value of specific surface area and also its standard
deviation approach to zero at the final stage of sintering. The RVE size of sintering
stress (Fig. 2.7(c)) increased with densification. The scatter of sintering stress in eachvolume element increases with the relative density ρ, because the sintering stress isinversely proportional to porosity 1− ρ which is close to zero in the final stage. Since
the sintering stress is dominated by the largest pores, the RVE size for sintering stress isrelated to the average distance between large pores in the final stage.

109

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Determination of the size of representative volume element for viscous sintering

5.4 Conclusions
The microstructural evolution of thin glass film during constrained sintering on arigid substrate, imaged by synchrotron X-ray microtomography, was analyzed toevaluate fundamental quantities, i.e., relative density, specific surface area, and sintering
stress. These parameters are important for the continuum mechanical description ofviscous sintering. The minimum size of RVE was defined as the edge length of cubic
volume elements above which the heterogeneity no longer affects the standard
deviation.
The RVE size was estimated to be from 11 to 17 times larger than the average
initial particle size. Alternatively the RVE size was defined from the value of
normalized standard deviation. The RVE size increased when the desired accuracy was
higher. The RVE size was dependent on relative density, and varied withmicrostructural evolution. The relationship between RVE size and relative density wasdependent on the physical property of interest.

110Chapter 5

Determination of the size of representative volume element for viscous sintering

References
1. R. Hill. Elastic properties of reinforced solids: Some theoretical principles. J. Mech.
Phys. Solids, 11, 357-372 (1963).
2. Z. Hashin, Analysis of Composite Materials—A Survey. J. Appl. Mech. 50, 481-505(1983).
3. W. J. Drugan and J.R. Willis, J. Mech. Phys. Solids. A micromechanics-based
nonlocal constitutive equation and estimates of representative volume element size forelastic composites. 44, 497-524 (1996).
4. K. Kanit, S. Forest, I. Galliet, V. Mounoury and D. Jeulin. Determination of the sizeof the representative volume element for random composites: statistical and numericalapproach. Int. J. Solids Structures, 40, 3647-3679 (2003).
5. I. M. Gitman, H. Askes and L.J. Sluys. Representative volume: Existence and size
determination. Engineering Fracture Mechanics 74, 2518-2534 (2007).
6. C. Pelissou, J. Baccou, Y. Monerie and F. Perales. Determination of the size of therepresentative volume element for random quasi-brittle composites. Int. J. Solids
Structures, 46, 2842-2855 (2009).
7. W. M. Harris and W.K.S. Chiu. Determining the representative volume element sizefor three-dimensional microstructural material characterization. Part 1: Predictivemodels. J. Power Sources, 282, 552-561 (2015).
8. D. Bernard, O. Guillon, N. Combaret and E. Plougonven. Constrained sintering ofglass films: Microstructure evolution assessed through synchrotron computed
microtomography. Acta. Mater. 59, 6228-6238 (2011).
9. S. Hara, A. Ohi, N. Shikazono. Sintering analysis of sub-micron-sized nickel
powders: Kinetic Monte Carlo simulation verified by FIB–SEM reconstructionJ. Power111

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Determination of the size of representative volume element for viscous sintering

Sources. 276, 105-112 (2015).
10. M. R. Razavi, B. Muhunthan, O. A. Hattamleh. Representative elementary volume
analysis of sands using X-ray computed tomography. Geotechnical Testing Journal, 30,
No. 3 (2007)
11. P. R. Shearing, L. E. Howard, P. S. Jorgensen, N. P. Brandon and S.J. Harris.
Characterization of the 3-dimensional microstructure of a graphite negative electrodefrom a Li-ion battery. Electrochem. Commun.12, 374-377 (2010).
12. J. Joos, M. Ender, T. Carraro, A. Weber and E. Ivers-Tiffee. Representative volume
element size for accurate solid oxide fuel cell cathode reconstructions from focused ion
beam tomography data. Electrochimica. Acta, 82, 268-276 (2012).
13. J. Laurencin, R. Quey, G. Delette, H. Suhonen, P. Cloetens and P. Bleuet.
Characterisation of Solid Oxide Fuel Cell Ni–8YSZ substrate by synchrotron X-ray
nano-tomography: from 3D reconstruction to microstructure quantification. J. Power
Sources. 198, 182-189 (2012).
14. W.M. Harris and W.K.S. Chiu. Determining the Representative Volume Element
Size for Three-Dimensional Microstructural Material Characterization—Part 2:
Application to Experimental Data. J. Power. Sources. 283, 622-629 (2015).
15. F. Wakai, O. Guillon. Evaluation of sintering stress from 3-D visualization of
microstructure: case study of glass films sintered by viscous flow and imaged by x-ray
microtomography. Acta. Mater. 66, 54-62 (2014).
16. F. Wakai. Mechanics of viscous sintering on the micro-andmacro-scale. Acta. Mater.
61, 239-247 (2013).

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Determination of the size of representative volume element for viscous sintering

Fig.5.1 Evolution of pore space in sintering: (a) stage1 (relative density, ρ=63.5%);(b) stage2 (ρ=87.8%); (c) stage3 (ρ=94.1%); (d) stage4 (ρ=98.4%).

113

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Determination of the size of representative volume element for viscous sintering

Fig.5.2 Section of a sample (stage1, relative density ρ =63.5%). The size of cubic
volume elements are illustrated for comparison to the initial particle diameter.

114

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Determination of the size of representative volume element for viscous sintering

Fig.5.3. Relative density of each volume element : (a) stage1; (b) stage2; (c)stage3; (d) stage4. The dotted line shows the mean value.

115

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Determination of the size of representative volume element for viscous sintering

Fig.5.4. Specific surface area of each volume elements: (a) stage1; (b) stage2; (c)stage3; (d) stage4. The dotted line shows the mean value.

116

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Determination of the size of representative volume element for viscous sintering

Fig.5.5. Sintering stress of each volume element: (a) stage1; (b) stage2; (c) stage3;(d) stage4. The dotted line shows the mean value.

117

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Determination of the size of representative volume element for viscous sintering

Fig.5.6. Standard deviation as a function of the edge length of cubic elements: (a)relative density; (b) specific surface area; (c) sintering stress.

118

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Determination of the size of representative volume element for viscous sintering

Fig.5.7. RVE size as a function of relative density. (a) relative density (Lr,d), (b)specific surface area of pores (Lr,a), and (c) sintering stress (Lr,s) .

119

Chapter 6

Summary

Chapter 6 Summary

The purpose of this doctoral dissertation is to construct the experimental andtheoretical methods to estimate sintering parameters, such as sintering stress that is thethermodynamic driving force for sintering, and bulk viscosity, which are fundamental
quantities for predicting the macroscopic shrinkage behavior in sintering, from 3D
visualization of microstructures observed by X-ray microtomography. Glass was chosen
as a model material, because the mechanism of material transport during sintering ofglass is much simpler than that of crystalline particle, and hence, we only consider
about viscous flow mechanism. The speedy and reliable method of predicting thesintering stress and bulk viscosity was proposed in this research.
This doctoral dissertation was divided into 6 chapters, and abstract of eachchapter was described as below;

Chapter 1: Introduction
The macroscopic continuum mechanics of sintering and the sintering process in
microscopic scale were described, followed by the objective of this doctoral
dissertation.

Chapter 2: Interface topology for distinguishing stages of sintering
Sintering is a common process during which nanoparticles and microparticles
are bonded, leading to the shrinkage of interstitial pore space. Understanding
morphological evolution during sintering is a challenge, because pore structures are
elusive and very complex. A topological model of sintering is presented here,120

Chapter 6

Summary

providing insight for understanding 3-D microstructures observed by X-ray
microtomography. It is found that the topological evolution is described by Euler
characteristics as a function of relative density. The result is general, and applicable
not only to viscous sintering of glasses but also to sintering of crystalline particles. It
provides criteria to distinguish the stages of sintering, and the foundations to identify
the range of applicability of the methods for determining the thermodynamic driving
force of sintering.

Chapter 3: Computation of sintering stress and bulk viscosity for each stage ofsintering
Sintering stress and bulk viscosity were derived as functions of relative densityfrom microtomographic images in viscous sintering of glass particles. Three methods
were proposed to estimate the sintering stress from relative density, specific surface area,
and average of curvature on pore surface, which were directly measured by X-ray
microtomography. These methods for determining sintering stress should be selected
depending on sintering stage, which was determined in chapter 2. The surface energy
method gave valid value in the final stage of sintering, while the mixed method gave
better estimation in the intermediate stage. For the initial stage of sintering, the sintering
stress was calculated from the average contact radius and the average coordination
number observed by X‐ray microtomography. The sintering stress at the final stage
increased in free sintering, but it decreased in constrained sintering due to porecoarsening. The bulk viscosity was calculated from the shrinkage rate and the sintering
stress.

121

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Summary

Chapter 4: Experimental verification of sintering stress and bulk viscosity
estimated from X-ray microtomography
The macroscopic sintering parameters, sintering stress, bulk viscosity, and shear
viscosity, were measured by the discontinuous sinter forging experiment for viscous
sintering of calcium aluminosilicate (CAS) glass. The calculated results were comparedwith experimental values from the microstructural evolution during viscous sintering of
spherical soda lime glass particles by X-ray microtomography in chapter 3. The
sintering stress of CAS glass normalized by surface energy and the initial particle size
was in good agreement with values estimated from the microtomography data of soda
lime glass, despite the differences in particle shape and chemical composition. The bulk
viscosity obtained by discontinuous sinter forging agreed fairly well with that estimatedby X-ray microtomography observation, when they were normalized by glass viscosity.

Chapter 5: Determination of the size of representative volume element for viscous
sintering
The representative volume element (RVE) is a basic concept in the continuum
mechanics of sintering of random heterogeneous porous materials. A quantitative
determination of its size was performed by using synchrotron X-ray microtomography
data of constrained sintering of thin glass film on a rigid substrate. A RVE size isassociated with a property of interest; I determined it for relative density, specificsurface area, and hydrostatic component of sintering stress. The RVE size was estimatedto be from 11 to 17 times larger than the average initial particle size. The RVE size wasassociated with a given precision of the property. It depended on the volume fraction of
porous structure, or, relative density, so that it varied with microstructural evolution.
122

Chapter 6

Summary

List of publications(1) Journal Paper (Reviewed)
1) G. Okuma, J. Gonzalez-Julian, O. Guillon, F. Wakai, Comparison between sinterforging and X-ray microtomograhy methods for determining sintering stress and bulk
viscosity, J. Euro. Ceram. Soc, 38, 4, 2053-2058, (2018).
2) G. Okuma, D. Kadowaki, T. Hondo, S. Tanaka, F. Wakai,Interface topology for distinguishing stages of sintering, Sci Rep,7, 11106 (2017).
3) G. Okuma, D. Kadowaki, T. Hondo, A. Sato, S. Tanaka, F. Wakai, Computation ofSintering Stress and Bulk Viscosity from Microtomographic Images in Viscous
Sintering of Glass Particles, J. Am. Ceram. Soc, 100, 867-875 (2017).
4) G. Okuma, D. Kadowaki, Y. Shinoda, T. Akatsu, O. Guillon and F. Wakai,
Determination of the size of representative volume element for viscous sintering, J.
Ceram. Soc. Japan, 124 [4], 421-425 (2016).
5) F. Wakai, O. Guillon, G. Okuma, N. Nishiyama, Sintering Forces Acting among
Particles during Sintering by Grain Boundary/Surface Diffusion, J. Am. Ceram. Soc.
DOI10.1111/jace.15716(2) Other publication - Review article6) G. Okuma, F. Wakai, S. Tanaka, T. Hondo, J. Gonzalez-Julian, O. Guillon, ,Determination of sintering stress and bulk viscosity from sinter-forging and X-ray
microtomography methods, Materials Today Proceedings, ISAC-6 (2018). submitted7) 大熊 学, 菅近 駿, 田中 諭, 若井 史博, X 線マイクロトモグラフィーによる焼結過程の微構造の 3 次元可視化と巨視的力学量の評価, 粉体粉末冶金 講演特集 第 64 巻 第 9 号 495-500 (2017). 査読有
8) 大熊 学, 若井 史博, 粉体成形プロセスにおける微視的な粒子充填構造と焼結における巨視的な収縮との関連の 3 次元トモグラフィーによる解明, ホソカ
ワ粉体工学振興財団年報 No.24, 144-147 (2016).

123

査読無

Chapter 6

Summary

List of presentations(1) International conference1) G. Okuma, S. Tanaka and F. Wakai, Distinction of stages of sintering from 3D
visalization of microstructure in sinteting by using X-ray microtomography, 42ndInternational Conference and Exposition on Advanced Ceramics and Composites,
Daytona Beach, Florida, USA, January (2018). (Invited)2) G. Okuma, F. Wakai , J. Gonzalez-Julian, O. Guillon, Comparison between sinterforging and X-ray microtomograhy methods for determining sintering stress and bulk
viscosity, 6th International Symposium on Advanced Ceramics (ISAC-6) March (2018).
(Reviewed)
3) G. Okuma, F. Wakai , J. Gonzalez-Julian, O. Guillon, Sinter forging and X-ray
microtomograhy methods for determining sintering stress and bulk viscosity, The 6thInternational Conference on Characterization and Control of Interface for High Quality
Advanced Materials (ICCCI2018) July (2018). (Student Poster Award)4) Gaku Okuma, Satoshi Tanaka, Fumihiro Wakai. Mechanics and evolution of
microstructural topology in viscous sintering observed by X-ray microtomography,
Sintering2017, Nov (2017). (Reviewed)5) Gaku Okuma, Daiki Kadowaki, Satoshi Tanaka, Fumihiro Wakai. Mechanics andevolution of interfacial topology for distinguishing stages of sintering, The Sixth
International Education Forum, Dec (2017). (Reviewed)6) Gaku Okuma, Daiki Kadowaki, Tsuyoshi Hondo, Satoshi Tanaka, Fumihiro Wakai.
Evolution of interface topology from microscopic observation of viscous sintering ofglass particles by using X-ray microtomography, The 34th International Japan-Korea
Seminar on Ceramics, Nov (2017).
7) Gaku Okuma, Daiki Kadowaki, Satoshi Tanaka, FUMIHIRO WAKAI. Distinctionof sintering stage by interfacial topology, The 10th International Conference on the
Science and Technology for Advanced Ceramics (STAC-10), Aug (2018). (Poster
Award)
8) G. Okuma, D. Kadowaki, S. Tanaka, F. Wakai. Bulk viscosity and sintering stress inviscous sintering of glass particles estimated from microscopic observation by using
X-ray microtomogtaphy, The Fifth International Education Forum, Dec (2016).
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Chapter 6

Summary(Reviewed)
9) G. Okuma, D. Kadowaki, S. Tanaka, F. Wakai. Estimation of bulk viscosity andsintering stress from microscopic observation of viscous sintering of glass particles byusing X-ray microtomography, The International Symposium on the Science ofEngineering Ceramics (EnCera 2016), May (2016). (Best Poster Award)10) G. Okuma, D. Kadowaki, F. Wakai. Size Determination of Representative Volume
Element in Microstructure Evolution during Sintering, The Fourth International
Education Forum, Dec (2015). (Reviewed)11) G. Okuma, D. Kadowaki, F. Wakai. Size determination of representative volume
element in microstructure evolution during sintering, The 9th International Conference
on the Science and Technology for Advanced Ceramics (STAC-9), Oct (2015).
12) Fumihiro Wakai, Gaku Okuma, Norimasa Nishiyama. Role of microscopic forces
in densification of particles in solid state sintering, The 6th International Symposium on
Advanced Ceramics (ISAC-6), Mar (2018).
13) Fumihiro Wakai, Shun Kanchika, Gaku Okuma, Norimasa Nishiyama.
Microscopic sintering forces acting among particles during sintering by grainboundary/surface diffusion, Sintering2017, Nov (2017).
14) Fumihiro Wakai, Gaku Okuma, Satoshi Tanaka. Microscopic Sintering Forces
behind Macroscopic Continuum Theory of Sintering, Japan-India YNU Symposium
2017, Dec (2017).

(2) Domestic conference15) 大熊 学, Jesus Gonzalez-Julian, O. Guillon, 若井 史博. X 線トモグラフィーにより推定した焼結応力の焼結鍛造試験による実験的検証, 平成 30 年度春季大会(第 121 回講演大会) 粉末冶金協会, May (2018).
16) 大熊 学, Jesus Gonzalez-Julian, O. Guillon, 若井 史博. 焼結鍛造試験による焼結応力および粘性係数の決定と X 線マイクロトモグラフィーによる微視的構造観察からの推定結果との比較, 日本セラミックス協会 2018 年年会, Mar(2018).
17) 大熊学, 門脇大騎, 田中諭, 若井史博. 焼結中の微構造の 3 次元可視化による焼結段階の分類, 第 56 回セラミックス基礎科学討論会, Jan (2018).
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Chapter 6

Summary18) 大熊 学, 門脇 大騎, 田中 諭, 若井 史博. 焼結段階を区別するための界面トポロジー, 第 30 回日本セラミックス協会秋季シンポジウム, Sep (2017).
19) 大熊 学, Jesus Gonzalez-Julian, Olivier Guillon, 若井 史博. 焼結鍛造試験による焼結応力および粘性係数の決定と X 線マイクロトモグラフィーによる微視的
構造観察からの推定結果との比較, 第 55 回セラミックス基礎科学討論会, Jan(2017).
20) 大熊 学, 門脇 大騎, 田中 諭, 若井 史博. 焼結段階を区別するための界面トポロジー, バルクセラミックスの信頼性に関するワークショップ（第 6 回バル
クセラミックス若手セミナーを兼ねる）, Sep (2017).
21) 大熊 学, 門脇 大騎, 田中 諭, 若井 史博. X 線マイクロトモグラフィーによるガラス粒子の粘性焼結における微視的構造観察からの巨視的粘性係数と焼結応力の推定, 第 29 回日本セラミックス協会秋季シンポジウム, Sep (2016).
22) 大熊 学, 門脇 大騎, 田中 諭, 若井 史博. X 線マイクロトモグラフィーによるガラス粒子の粘性焼結の微視的構造観察からの巨視的粘性係数と焼結応力の推定, 平成 28 年度春季大会 (第 117 回講演大会) 粉末冶金協会, May (2016).
23) 大熊 学, 門脇 大騎, 篠田 豊, 赤津 隆, 若井 史博. 焼結中の微構造変化における代表体積要素の寸法決定, 第 28 回日本セラミックス協会秋季シンポジウ
ム, Sep (2015).
24) 若井 史博, 大熊 学, 西山 宣正, Olivier Guillon, 3 粒子の固相焼結の背後にある焼結力, 日本セラミックス協会 2018 年年会, Mar (2018).
25) 若井史博, 大熊学, 田中諭. 焼結プロセスにおける内部応力の起源のミクロ
スケール力学解析, 第 37 回エレクトロセラミックス研究討論会, Oct (2017).
26) 若井 史博, 大熊 学, 田中 諭, Olivier Guillon. 理論とシミュレーションに基づく X 線マイクロトモグラフィー観察からの粘性焼結の構成式の推定, 平成 28
年度秋季大会 (第 118 回講演大会) 粉末冶金協会, Nov (2016).
27) 門脇 大騎, 大熊 学, 若井 史博, 本堂 剛, 佐藤 暁洋, 田中 諭. X 線マイク
ロトモグラフィーによるガラス粒子の粘性焼結における気孔形態の 3 次元解析,
第 28 回日本セラミックス協会秋季シンポジウム, Sep (2015).

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Summary

Achievement(1) Grant1) 日本学術振興会(JSPS) 特別研究員奨励費 - DC2
研究課題名: 3 次元可視化による焼結の熱力学的駆動力の解明と制御
2) ホソカワ粉体工学振興財団 研究者育成のための援助
研究課題名: 粉体成形プロセスにおける微視的な粒子充填構造と焼結における巨視的な収縮との関連の 3 次元トモグラフィーによる解明
3) SPring-8: 2018A 大学院生提案型課題（旧名：萌芽的研究支援課題）
研究課題名: 結像型 CT による焼結プロセスのナノスケール 3 次元観察

(2) Scholar ship4) 日本学生支援機構(JASSO) 特に優れた業績による全額返還免除 (修士課程)
5) 日本学生支援機構(JASSO) 特に優れた業績による返還免除 (博士課程-二年次)(3) Awards6) Student Poster Award – Sinter forging and X-ray microtomography methods fordetermining sintering stress and bulk viscosity, ICCCI2018, Kurashiki, July (2018)
7) Poster Award - Distinction of sintering stage by interfacial topology, The 10th
International Conference on the Science and Technology for Advanced Ceramics
STAC10, Yokohama, Aug (2017)
8) Best Poster Award - Estimation of bulk viscosity and sintering stress frommicroscopic observation of viscous sintering of glass particles by using X-ray
microtomography, The International Symposium on the Science of Engineering
Ceramics, EnCera2016, Niigata, May (2016)(4) Invited lecture9) Distinction of stages of sintering from 3D visalization of microstructure in sintetingby using X-ray microtomography, 42nd ICACC, Daytona Beach, Florida, USA, January(2018)

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Acknowledgements

Acknowledgements
This work was mainly supported by Prof. Fumihiro Wakai. I would like to express
my sincere gratitude to Prof. Wakai. In addition, Associate Prof. Norimasa Nishiyama,
Dr. Yutaka Shinoda, Prof. Satoshi Tanaka (Nagaoka University of Technology), Daiki
Kadowaki and laboratory roommates give me some advice. Thanks a lot.

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