parameters for inherently homogeneous sintering processesœber uns... · surface diffusion on...

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DOI: 10.1002/adem.201400489
Parameters for Inherently Homogeneous SinteringProcesses

By Friedrich Raether*

Thermodynamic criteria are introduced to calculate the driving force for the formation of heterogeneousmetastable states during sintering. As already shown for one special case of liquid phase sintering,these metastable states significantly affect the reliability of sintered ceramics. Inherently safe sinteringprocesses are proposed, providing a homogeneous evolution of microstructure. A desintering parameteris defined which can be used for a quantitative evaluation of different sintering processes in terms oftheir tendency for formation of heterogeneity. The desintering parameter has been calculated usingsintering simulations in 3D representative volume elements. For that, a deterministic model has beendeveloped. It considers interface energy minimization and grain growth for random arrangements ofparticles. Computational effort could be significantly reduced by periodic continuation of themicrostructure at the sides of the RVE. Basic parameters controlling inherently homogeneous sinteringare presented. Surface diffusion is positive at large dihedral angles. Unexpectedly, grain growth can beadvantageous as well, provided that it acts continuously during the entire sintering cycle.

1. Introduction

Coarsening of the microstructure and grain growth is a wellknown phenomenon in sintering science. More than 4000papers were published on these topics during the last 4decades. Much less attention was paid to differential sintering:30 papers. However, differential sintering can significantlydeteriorate material properties of sintered products. Graingrowth starts earlier in regions which are denser than theresidual material. Also, large pores are formed between thedense regions which are difficult to eliminate in final stagesintering. Even local desintering is observed, leading to theopening of particle contacts which have already been formed,while the main part of the compact is densifying.[1,2] Theinhomogeneity caused by differential sintering leads todecrease of strength and reliability and is considered as oneof themost importantobstacles forawideruseof sinteredparts.

Constraints affecting local shrinkage during sintering, e.g.,acting during co-firing, have frequently been shown to causedifferential sintering phenomena (compare, e.g., ref.[3]). Withfree sintering, differential shrinkage was intensively studiedby F.F. Lange since the 1980s.[1,4–7] He showed that

[*] Dr. F. RaetherFraunhofer-Center for High Temperature Materials andDesign, Gottlieb-Keim-Str. 60, Bayreuth 95448, GermanyE-mail: [email protected]

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inhomogeneity, which was already present in the greencompact, led to differential shrinkage during sintering.[5]

However, desintering was reported even for partially sinteredperiodic arrays of polymer spheres[5] and glass and copperspheres.[2] Therefore, random arrangements of particles witha size distribution were considered favorable to avoiddifferential sintering.[5] Lin and De Jonghe suggested aprecoarsening treatment at relatively low temperatures,todecrease differential sintering effects.[8] Since surface diffu-sion is usually more active at lower temperatures thandensifying mechanisms, the microstructure can be stabilizedby this treatment before shrinkage sets in and createsdifferential stresses. They showed that homogeneity ofsintered compacts could be improved by the precoarseningprocess.[8] On the other hand, a two-step sintering processwasrecently proposed starting at a short holding period at hightemperature followed by a longer holding period at lowertemperature.[9] It was reported that this process can also leadto more homogenous microstructures.

A measure is required to discuss heterogeneity duringsintering on a quantitative base. For that, the scaled varianceof pore area distribution was proposed.[10] The pore area ismeasured in polished sections of green or partially sinteredcompacts using scanning electronmicroscopy (SEM). For eachsample, the variance s2 of pore area distribution is obtainedfrom the data of several sections of the same size. Note that sis the corresponding standard deviation of pore area

KGaA, Weinheim ADVANCED ENGINEERING MATERIALS 2015, 17, No. 9

Fig. 1. Polished sections of liquid phase sintered aluminum nitride ceramics: a) beforeand b) after optimizing sintering parameters according to thermodynamic model (detailsin ref. [18,19]).

F. Raether/Parameters for Inherently Homogeneous Sintering Processes

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distribution. The area A of these sections is varied toinvestigate homogeneity on different length scales. Since s2

strongly decreases with area A, a simple scaling wasintroduced considering the variance r2ran expected for arandom distribution of mono-sized circles with the radius ofaverage pore chord length lp and average pore fraction p:

s2scaled ¼

s2

s2ran

¼ s2A

pl2pð1Þ

It was shown that r2scaled drastically increased during solidstate sintering of alumina.[10] An alternative measure forheterogeneity is the density variation in small test volumesobtained from microfocus computer tomography (mCT).[11,12]

Whereas the strength of the SEM-based method is themeasurement of short-range disorder on the submicron andmicron scale, mCT and CT are favorable for larger scales fromsome tens of microns up to centimeters.

The minimization of Gibbs free energy G during sinteringat ambient pressure is mainly driven by decrease of interfaceenergy – provided that no chemical reactions between theconstituents occur.[13] Application of thermodynamic princi-ples to sintering studies was restricted to relatively simplegeometries and/or special topics. Kellett and Lange used athermodynamic approach to describe neck formation andpore elimination for different dihedral angles in the initialrespectively final sintering stage.[6,7] Exner used a simplenumerical model to study the effect of grain boundary andsurface diffusion on equilibrium shape during sintering.[14]

Delannay presented an average Voronoi cell and sophisticatedanalytical solutions to study equilibrium shapes in a widerportion of the sintering cycle.[15] A similar analytical approachwas used byHirata et al. to describe density evolution in termsof dihedral angle and number of particle contacts.[16] Kangproposed a pore filling theory explaining the redistribution ofmelt in liquid phase sintering. Driving force is the decrease ofinterface energy when smaller pores are filled with meltphase.[17] For the final stage of liquid phase sintering, theauthor has proposed another thermodynamic model.[18] Itexplains the formation of metastable states formed bysegregation of liquid phase within the ceramic (Figure 1a).After adapting the sintering conditions according to thethermodynamic requirements a homogenous microstructurewas achieved (Figure 1b). Weibull modulus was increasedfrom 9 to 22, demonstrating the large effect of homogeneity onreliability of ceramics.[19] The thermodynamic model wasbased on interface energy minimization in a two-phase solid–liquid system using a simple periodic microstructure.

A generalization of the model is presented in this paper.First, the thermodynamic criteria for inherently homogeneoussintering processes are explained. Then, a microstructuresimulation is presented which can handle more complexgeometry than the previous models. First results arepresented showing parameters which affect formation ofheterogeneous structures during sintering.

ADVANCED ENGINEERING MATERIALS 2015, 17, No. 9 © 2015 WILEY-VCH Ver

2. Thermodynamic Criteria

The driving force for sintering is the decrease in interfaceenergy EM of the sintered compact (indexM denotes scalingwith mass). Whereas interface energy before sintering isdominated by particle surfaces, grain boundary energytypically controls the interface energy in the final state.Figure 2 shows interface energy EM versus specific volume vfor a solid state sintering process. The specific volume is theinverse of the density r. It is used in Figure 2 instead of densitybecause it simplifies the geometric construction of mixingenergies (see below).

In the most general case, EM is the sum of variouscontributions:

EM rð Þ ¼ gslSsl rð Þ þ gsgSsg rð Þ þ g lgSlg rð Þ þ ggbSgb rð ÞM

; ð2Þ

withM¼mass, S¼ interface area, g¼ specific interface energyand indices s, l, g, and gb denoting the solid, liquid, andgaseous phase and grain boundaries, respectively. In manysystems, Equation 2 can be simplified as follows: with solidstate sintering, Ssl and Slg are zero; with viscous sintering, Sgbvanishes; and in the final state of liquid phase sintering, Ssgand Slg can be neglected. On the other hand, specific interface

lag GmbH & Co. KGaA, Weinheim http://www.aem-journal.com 1375

m1

i

∆EM

f

m

a)

Inte

rface

ene

rgy

EM

vf

Inte

rface

ene

rgy

EM

Specific volume vvm vi

∆EMm2

m

b)

Fig. 2. a) Concave interface energy curve showing the energy gain DEM by separationof dense and porous phase at average specific volume vm and b) energy gainDEM due tophase separation to intermediate states m1 and m2.


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energy g may also change with sintering state and tempera-ture: g¼ g (r, T).[20]

If the curve EM rð Þ has a concave shape – as indicated inFigure 2–the interface energy in an intermediate state ofsintering can be decreased by a separation of denser andmoreporous regions. We consider a sintered body with an averagespecific volume vm and assume that the mass in a volume Vofthis body is redistributed. In a partial volume Vi, the specificvolume shall correspond to the initial value vi and in theresidual volume Vf¼V�Vi, the specific volume shallcorrespond to the final value vf. The mass fractions of theporous respectively dense regions are given by the so calledlever rule,[21], which simply reflects mass conservation:

Mi

M¼ vm � vf

vi � vf;Mf

M¼ vi � vm

vi � vf: ð3Þ

The decrease of energyDEM due to this phase separation isdescribed by

DEM ¼ EMm � ðvm � vf ÞEMi þ ðvi � vmÞEMf

ðvi � vf Þ ð4Þ

with indices i, m, and f denoting the initial, intermediate, andfinal state of sintering, respectively. EM, M, and v are thecorresponding interface energies, masses, and specific vol-umes. DEM is indicated in Figure 2a. It can be constructed byconnecting the points i and f with a straight line andintersecting this line with the vertical line at vm. Figure 2bdemonstrates that any separation to other states than initialand final state also leads to a decrease of total energy if EM rð Þis concave. The respective energy decrease DEM can becalculated by Equation 4 as well or constructed graphically if

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the initial and final states are replaced by the two intermediatestates m1 and m2 with higher respectively lower specificvolume. If interface energy is scaled with volume V instead ofmass M, Equation 4 can be written in a similar way byreplacing the specific volumes by densities or pore fractions.This representation has been selected in our previouspublications.[18,20] On the first view, this looks more conve-nient since specific volumes are rarely used in sinteringscience. But EV(r) does not represent a closed system withconstant mass. Therefore, dEV(r) is not necessarily negativeduring sintering. To avoid confusion arising from increasingEV(r), the representation shown in Equation 4 is preferred.With liquid phase sintering, the specific volume can be used aswell if phase separation between solid and liquid phase, onthe one hand, and gaseous phase, on the other hand, shall beinvestigated. If phase separation between solid and fluidphases shall be discussed, the corresponding partial specificvolume vs¼V/Ms has to be used instead of v.

Note, that phase separation is well known from binaryphase diagrams. If the free energy curves show concavesections, a miscibility gap exists.[21] A homogenous mixture oftwo components is not stable in the concentration range of themiscibility gap. So, different phases coexist in the equilibriumstate. Similarly, during sintering, the energy decrease DEM

resulting from a concave curve EM rð Þ creates a driving forcefor the separation of more or less dense regions in theintermediate stage. The driving force increases if themagnitude of the curvature rises. This must not necessarilylead to heterogeneities since densification can be faster thanthe separation process. Especially, a complete separation ofinitial and final state of sintering is not realistic. But, a concaveinterface energy curve always involves the risk of phaseseparation. Once these metastable heterogeneities are formed,it will be difficult to obtain a homogenous material in the finalstate, since many mechanisms – like grain growth – proceeddifferently in the respective regions. On the other hand, astraight or convex curve EM rð Þ does not bear such a risk as nothermodynamic driving force for the formation of hetero-geneities during sintering exists. In that sense, inherentlyhomogeneous sintering processes require convex or at leaststraight interface energy curves!

The thermodynamic driving force for the formation ofheterogeneities during sintering – as described in the previoussection – is considered essential for the evaluation of sinteringprocesses. Therefore, it is necessary to introduce a measurewhich enables comparison of different sintering processes.The determination of interface energy with high enoughaccuracy and at small enough density intervals to allow for asatisfactory calculation of its curvature is very tedious. Insteadit is suggested to use the area A between the curve EM rð Þ andthe straight line connecting initial and final state, i.e., thedashed area in Figure 2a, as measure for the driving force forphase separation. Since phase separation and densificationcompete, area A should be scaled by the driving force forsintering. The latter is roughly represented by the slope sl ofthe straight line connecting initial and final state of the

o. KGaA, Weinheim ADVANCED ENGINEERING MATERIALS 2015, 17, No. 9

Particle generationà Initial diameterà Initial positionà Grain boundary energyà Particle displacementà Particle growth

Interface energiesà Voxel interfacesà Local curvatureà Voxel displacement

Sinteringà Particle displacementà Grain growthà Shrinkage

RVE generationà Voxel assignmentà Overlap handling

Equilibrium?

Final density?

Initial density?

Yes

No

Yes

Yes

No

No

Fig. 3. Work flow during sintering simulation using deterministic model described inthe text.


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interface energy curve EM:

D ¼ �Asl¼ A vi � vf

� �

EMi � EMf

: ð5Þ

D is denoted desintering parameter. A largeD indicates a highrisk of formation of heterogeneity during sintering, whereas anegative D characterizes an inherently homogeneous sinter-ing process.

The desintering parameter D can be obtained experimen-tally or from computer simulation. For that, area A anddensity r respectively specific volume v have to be deter-mined in the initial, intermediate and final stages of sintering.Experimentally this can be done using partially sintered andsubsequently quenched specimen. From polished – and ifnecessary etched – sections of these specimens, interface areasS can be determined using SEM images and linear interceptanalysis. The interface area Sij between two phases iscalculated from the number of intersections nij of the lineswith the respective phase boundary and the total length L ofthe testing lines:[22]

SijV

¼ 2nijL

ð6Þ

with indices ij¼ sl, sg, lg, or gb. The ratios between specificinterface energies gij are determined from measurements ofdihedral angle and contact angle.[23] Dihedral angles can bemeasured geometrically in SEM images of the polishedsections and contact angles by heating microscopes.[18]

Density r is usually measured by Archimedes method. UsingEquation 2, interface energy curve EM rð Þ is constructed at anumber of discrete points. Then area A is obtained bynumerical integration of this curve and finally the desinteringparameter D is calculated according to Equation 5. Note thatusually an unknown scaling factor remains in the determina-tion of specific interface energies gij. But this factor iseliminated when area A is divided by the slope sl in Equation5. Similarly, the desintering parameter D is obtained frommicrostructure models of the sintering process. With voxel-based models, artefacts due to the stepped interfaces can beavoided using the linear intercept method and orienting thetest lines parallel to the coordinate axes.[24]

In the following section, a voxel-based model is presentedwhich allows the calculation of interface energies duringsintering. For purposes of simplification, we concentrate onsolid state sintering processes.

3. Microstructure Modeling

A voxel-based 3D representative volume element (RVE)was constructed using the in house software GeoVal.[24] Thework flow of the simulation is shown in Figure 3. It startedwith spherical particles. Typically 50 particles were used. Theparticles were placed in the RVE using a Poisson distributiongenerated by random numbers. An individual – small –


particle diameter was attributed to each particle obtainedfrom a Gaussian distribution. Specific interface energies gsgwere assigned to the particle surfaces and the grainboundaries ggb. Each particle was considered as a singlecrystal. The grain boundary energy between two individualparticles i and j could be set according to a Gaussiandistribution with the center ggb and a specified width. Usuallythe grain boundary energy was constant.

The green microstructure was generated by a procedureroughly considering a wet forming process: all particlediameters were increased in small steps by the same scalingfactor until particles touched. Particle centers were shiftedaccording to hard sphere repulsion. The model was strictlyperiodic at the faces, edges and corners of the RVE. If particleswere pushed out on one side of the RVE they were continuedat the opposite side. If density was so high, that adisplacement due to the hard sphere model was not possibleany longer, the elastic energy corresponding to a Hertziancontact at the particle surfaces[25] was minimized by furthersmall displacements of the particle centers. Particle growthwas stopped when a specified “green” density was achieved –

typically 60%. The RVE was constructed by assigning eachvoxel to the corresponding particle or to the pore spacerespectively. If overlap between particles occurred voxelswere attributed according to a Voronoi tessellation of the



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overlap region. Particles with different radii were considered
according to the radical plane technique.[26] A typical size ofthe RVE was 643 voxels. Figure 4 shows a RVE at fractionaldensity of 60% containing 50 spherical particles with aGaussian distribution of diameters (1� 0.2) mm after rear-rangement of particles according to hard sphere repulsion.The computer program ParaView has been used for graphicalrepresentation of the RVE.[27] Note that smooth surfaces areintersections of the spherical particles with the faces of theRVE, whereas stepped structures reflect its porous interior.

During sintering, it was supposed that surface diffusionwasmuch faster than grain boundary or volume diffusion as itwas confirmed in many experimental studies.[23] It waspresumed that each particle surface had achieved a shapewith minimal interface energy. To identify this equilibriumshape, local energy of voxels was determined in two steps.First, the voxel energy due to the next neighbors of each voxelwas summed up. Second, the interface energy of each voxelwas calculated considering the curvature and thevoxel energy of the neighbor voxels up to the forth neighborshell. Whereas the first step corresponds to the procedureused in Monte Carlo simulations of sintering,[28] the secondstep enables considering interface tension on a larger scalethereby smoothing the effects of the angular voxel structure. Ifgrain boundaries occurred within the fourth neighbor shell,they were used for the calculation of local interface energy aswell to enable the formation of realistic dihedral angles at theedges of sintering necks.

The energy decrease due to the redistribution of surfacevoxels was measured using the linear intercept methodmentioned in the previous paragraph. If energy minimizationhad converged, a sintering step was performed. For that,forces between particles were determined from interface

Fig. 4. RVE with 50 particles before sintering (fractional density 60%).


tensions in the grain boundaries. These forces were summedup for each particle. Residual forces on the particle centerswere used to calculate particle displacements for the nextiteration step. Numerically, the displacements were per-formed in small steps using virtual springs connecting theparticle centers until elastic energy had relaxed. Frominterface tensions in the grain boundaries, tensile andcompressive stresses on the particles were also obtained.From these stresses, a parameter was derived whichcontrolled individual grain growth or shrinkage of particlesin the following iteration. If particle diameter was smallerthan a tenth of mean diameter, the respective particle wasremoved. Thereafter, particle diameterswere increased evenlyuntil the set density of the next iteration step was obtained(Figure 3). The voxel dimensions were decreased accordinglyto consider adequate shrinkage of the RVE and the interfaceenergy was minimized again. The process was repeated untilfinal density was achieved. Typically, a 1 or 2% increase indensity was used in one iteration step. During all steps of thesintering simulation, the periodic continuation at the sides ofthe RVE was strictly observed. Note that diffusion processesare not implemented in the model. It provides simply asequence of local interface equilibria.

The evaluation of the microstructure data has already beendescribed in the previous paragraph. The whole process ofparticle generation and sintering was repeated typically 5 to10 times to account for statistical effects. Figure 5 shows anexample of microstructure evolution during sintering at adihedral angle of 120° using the initial structure shown inFigure 4. Besides densification, coarsening of the microstruc-ture is also observed.

4. Results and Discussion

The interface energy during sintering has already beeninvestigated for periodic structures using triangularmeshes.[18] Interfaces with minimal energy were calculatedby the computer program surface evolver,[29] which had beenintensively utilized by Wakai.[30] To compare results obtainedwith the voxel model with results from the triangular meshmodel, some simulations are presented below using a simplecubic arrangement of particles. Figure 6 shows particle shapesobtained with the voxel model for different dihedral angles ofone truncated sphere located in the center of the cubic RVE.Size of the RVE was 643 voxels. Whereas a small dihedralangle leads to a convex particle shape, a large dihedral anglecauses the expected mixture of convex and concave surfaces.The same structures were obtained using surface evolver(compare Figure 4 in ref.[26]).

Figure 7 shows interface energy EM versus specific volumeduring sintering of the simple cubic structure for a dihedralangle of 120° and different sizes of the RVE. Note thatboth quantities EM and vwere scaled by the respective valuesof the fully dense state. It can be seen that a fairly goodconvergence was already achieved for sizes between 163 and323 voxels.


Fig. 5. RVE with 50 particles during sintering at different fractional densities: a) 60%, b) 70%, c) 80%, and d) 90%.


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The effect of dihedral angle on the interface energy curvesis shown in Figure 8 using RVEs with 643 voxels. The curvesbecome steeper with increasing dihedral angle and curvatureincreases. This was already shown for simple cubic and facecentered cubic structures using triangular meshes.[20,18] Usingtriangular meshes, curvature of the interface energy curve

Fig. 6. Particles in a simple cubic arrangement at fractional density of 75% and dihedra


becomes slightly positive at dihedral angles smaller than 90°.In contrast, curvature is negative for all dihedral angles usingthe voxel model (compare Figure 8).

The difference is attributed to the coarser representation ofthe neck region in the voxel model which leads to inaccuraciesespecially at small dihedral angles. On the other hand, the

l angles of a) 60° and b) 150°.


1.0 1.2 1.4 1.61.0

1.1

1.2

1.3

1.4E

M /

E dens

v / vdens

# voxels 163

323

643

1283

Fig. 7. Convergence of the cubic model at different sizes of the RVE.

2

4

6

1.0 1.2 1.4 1.6 1.8

1.0

1.5

2.0

a)

EM

/ E

dens

120°

90° 60° 30°

150°

b)

d par

ticl

e in

µm

v / vd en s

D ih edral angle 30° 60° 90° 120° 150°

Fig. 9. a) Scaled interface energy EM/Edens versus scaled specific volume v/vdens forRVEs with 50 spheres and different dihedral angles, b) evolution of particle diametersdparticle during sintering.


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voxel model allows the calculation of much larger models atreasonable computer times. Sintering of an RVE with 50particles requires typically 1 or 2 h on a standard PC, whereassimulations using surface evolver are slower by about twoorders of magnitude and stability of the algorithm is muchworse. So, the voxel model is preferred for the simulation ofcomplex structures. However, results with dihedral anglessmaller than 90° obtained with the voxel model should beinterpreted with care. The interface energy in the initial stageof sintering can also be compared with the Coble model usinga dihedral angle of 180°, grain boundary diffusion and closedform equations.[31] Interestingly, the Coble model provides aslightly convex interface energy versus specific density curvewhich is attributed to artefacts in the Coblemodel consideringstructures far away from local equilibrium.

Figure 9a shows interface energy curves for randomarrangements of spherical particles sintered at differentdihedral angles. Starting configurations in each case wereseveral RVE containing truncated spheres. The particlediameters had a Gaussian distribution of (1� 0.2) mm. An

1.0 1.2 1.4 1.6 1.81

2

3

4150°

30°60°90°

EM /

E dens

v/vdens

120°

Fig. 8. Scaled interface energy EM/Edens versus scaled specific volume v/vdens for simplecubic symmetry and different dihedral angles.


example is shown in Figure 4. Moderate grain growth andparticle displacement were allowed during sintering (com-pare Section 3). The interface energies are mean values fromsimulating 5 to 10 RVE per dihedral angle. The root meansquared errors are indicated in Figure 9a. They are hardly seendemonstrating high reproducibility of the simulation.

The good reproducibility is attributed to the periodiccontinuation at the sides of the RVE. Interface energy curvesshow concave shape at a dihedral angle of 90° and below. Athigher dihedral angles, convex and concave regions of thecurves occur (compare Figure 9a). The evolution of grain sizeduring sintering is shown in Figure 9b. Moderate graingrowth occurs for all dihedral angles. However, grain growthis significantly larger for large dihedral angles of 120° and 150°compared to dihedral angles of 90° and below. This increase isattributed to the larger grain boundary areawhich reduces thepinning of grain boundaries by pores. Figure 10 shows thedesintering parameter (compare Equation 5) for the curvesshown in Figure 9a. The desintering parameter significantlydecreases at dihedral angles above 90°. This decreasecorresponds to the lower curvature of interface energy curvesin Figure 9a. Interestingly, the scaled variance also plotted inFigure 10 shows a similar decrease. This underlines theimportance of the desintering parameter. It was not expectedthat disorder could be observed already on the scale of suchsmall RVEs. Note that the thermodynamic criteria, presentedin Section 2, do not allow conclusions on the size of the regionswhich are separated.


50 100 1500.00

0.05

0.10

0.15

0.20

D

θ in °

Desintering0.15

0.20

σ2 80 / σ2 ra

n

Variance

Fig. 10. Desintering parameter D (left axis) and scaled variance at fractional density of80% s80/sran (right axis) versus dihedral angle u for the simulations shown in Figure 9.


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Figure 11 shows the individual effects of surface diffusionand grain growth on the interface energy curves at a dihedralangle of 120°. Curvature is largest if neither surface diffusionnor grain growth occurs (curve A in Figure 11). Surfacediffusion leads to a significant decrease of initial interfaceenergy and a flattening of the interface energy curve (curve Bin Figure 11). If grain growth is additionally switched on, theinterface energy curve with concave and convex sectionsalready shown in Figure 9a is obtained (curveC in Figure 11).Final interface energy decreases significantly compared tocurve A and B. This is simply caused by the reduction of grainboundary energywith increase of grain size. Note that scalingwas always done with the interface energy of dense RVEsobtained with grain growth. In addition also the effect of adistribution of grain boundary energies for individual grainboundaries was tested. But no effect on the interface energycurve was obtained.

Apparently, surface diffusion and grain growth can reducethe curvature of interface energy curves and thereby the

1.0 1.2 1.4 1.6 1.81

2

3

4

EM /

E dens

v / vdens

A

B

C

Fig. 11. Scaled interface energy EM/Edens versus scaled specific volume v/vdens for RVEswith 50 spheres and dihedral angle of 120° for different combinations of grain growthand surface diffusion, curve A: no grain growth, no surface diffusion, curve B: no graingrowth, surface diffusion, curve C: grain growth, surface diffusion.


tendency for differential sintering. This is an unexpectedresult especially for grain growth. For surface diffusion, thepresent results provide another explanation for the advan-tages of the above-mentioned precoarsening treatmentsuggested by De Jonghe.[8] So, besides the mechanicalstabilization of the microstructure, surface diffusion has athermodynamic effect reducing the driving force for differen-tial sintering. More differentiation is required to understandthe effect of grain growth. Looking again at Figure 9b, itbecomes clear that a drastic grain growth only in the finalsintering stage – as it was obtained for small dihedral angles –has a detrimental effect since the final part of the interfaceenergy curve is pulled down and becomes concave. Usually alate onset of grain growth is considered desirable in sinteringstudies, because final grain size can be reduced.[23] Thepresent paper suggests that grain size and microstructurehomogeneity may be complementary in sintered systems. So,a tradeoff situation may exist where a moderate grain growthduring the entire sintering cycle is helpful to improvereliability of sintered ceramics. One can speculate that someinitial disorder is helpful to achieve this early onset of graingrowth.

5. Conclusions and Outlook

Sintering involves billions of individual particles. Due tothe statistical nature of many parameters affecting particlearrangement during forming, some heterogeneity will beinevitable. Heterogeneity can increase during sinteringleading to large flaws in sintered compacts. These flawsdeteriorate strength and reliability and have to be carefullyavoided. To achieve better sintering processes, thermody-namic criteria should be considered. As shown in Section 2,the concave or convex shape of the interface energy curveduring sintering provides a criterion whether a sinteringprocess is inherently homogeneous. A desintering parameterhas been derived which can be used to compare differentsintering processes regarding their tendency for the formationof heterogeneous structures. The desintering parameter wascalculated for a voxel-based model, already showing someparameters which are important in achieving inherentlyhomogeneous sintering processes. Both surface diffusion andgrain growth significantly affect the desintering parameter.

Besides the thermodynamic driving force, kinetic effectscontrol the actual evolution of microstructure. Therefore, anevaluation of kinetic models with respect to the evolution ofinterface energy is required. Fortunately, many existingsintering models can be easily adapted to that purpose.Analytical sinteringmodels are based on a simple geometry ofthe particle and pore geometry (compare, e.g., ref. [6,16]). Theyallow an easy extraction of interface energy but are restrictedto sections of the sintering cycle. With the extension ofDelannay, larger portions of the sintering cycle can beevaluated.[32] Also, the geometric model of Wakai allows asimple extraction of surface energy.[30] Obviously, thedesintering parameter can be easily obtained from the Monte



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Carlo models on sintering (compare, e.g., ref. [28]). In addition,
it was pointed out in Section 2 that the desintering parametercan be derived experimentally using partially sinteredsamples. More investigations on the formation of heteroge-neity during sintering are considered essential for theimprovement of sintered material properties.

Received: October 27, 2014Final Version: February 24, 2015Published online: March 17, 2015

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