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An experimental and numerical study of particle size distribution

An experimental and numerical study of particle size distribution

effects on the sintering of porous ceramics

Ken Darco v ich a,*,Floyd Toll a ,Pierre Hontanx b ,Virginie Roux c ,

Kazunari Shinagawa d

a

Institute for Chemical Process and En v ironmental Technology,National Research Council of Canada,Ottawa,Ont.,Canada K1A 0R6

b

ICAM-Toulouse,75,a v .de Grande Bretagne,31300Toulouse,France c

ICAM-Nantes,35,a v .du Champ des Manoeu v res,44470Carquefou,France

d

Department of Ad v anced Materials Science,Faculty of Engineering,Kagawa Uni v ersity,Hayashi-cho 2217-20,Takamatsu 761-0396,Japan

Recei v ed 7December 2001;recei v ed in re v ised form 28August 2002;accepted 9September 2002

Abstract

A single-step processing method has been established to prepare asymmetric porous alumina microstructures by a controlled sedimentation technique.Fine powder from an aqueous suspension is consolidated o v er a casting slab.Metastable surface chemical control of the suspension properties was able to induce a highly porous flat disc structure with a continuously increasing mean pore size from top to bottom.Formation of this gradient structure was facilitated by using a powder with a v ery broad particle size distribution.These structures can be used as either ultrafiltration media or as substrates for inorganic membrane making.Sintering can readily introduce defects into functionally gradient ceramics.Despite these problems,the asymmetric structures considered in this paper can be readily sintered without warpage or cracking.In this regard,a finite element method numerical simulation had been de v eloped to model the sintering characteristics of functionally gradient ceramic structures.The key for being able to predict a non-warped structure was the incorporation into the model of the powder particle size distribution as a field v ariable.Across the v ertical section of the structure,the distributions were broad and o v erlapping,all with a significant fines tail.These characteristics accelerate and homogenize local sintering rates,such that the net result is a non-warped fused structure.This paper presents recent ad v ances with the simulation,where sample geometry,porosity and particle size distribution e v olutions were traced alongside measurements made on physical specimens.In general the model corresponded well with the experimental obser v ations.The correct accounting of obser v ed trends lends confidence to the underlying sintering mechanisms incorporated into the model.#2002Else v ier Science B.V.All rights reserved.

Keywords:Numerical study;Particle size distribution;Porous ceramics;Sintering

1.Introduction

As part of an on-going project,porous structures expressly intended as membrane substrates ha v e been prepared v ia a polydisperse slurry sedimentation method which yields a functionally gradient material.The benefit of creating an asymmetric microstructure is to produce a smaller substrate pore size o v er a thinner

region,thereby imparting superior permeation proper-ties.

Preliminary work has demonstrated the v iability of this method for preparing functionally gradient samples made from a -alumina which retain a high porosity after sintering [1].Hardness testing was used to demonstrate that the bodies fused and strengthened without signifi-cant densification,and hardness gradients measured across the cross-section reflected a functionally gradient microstructure [2].The dispersion of a small amount of fines throughout the body,promoted through the metastable nature of the suspensions,ser v ed as localized sites from which sintering was enabled at lower tem-peratures,sufficient to fuse the sample into a contiguous hardened porous structure.SEM images demonstrated

NRCC No.44390.

*Corresponding author.Tel.:'1-613-993-6848;fax:'1-613-941-2529

E-mail address:ken.darco v ich@nrc.ca (K.Darco v ich).

Materials Science and Engineering A348(2003)76á

83

www.else v https://www.wendangku.net/doc/e63011846.html,/locate/msea

0921-5093/02/$-see front matter #2002Else v ier Science B.V.All rights reserved.PII:S 0921-5093(02)00634-2

the continuously increasing mean particle size across the cross-section of the samples[3].Porosimetric measure-ments on horizontal sections of the samples corrobo-rated the pre v ious findings[1].

It has been shown that the colloidal state or suspen-sion microstructure can be controlled with pH and polyelectrolyte stabilizing additi v es.Rele v ant stability criteria for systems such as presently under considera-tion here are a v ailable in the literature[4].The colloidal phase state of a suspension which forms the consoli-dated green body has a direct bearing on the e v entual microstructure of the sintered solid object[5,6].By controlling the dispersity of a suspension,slight aggre-gation and/or hierarchical clusters can contribute to o v erall porosity increases while at the same time producing a relati v ely fine pored top surface.Warping and cracking problems were resol v ed by controlling the particle zeta potential in the suspension so that v ery fine particles were associated with all size classes,and were present in all v ertical positions of the consolidated structure[2].

The sintering of functionally gradient materials has also been a topic of recent interest.Numerical simula-tions of sintering ha v e been de v eloped to assess mechan-istic theories of sintering based on powder compact properties[7,8].It was known from experiments that local distribution effects can o v erride tendencies that would otherwise be predicted considering only mean powder properties.Our pre v ious paper presented a model formulation to explicitly account for particle size distribution effects[9].

In continuing this thread,it is therefore the main objecti v e of this paper to detail a v erification of our simulation output against experimental measurements taken from functionally gradient ceramic structures in both the green and sintered states.

2.Constitutive model for sintering

A constituti v e model for sintering ceramic powders originally de v eloped by Shinagawa and Hirashima[10] is outlined briefly below.Full details of the formulation are gi v en elsewhere[9].

The principal components of the sintering model are as follows.

An expression from Coble[11]was employed as the basis for sintering deformation under grain boundary diffusion:

˙o047V hD

b

s

kTd3

s

3h

(1)

where V is the atomic v olume,h is the width of the grain boundary,D b is the grain boundary diffusion coeffi-cient,k is Boltzmann’s constant,T is the absolute temperature and d is the grain size.From Eq.(1),a number of the parameters can be lumped to gi v e h,the sintering v iscosity,which can be considered as a measure of a material’s resistance to sintering.

As gi v en by Shinagawa,[7]the constituti v e equation for the strain rate˙o

ij

;is,

˙o

ij

1

2h

1

r2n(1

s?

ij

'd

ij

2

9f2

(s

m

's

s

)

(2)

Abo v e,r is the relati v e density,s?is the de v iatoric stress,s m is the hydrostatic stress,s s is the sintering stress,and f and n are empirical parameters.

This strain rate is related to the induced sintering stress,[8]

s

s

0r N

2g

z R

r

r

(1(r

)1=3

(1(r)

(3)

where g is the solid surface tension,R is the particle radius,r0is the initial relati v e density,and z and N are constants.

The flow stress of powder particles during sintering can be expressed from Eqs.(1)á(3),leading to the stressástrain rate relations,

f s g0[D]f˙o g0[D][B]f u g(4)

where[B]is the strain rate matrix,elements of[D]are strain coefficients and{u}is the nodal v elocity v ector. Eq.(4)can be discretized for the finite element method. The local influence of powder distributions was incorporated into the model with the relation[12]:

@F

@t

'

@

@r

C

G

F

r n

1

r

c

(

1

r

00(5)

Eq.(5)enables each radius r and its corresponding frequency F to be sol v ed together in a marching algorithm tracking local particle size distribution e v olu-tion.The parameter C G is described by Greenwood[13]. It can be expanded as:

C

G

2D

b

S gV

kT

(6)

Abo v e,S is the solubility of a particle of infinite radius, referring to the solidágas sintering system and D b is the grain boundary diffusion coefficient.For alumina at 15008C,C G was estimated as0.02283m m3min(1for the grain boundary diffusion case[9].

To account for all pair interactions throughout the entire particle size distribution,the parameter h from Eq.(1)is adjusted according to:

K.Darco v ich et al./Materials Science and Engineering A348(2003)76á8377

h PSD 0

g r L

r S

g r b

r S

h(r

i

;r

b

)F(r

i

)F(r

b

)d r

i

d r

g r L

r S

g r b

r S

F(r i)F(r b)d r i d r

C

N AVG

C

N PSD

(7)

Abo v e C N refers to the coordination number of the powder pack.The particle pair interaction term:

h(r

i ;r

b

)08K

E

r3:4725

i

r0:4725

b

(8)

comes from a deri v ation by Pan et al.[14]where K E is a parameter deduced from the Coble equation,equal to kT/47V hD b s.

In order to allow the particle size distribution effects to manifest themsel v es in a two-dimensional fashion,it was necessary to introduce a temperature field o v er the sample domain in the simulation.The structure was modeled in two dimensions on an11by35grid shown in Fig.1.For a rectangular slab,heating by conduction was considered.The well known conduction equation, gi v en below was employed.

@2T @x2'

@2T

@y2

r c

k

T

@T

@t

(9)

Abo v e,x and y refer to horizontal and v ertical

coordinates,respecti v ely,c is the heat capacity and k T

is the thermal conducti v ity.Data from the literature was

found for alumina for c and k T across the rele v ant

temperature range[15,16].

As a boundary condition,the outside surface tem-

perature of the structure was considered to be equal to

the temperature according to the sintering timeátem-

perature profile.As an initial condition,the entire

structure was considered to be at208C at the start of

the sintering program.The timeátemperature sintering

profile used for these simulations and in the experiments

is gi v en in Fig.2.

3.Experimental

The Ceralox APA-0.2alumina powder was chosen for

this work.It had a broad particle size distribution,and

was known to produce consolidated ceramic pieces with

a functionally gradient structure.The APA-0.2powder

had a specific surface area of40.0m2g(1,a high v alue

since it has a substantial tail of fines in the distribution,

and it occurs in the form of aggregates of ultra-fine

particles.The data for specific surface area were taken

from product technical information pro v ided by the

suppliers.Powder size distributions were

determined

Fig.1.FEM grid used for numerical

simulations.

Fig.2.Ramp and soak sintering pro?le.

K.Darco v ich et al./Materials Science and Engineering A348(2003)76á83

78

using a Horiba model LA-920Particle Size Analyzer,which is a laser diffraction de v ice.

Full details on the sample preparation and character-ization are gi v en in a pre v ious paper [1].A brief outline of procedures is gi v en below.Aqueous alumina suspen-sions were prepared at solids loadings of 5v olume percent solids (v /o),and a steric effect was pro v ided by the addition of an ammonium polymethylacrylic acid

electrolyte at a le v el of 0.5g m (2(NH '

4PMA ()of molecular weight of approximately 15000(Dar v an C,R.T.Vanderbilt Co.Inc.,Norwalk,CT).

The ceramic structures were then produced by sedi-mentation of the slurries in 45mm diameter tube sections o v er milled gypsum slabs,such that they consolidate to form flat discs 3.3mm thick.Prior to

sintering,the green bodies were dried in an o v en at 508C.The sintering ramp and soak profiles followed in the experiments are shown in Fig.2.

Pore size distributions of the sintered ceramic struc-tures were obtained with a Quantachrome Pore Master 60mercury porosimeter.

3.1.Sample series

In order to compare the e v olution of samples under-going sintering with the progress predicted by the numerical simulation,a number of different samples were prepared for analysis.In reference to the ramp and soak profile,(Fig.2),samples were obtained as follows:.three samples of the green body (non-sintered);.samples at 8508(^on Fig.2);.samples at 12008(2on Fig.2).

To obtain these samples,the sintering o v en was briefly opened and the discs were remo v ed in the middle of the heat treatment

program.

Fig. 3.Sectioning method to obtain v ertical resolution for local sample microstructure

data.

Fig.4.Particle size distribution e v olution from both the numerical simulation and the experimental data.

K.Darco v ich et al./Materials Science and Engineering A348(2003)76á8379

A key feature of these samples is their asymmetric cross-section.To pro v ide some spatial resolution to the data,a v ery thin diamond saw with a circular blade about 0.3mm thick was used to section the discs into four layers as depicted schematically in Fig.3.

Particle size distribution measurements with both the green bodies and the sintered samples likely included an error caused by either adhering or bonded grains.Both the consolidation and the bonding in the sample necessitated a redispersion of the indi v idual grains to asses their e v olution through the process.A mortar and pestle pre-treatment was made and then high ultrasonic agitation was used during the particle size distribution measurements to attempt to reduce the powder samples to a state representing the indi v idual grains.

4.Results and discussion

4.1.Sample analysis and simulation inputs

The alumina samples were prepared as described pre v iously.The sectioned pieces from the green body were analyzed to pro v ide data which ser v ed as initial conditions for the simulation.To obtain this,particle size distributions from the different v ertical positions were superimposed on the calculational grid by a bilinear interpolation to go from a set of four v alues to a 10)34array corresponding to the finite element grid.The area of the distributions had to be normalized in proportion to respecti v e sample v olumes.In a like-wise manner,the porosity data for the different v ertical positions were interpolated in one dimension for the numerical simulation.

4.2.Measured results and model predictions

Fig.4shows the particle size distributions from the measurements taken on the powder from the structures at the different of points along the sintering path.Ostwald ripening is e v ident from the flattening and broadening of the cur v es,showing that finer particles are consumed into coarser ones.The thinner cur v es in this same figure show the simulated data.The forms and trends indicated match fairly well,in support of the models incorporated into the simulation which account for the sintering mechanisms.The experimental data show a weaker peak and broader distribution compared with the simulation https://www.wendangku.net/doc/e63011846.html,parati v ely ad v anced coalescence between coarse and fine particles could ha v e pre v ented the grinding and ultrasonic treatment from completely redispersing the grains,thereby leading to this discrepancy.Some micrographs shown in Fig.5demonstrate some of the abo v e discussion.In Fig.5a,a fracture surface from the top layer is shown.The particles are comparati v ely fine and well sintered.Fig.5b shows a fracture surface from the third cut layer,where the extent of sintering is seen to be less,and the a v erage grain sizes are larger,corresponding to their particle size distributions in Fig.4.

The sample sections from the different sintering stages were also tested by porosimetry.These measurements are shown in Fig.6.The pore size distributions e v ol v e under sintering in a manner analogous to the particle size distributions.In general,as the sintering ad v ances,the mean pore sizes become smaller and the areas under the cur v es also decrease,which is indicati v e of sample densification.The range of bulk porosities measured ranged from 46.7%at the top of the samples taken at the end of the sintering program,to 67.5%for the bottom sections of the green bodies.For each section,the cur v es in Fig.6show the same qualitati v e trends as cur v es showing the pore size distribution e v olution

under

Fig.5.SEM images of the v ertical faces of fracture planes from the samples sintered at 12008C.(a)Top layer (b)third layer.

K.Darco v ich et al./Materials Science and Engineering A348(2003)76á83

80

sintering for lithium fluoride samples [17].The mean pore diameters increase as the point of reference is mo v ed from top to bottom in the sample.The effect of the saw cuts on the measurements could be considered negligible,in the sense that only the surface of the sample would be affected,while

the

Fig.6.Porosimetric data showing pore size distribution e v olution in the

samples.

Fig.7.Sample shape e v olution o v er the sintering period showing density contours.Simulated results are on the left.Experimental measurements are on the right.

K.Darco v ich et al./Materials Science and Engineering A348(2003)76á8381

measurements of pore and grain size distribution were for bulk properties.Pre v ious work which measured pore size distributions in horizontally sectioned top and bottom layers of a sample,as well as the o v erall structure,confirmed this claim [1].

4.3.Shrinkage,densi?cation and deformation

Structural information was traced by the simulation.Fig.7shows cross-sections of the structure at ad v ancing times o v er the sintering profile,along with density contours.In general the correspondence is good.This suggests that the grain boundary diffusion sintering mechanism is the principal agent for structural e v olution under the conditions imposed in this case.

Specifically,the displacement of the bottom right point of the structure is plotted in Fig.8for the theoretical and simulated cases.At the end of the sintering a displacement of 1.6%in the radial direction and 4.4%in the horizontal direction was obser v ed in the sample,compared with respecti v e v alues of 1.3and 3.9%by simulation.Note that the displacement was about three times as great in the horizontal direction,indica-ti v e of the finer grain sizes along this axis compared with the plane along the bottom of the structure.This une v en shrinkage causes a small warpage in the structure,but at le v els of only a few percent (in real dimensions,around 0.3mm),the de v iation from being a perfectly flat piece is acceptable for practical use.The good qualitati v e and quantitati v e correspondence between these v alues in-

dicated that a sufficient and suitable amount of mechan-istic beha v ior is incorporated into the sintering model to enable it to properly trace structural e v olutions of porous ceramics made from broad powder size distribu-tions.

The present results can be further discussed in the context of related work a v ailable in the literature.A sintering simulation based on a similar constituti v e model was presented by Moritz et al.[18].In this case,the functionally gradient character was modeled as a four layer laminate,no particle size distribution effects were considered and the simulation proceeded to much higher theoretical densities,as the aim of the work was more focussed on sample warpage.

A paper by Subbanna et al.[19]pro v ides a functional mathematical model incorporating pore shrinkage and grain growth in systems densifying under sintering.In this case,no physical mechanism in ascribed,thus exponential parameters in the model must be allowed to float to match other published data.The present simulation is a has a more direct application to real systems because its parameters are deri v ed from experi-mental work,and were used in tracking the e v olution of grain growth and density within the confines of a physically defined porous structure.

5.Conclusions

Upon formulation and numerical implementation of a model designed to demonstrate the effects of local particle size distribution effects in a porous asymmetric ceramic structure,it can be said that these effects account for obser v ed sintering beha v ior in real and substantial ways.The calculated predictions corre-sponded well to the results obtained when physical samples with such characteristics were sintered.The presence of fines throughout the compact produced a sintered piece which bonded at lower temperatures and with less warpage.Accounting for the particle size distribution as a field v ariable helps to more realistically simulate the sintering results of functionally gradient materials prepared by colloidally metastable sedimenta-tion.For future work,this simulation tool will pro v e v aluable in understanding microstructural effects on sintering beha v ior,and ultimately can aid in fabrication parameter optimization.

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