Shrinkage of Pores Located at Grain Corners byGrain-Boundary Diffusion

Byung-Nam Kim,w Keijiro Hiraga, Koji Morita, Hidehiro Yoshida and Haibin Zhang

National Institute for Materials Science, Ibaraki 305 0047, Japan

The densification rate of pores located at grain corners is ob-tained by constructing and analyzing a simple model with atransport mechanism of grain-boundary diffusion. The densifi-cation rate obtained for a microstructure of tetrakaidecahedralgrains decreases with densification and is nearly consistent withthe Wilkinson model at low porosities. The effect of grain-sizedistribution on the densification rate is also estimated under thetwo limiting conditions of constant stress and constant strainrate. At constant strain rate, the densification rate with grain-size distribution is lower than the value for the average grainsize, whereas it is identical to the value at constant stress.

I. Introduction

MOST ceramic materials and an increasing number of metalmaterials are fabricated by sintering powder compacts at

elevated temperatures. The sintering of powder compacts canoccur by the mechanism of plastic yielding, power-law creep,lattice diffusion, and/or grain-boundary diffusion.1 The exactmechanism is dependent on the temperature, external pressure,particle size, and the stage of sintering. In the final stage ofsintering, the densification is characterized by the shrinkage ofisolated pores located at grain corners and grain-boundary fac-ets. The shrinkage kinetics was analyzed in several studies withthe mechanism of lattice/grain-boundary diffusion.2–6 In partic-ular, grain-boundary diffusion is a dominant transport mecha-nism in the high-temperature deformation and sintering of someceramic materials including alumina.7,8 With the mechanism,Pan and Cocks4 and Riedel et al.5 conducted elaborate analysison the shrinkage of pores in the final stage of sintering. In thepresent study, a new simple model is constructed for the shrink-age of the corner pores with the transport mechanism of grain-boundary diffusion, and the effect of grain-size distribution isestimated on the densification rate. The present densificationrate is also compared with the results of the previous models.

II. Shrinkage of Corner Pores

The material analyzed in the present model is a polycrystallinesolid where tetrakaidecahedral grains with spherical pores con-fined to each of the grain corners, as shown in Fig. 1, are uni-formly distributed in all orientations. It is assumed that thegrains are rigid and the shrinkage of the pores occurs by diffu-sion from grain boundaries into pores. Each pore is connectedto six grain-boundary facets, on which radial diffusion occursinto pores. Figure 2 shows the grain-boundary diffusion unit

employed in the present model. In the unit, the angle betweentriple lines is B1101, which is an equilibrium angle when fourtriple lines meet at one point, and the radius rc of the unit isrelated to the size of the boundary facet. Hence, one grain-boundary facet of the tetrakaidecahedron is divided into multi-ple diffusion units, the number of which is equal to the numberof pores on the facet.

From the configuration of a tetrakaidecahedron composed ofsix square and eight hexagonal facets, the grain volume V in-cluding pores is 8O2l3 and the porosity r is given by p/O2(a/l)3,where l is the edge length of the tetrakaidecahedron and a is theradius of the pore. The grain volume can also be representedusing the radius R of the sphere having the same material vol-ume as has the grain, i.e. (1�r)V5 4pR3/3. The total surfacearea of the tetrakaidecahedron is 6(112O3)l2 and the averagesurface area of the grain-boundary facet Ab is 3(112O3)l2/7.Dividing the average facet area Ab with the average number ofpores on the facet Np (5 36/7), we obtain the area of the diffu-sion unit including the pore area as (112O3)l2/12. Because thearea of the diffusion unit in Fig. 2 is equal to 110prc

2/360, weobtain the radius of the unit as rc

25 3(112O3)l2/(11p).On the other hand, the rate of the volume changeV due to the

pore shrinkage can be obtained from V5NbAbnn 53(112O3)l2nn, where Nb (5 7) is the number of grain bound-aries belonging to the tetrakaidecahedral grain and nn is therelative velocity of grain boundary. Because the volume changeis related to the mean strain rate _em as V/V5 3_em, the relativevelocity of the boundary facet in the present model is repre-sented as follows:

Vn ¼ 4ð1þ 2p3Þ�1ð2VÞ1=3 _em (1)

The constant value 4(112O3)�1 (5 0.896) in Eq. (1) is betweenthe values for square (51) and hexagonal facets (5O3/2)of the tetrakaidecahedral grain,3 which may indicate a reason-able simplification of the present model.

The radial flux J in the diffusion unit of Fig. 2 is related to therelative velocity of grain boundary nn by (qJ/qr1J/r)5�nn/O,where r is the distance from the pore center and O is the mo-lecular volume. Solving this equation with the boundary condi-tion of J5 0 at r5 rc, we obtain the radial flux J as follows:

J ¼ � r

2þ 0:195l2

r

� �vnO

(2)

From the thermodynamic principle that the external workmust be equal to the energy dissipation by grain-boundarydiffusion,

ðsm � ssÞ _V ¼ NpNbOkTdDb

ZJ2dS (3)

R. Riedel—contributing editor

This work was partly financially supported by the Grant-in Aid for Scientific Research (C-22560675) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

wAuthor to whom correspondence should be addressed. e-mail: kim.byung-nam@nims.go.jp

Manuscript No. 28637. Received September 21, 2010; approved December 16, 2010.

Journal

J. Am. Ceram. Soc., 94 [4] 982–984 (2011)

DOI: 10.1111/j.1551-2916.2011.04407.x

r 2011 The American Ceramic Society

982

we obtain the bulk viscosity K as

K ¼ðsm � ssÞ3 _em

¼ kTR3

OdDb� ð4:96r

2=3 � 1:87r4=3 � 1:10 lnr� 3:14Þ54ð1� rÞ

(4)

where sm is the externally applied hydrostatic stress, ss is theinternal sintering stress, k is Boltzmann’s constant, T isthe absolute temperature, d is the thickness of grain boundary,Db is the diffusion coefficient of grain boundary, and S is hegrain-boundary area in the diffusion unit of Fig. 2. Ignoringthe sintering stress under the compressive hydrostatic pressureP(5�sm � ss), we can also obtain the densification rate D asfollows:

_D ¼DP

K

¼OdDbP

kTR3� 54ð1� rÞ2

ð4:96r2=3 � 1:87r4=3 � 1:10 lnr� 3:14Þ(5)

where D (5 1�r) is the relative density.For the same problem of the tetrakaidecahedral grain with

corner pores, different solutions on the densification rate wereobtained as follows:

_D ¼ OdDbP

kTR3� 54ð1� r2=3Þð3r2=3 � ð1þ r2=3Þ lnr� 3Þ (6)

and

_D ¼ OdDbP

kTR3� 54ð1� rÞ2

ð4:98r2=3 � 1:92r4=3 � 1:28 lnr� 3:20Þ (7)

by Wilkinson1 and Pan and Cocks,4 respectively. Despite thedifferent method for analysis, the rate equation of the presentmodel (Eq. (5)) has the same form with that of Pan and Cocks(Eq. (7)): only the numerical constants are slightly different.These densification rates are compared in Fig. 3, where the ratedecreases with decreasing porosity or increasing density. InFig. 3, the present densification rate is close to that of Wilkin-son, and at ro0.03, the two rates are nearly identical. Thedensification rate of Pan and Cocks, however, is considerablylower than the present rate. Regarding the model of Pan andCocks, some problems were pointed out by Riedel et al.,5 i.e. theresults of Pan and Cocks4 were derived for a special orientationand the cubic anisotropy of the tetrakaidecahedral grain wasignored. Riedel et al.5 analyzed the deformation of tetrakaide-cahedral grains with pores on grain-boundary facets, which isdifferent from the present corner-pore problem. From the com-parison of their results for a fully dense material (r5 0), theyalso noted that the uniaxial strain rate is lower in the Pan–Cocksmodel than in their model by 153/272 times. Therefore, when weapply the method of Riedel and colleagues to the corner-poreproblem, the densification rate in the Pan–Cocks model (Eq. (7))may be increased by about 272/153 times, which then becomesalmost consistent with the present rate (Eq. (5)).

III. Effect of Grain-Size Distribution

The densification rate obtained in the above analysis assumesthe same grain and pore sizes. During sintering, however, actualpolycrystals consist of different sizes of grains and pores.Among these various parameters, in this section, we crudelyestimate the effect of grain-size distribution on the densificationrate, using two limiting conditions.

We define the average grain radius �R by �R3 ¼ 3VT=ð4pNTÞ,

where VT is the total grain volume and NT is the total number ofgrains. In a polycrystal, the grain-size distribution can be de-scribed by the probability density function F(u), where u is thenormalized grain radius defined as R= �R. During sintering, it isassumed that F(u) is self-similar and

RF(u)du5 1, as in steady-

state grain growth. It is also assumed that the polycrystal iscomposed of groups of grains with identical size. Each grouphas the same porosity and is distributed randomly in the poly-crystal. In the present corner-pore model (Fig. 1), the ratio of thepore radius to the grain radius a/R is directly related to theporosity as r/(1�r)5 6(a/R)3, so that the ratio a/R is constantfor all the groups.

Now, we consider the energy dissipation by diffusional pro-cess in the respective grain groups. The volume fraction of thegroup u is u3F(u)du and the energy dissipation is given by_emusmuVTu

3F(u)du, where _emu and smu are the mean strainrate and the hydrostatic stress of the group u, respectively. From

Fig. 3. Densification rate in the corner-pore models.

Fig. 1. Tetrakaidecahedral grain with spherical pores at corners.

Fig. 2. The grain-boundary (GB) diffusion unit.

April 2011 Rapid Communications of the American Ceramic Society 983

the summation for all the groups, we obtain

_emsmVT ¼Z

_emusmuVTu3FðuÞdu (8)

In the present study, two limiting conditions are employed,that is, constant stress and constant strain rate in the graingroups, which predict the upper-bound and lower-bound dens-ification rates, respectively. In the upper-bound, sm5smu andthen Eq. (8) becomes _em5

R_emuu

3F(u)du. Considering that_emu ¼ sm=3KðRÞ and K(R)BR3 in Eq. (4), we obtain _em ¼sm=3Kð �RÞ and _D ¼ DP=Kð �RÞ, which are dependent only onthe average grain radius �R and independent of the grain-sizedistribution F(u). Under the condition of constant stress (upper-bound), the densification rate of the polycrystal with grain-sizedistribution is thus identical to the value for the average grainradius �R.

In the lower-bound or at constant strain rate, _em5 _emu andthen Eq. (8) becomes sm 5

Rsmuu

3F(u)du. Considering thatsmu ¼ 3KðRÞ _em and K(R)BR3, we obtain _D ¼ DP=½Kð �RÞRu6FðuÞdu�, which is dependent on the grain-size distribution

F(u). The value of the integral is 1.705 and 3.390 for the Hillertand the Rayleigh distributions of the grain size, respectively.9–11

The integral value 41 indicates that the grain-size distributioncontributes to reduce the densification rate. It is thus concludedthat the densification rate of the polycrystal with grain-sizedistribution, which value would be between the two bounds, islower than that calculated with the average grain size.

IV. Conclusions

The shrinkage of the pores located at grain corners was analyzedwith a mechanism of grain-boundary diffusion. The densificat-ion rate obtained for a microstructure of tetrakaidecahedral

grains decreases with increasing density, and is nearly identicalto the Wilkinson prediction, when the porosity is o0.03. Theestimation of the effect of grain-size distribution under the twolimiting conditions of constant stress and constant strain rateshowed that the densification rate of the polycrystal with grain-size distribution is lower than that calculated with the averagegrain size. The probability density function of grain size alsoaffects to the densification rate, which should be considered formore correct understanding of final-stage sintering.

References

1A. S. Helle, K. E. Easterling, and M. F. Ashby, ‘‘Hot-Isostatic Pressing Dia-grams: Ner Developments,’’ Acta Metall., 33 [12] 2163–74 (1985).

2R. L. Coble, ‘‘Sintering Crystalline Solids. I. Inermediate and Final StateDiffusion Models,’’ J. Appl. Phys., 32 [5] 787–92 (1961).

3H. Riedel, H. Zipse, and J. Svoboda, ‘‘Equilibrium Pore Surfaces, SinteringStresses and Constitutive Equations for the Intermediate and Late Stages of Sinte-ring—II. Diffusional Densification and Creep,’’ Acta Metall. Mater., 42 [2] 445–52(1994).

4J. Pan and A. C. F. Cocks, ‘‘A Constitutive Model for Stage 2 Sintering of FineGrained Materials—I. Grain-Boundaries Act as Perfect Sources and Sinks forVacancies,’’ Acta Metall. Mater., 42 [4] 1215–22 (1994).

5H. Riedel, V. Kozak, and J. Svoboda, ‘‘Densification and Creep in the FinalStage of Sintering,’’ Acta Metall. Mater., 42 [9] 3093–103 (1994).

6S.-J. L. Kang and Y.-I. Jung, ‘‘Sintering Kinetics at Final Stage Sintering:Model Calculation and Map Construction,’’ Acta Mater., 52 [15] 4573–8 (2004).

7R. M. Cannon, W. H. Rhodes, and A. H. Heuer, ‘‘Plastic Deformation of Fine-Grained Alumina (Al2O3): I, Interface-Controlled Diffusional Creep,’’ J. Am.Ceram. Soc., 63 [1–2] 46–53 (1980).

8A. M. Thompson and M. P. Harmer, ‘‘Influence of Atmosphere on the Final-Stage Sintering Kinetics of Ultra-High-Purity Alumina,’’ J. Am. Ceram. Soc., 76[9] 2248–56 (1993).

9M. Hillert, ‘‘On the Theory of Normal and Abnormal Grain Growth,’’ ActaMetall., 13 [3] 227–38 (1965).

10N. P. Louat, ‘‘On the Theory of Normal Grain Growth,’’ Acta Metall., 22 [6]721–4 (1974).

11B.-N. Kim, K. Hiraga, and K. Morita, ‘‘Kinetics of Normal Grain GrowthDepending on the Size Distribution of Small Grains,’’ Mater. Trans. JIM, 44 [11]2239–44 (2003). &

984 Rapid Communications of the American Ceramic Society Vol. 94, No. 4