diffusion induced grain-boundary migration and enhanced grain growth in batio3

INTERFACE SCIENCE 8, 223–229, 2000c© 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

Diffusion Induced Grain-Boundary Migration and EnhancedGrain Growth in BaTiO 3

HO-YONG LEE AND JAE-SUK KIMDivision of Metallurgical and Materials Engineering, Sunmoon University, Chungnam, Asan, 336-840, Korea

SUK-JOONG L. KANGDepartment of Materials Science and Engineering, Korea Advanced Institute of Science and Technology,

Taejon, 305-701, Korea

Abstract. The effect of diffusion induced grain-boundary migration (DIGM) on grain growth has been studied ina model system of BaTiO3-PbTiO3. When sintered BaTiO3 samples of two different grain sizes were heat-treatedin contact with PbTiO3, DIGM occurred in the coarse-grained samples (∼200 µm in average size) while fastgrain growth was observed in the fine-grained samples (∼4 µm). Energy dispersive spectroscopy (EDS) analysisconfirmed that the fast growth of BaTiO3 grains was accompanied by the alloying of Pb and thus related to DIGM.A calculation of coherency strain energy for the BaTiO3-PbTiO3 system showed that the coherency strain energyof a coherent (Ba0.8Pb0.2)TiO3 layer on BaTiO3 was between 2 and 3 MJ/m3 depending on the surface orientation.The calculated coherency strain energy values are much higher than the capillary energy due to the grain boundarycurvature of 4µm grains in the fine-grained sample. The observed enhancement of grain growth appears thereforeto be a result of DIGM. Such grain growth enhancement by DIGM is thought to occur in materials processing underchemical inhomogeneity or inequilibrium, for example, in the sintering of powder mixtures and in the annealing ofchemically inhomogeneous polycrystals.

Keywords: DIGM, barium titanate, grain growth, chemical inhomgeneity, coherency strain energy

1. Introduction

For a polycrystalline solid in contact with a new solutesource, it has been commonly thought that the alloy-ing of the solute atoms into grains proceeds by latticediffusion and that the interfaces in the polycrystal areonly a rapid path for diffusion. In this case, the con-centration of solute atoms shows a symmetrical distri-bution across the interface and decreases exponentiallyfrom the interface [1]. In the 1970s, however, the equili-bration reaction between them was observed to occurby the migration of interfaces, forming behind thema new solid solution with the same crystalline struc-ture and orientation as those of the parent grain [2–4].When the alloying of solute atoms into grains proceedsby grain boundary migration, the distribution of thesolute atoms is asymmetric across the migratingboundary [1]. Since boundary migration has usually

been observed to occur at low temperatures where thegrain boundary diffusion becomes dominant and to pro-ceed at high rates, the boundary migration providesa fast and efficient means of polycrystal compositionchange. This phenomenon is called diffusion inducedgrain-boundary migration (DIGM) [5–8].

When a boundary is migrating at a velocityv, a so-lution of macroscopic diffusion equations shows thata solute diffusion zone of a thickness of aboutD/v(where D is the lattice diffusivity) exists in front ofthe boundary [9]. In case where the thickness is thin,the diffusion zone can remain coherent with the latticeof the solid being consumed. From this point of view,Hillert proposed that the driving force for DIGM arosefrom the coherency strain energy built in a thin coher-ent diffusion layer on a receding grain [10, 11]. Thevalidity of the coherency strain model was tested bytwo kinds of critical experiments: the lattice matching

224 Lee, Kim and Kang

experiments in Mo-Ni [12] and cubic ZrO2-Y2O3

[13], and the migration direction experiments in Al2O3

[14, 15].By introducing two solute species of different sizes

into Mo–Ni [12] and cubic zirconia [13], Yoon andco-workers showed that the migration did not occurwhen the expected coherency strains of both the posi-tive and negative signs were reduced to very small val-ues. These observations demonstrated critically that thediffusional coherency strain energy drove the migra-tion of the boundaries. Tests of the migration directionwere also made by observing many grain boundariesbetween alumina single crystals of known surface ori-entations [14, 15]. For all the grain boundaries studied,DIGM always occurred in the direction of the grainwith the surface orientation corresponding to highercoherency strain energy. These observations confirmedthe prediction of migration initiation by Cahn and co-workers [16], where the migration began in the direc-tion of the grain with the surface orientation of highercoherency strain energy.

Since the proposed coherency strain energy in a thindiffusional layer on grains was found to be the driv-ing force for DIGM, coherency strain energy may actas an additional driving force for grain growth in apolycrystalline solid. Then, the total driving force forgrain growth should be the sum of the energies comingfrom capillary pressure, as in normal grain growth, andcoherency strain. If the coherency strain energy has avalue comparable to or higher than the capillary energy,the total driving force for grain growth becomes muchhigher than that expected only as result of the capillaryeffect. Much faster grain growth is then expected tooccur.

For systems containing volatile components or withlocal inhomogeneity in chemical composition, thechemical composition of solid grains can change dur-ing sintering or heat treatment and thus may result inDIGM. Since DIGM may have an effect on both graingrowth and densification during sintering, it is impor-tant to understand how DIGM takes part in microstruc-tural evolution in order to control the microstructureand properties of polycrystalline solids.

In the present investigation, the effect of coherencystrain energy on grain growth has been studied in amodel system, BaTiO3. As a solute compound, PbTiO3

was selected because of its high solid solubility inBaTiO3 [17]. To examine more specifically the effectof the non-uniform distribution of the solute elementon microstructural evolution, PbTiO3 particles werescattered on the surface of sintered BaTiO3 samples.

Since DIGM in the BaTiO3-PbTiO3 system had not asyet been reported, a heat-treatment of coarse-grainedBaTiO3 samples was first done in the presence ofPbTiO3 to observe the possibility of DIGM and thus todiscover the appropriate experimental condition wherecoherency strain energy is operative. After the deter-mination of this experimental condition for DIGM inBaTiO3, fine-grained BaTiO3 was heat-treated in thepresence of PbTiO3. The experimental results werediscussed in terms of the estimated coherency strainenergy in the BaTiO3-PbTiO3 system and capillaryenergy.

2. Experimental Procedure

Coarse-grained and fine-grained BaTiO3 samples wereprepared from hydrothermally-produced commercialBaTiO3 powder (BT-01, Sakai Chemical Industry Co.,Osaka, Japan). The powder was ball-milled for 12 h ina Nalgene bottle with ethanol and zirconia balls. Afterdrying and sieving to 100µm, the ball-milled powderwas compacted uniaxially at 1 MPa into cylindricalpellets 10 mm in diameter and 5 mm high and thenpressed hydrostatically at 200 MPa. The powder com-pacts were sintered for 1 h in air at1350◦C or 1380◦C tomake fine-grained and coarse-grained samples, respec-tively. The heating and cooling rate was 4 K/min. Thesintered samples were cut and polished up to a 0.25µmfinish. The solute source of the PbTiO3 particles wasprepared by sintering PbTiO3 (99%, Aldrich ChemicalCo., Milwaukee, WI, USA) powder compacts at 850◦Cfor 1 h and crushing. The prepared PbTiO3 particleswere scattered on the polished surface of the sinteredBaTiO3 samples and heat-treated at 1250◦C. To reducePb volatilization from scattered PbTiO3 particles dur-ing the heat-treatment, fine PbTiO3 powder was placedaround the BaTiO3 samples, as atmospheric powder.After the heat-treatment, microstructural observationand energy dispersive spectroscopy (EDS) analysiswere made on the polished surface of the BaTiO3

samples.

3. Experimental Results and Discussion

3.1. Heat-treatment of Coarse-grained BaTiO3

Samples with PbTiO3

The BaTiO3 sample sintered at 1380◦C for 1 h consistedof large grains of about 200µm in average size, indi-cating that abnormal grain growth had been completedduring the sintering. When the PbTiO3 particles were

Diffusion Induced Grain-Boundary Migration and Enhanced Grain Growth in BaTiO3 225

Figure 1. Surface microstructure of the BaTiO3 sample sintered at1380◦C for 1 h in air andthen heat-treated with PbTiO3 particlesat 1250◦C for 5 h. PbTiO3 particles were scattered on a polishedsurface of the sample before heat-treatment.

scattered on the polished surface of the sintered BaTiO3

and heat-treated at 1250◦C for 5 h, the PbTiO3 particlesremained on the polished surface of the heat-treatedsample, as shown in Fig. 1. Between grains, doublelines, a typical microstructure of DIGM [5–8], wereobserved. The double lines reveal that the grain bound-aries migrated during the heat-treatment. The line in-dicated by an arrow with “O” is the original positionof the boundary in the sintered sample before the heat-treatment and that indicated by an arrow with “M” isthe boundary positioned after the heat-treatment. Dur-ing the heat-treatment, the grain boundaries migratedin a way to increase their area and independently oftheir curvature.

Figure 2 shows the grain boundary migration ob-served at a vertical cross section in the sample inFig. 1. The bright part between two grains is the re-gion swept by a migrating boundary. The considerablegrain boundary migration occurred at the regions closeto the surface, but not in the interior. EDS analysisalong the line fromA to B, indicated in Fig. 2, re-vealed a discontinuous increase in Pb concentration(Pb/(Ba+Pb) [at%]) in the migrated region, as shownin Fig. 3. This result indicates that the boundary migra-tion was accompanied by Pb alloying of about 20 at%(Pb/(Ba+Pb)) in the BaTiO3. The composition analy-sis also showed that the PbTiO3 particles on the surfacechanged to a solid solution of about (Ba0.6Pb0.4)TiO3

after the heat-treatment. When the BaTiO3 sampleswere heat-treated without the PbTiO3 particles on thesurface, boundary migration did not occur even inthe presence of the atmospheric PbTiO3 powder around

Figure 2. Vertical cross section of the BaTiO3 sample sintered at1380◦C for 1 h in air andthen heat-treated with PbTiO3 particles at1250◦C for 5 h.

Figure 3. Measured concentrations of Pb along the line fromA toB at the cross section shown in Fig. 2. The dotted and solid linesshow the initial and final positions of the migrating grain boundary,respectively.

the sample. Therefore, the Pb detected in migrated re-gions was supplied from scattered PbTiO3 particles viasurface diffusion to the BaTiO3 grain boundaries duringtheir migration. This boundary migration accompaniedby the alloying of Pb in the BaTiO3 shows that DIGMcan occur in BaTiO3 when adding PbTiO3.

3.2. Heat-treatment of Fine-grained BaTiO3

Samples with PbTiO3

Figure 4 shows the microstructure of the BaTiO3 sam-ple sintered at 1350◦C for 1 h and heat-treated at


Figure 4. Surface microstructure of the BaTiO3 sample sintered at1350◦C for 1 h in air andthen heat-treated at 1250◦C for 5 h in air.

Figure 5. Microstructure of the BaTiO3 sample sintered at 1350◦Cfor 1 h in air and then heat-treated with PbTiO3 particles at 1250◦Cfor 1 h in air.

1250◦C for 5 h in air.Unlike the BaTiO3 sample sin-tered at 1380◦C, this sample does not contain anyabnormally large grains, but consists of small grainsof about 4µm in average size. When PbTiO3 parti-cles were scattered on a polished surface of the sin-tered BaTiO3 and heat-treated at 1250◦C for 1 h, fastgrowth of the BaTiO3 grains occurred near the scatteredPbTiO3 particles, as shown in Fig. 5. In Fig. 5, the brightparticles are the scattered PbTiO3 particles. Figure 6shows the Pb concentrations measured along the lineA–B in Fig. 5, from a scattered particle, a growinggrain to a receding grain. The scattered particle changedto a solid solution of about (Ba0.6Pb0.4)TiO3 and thegrowing grain near the scattered PbTiO3 particle was

Figure 6. Measured concentrations of Pb (in (Pb/(Ba+Pb))× 100[at%]) along the line fromA to B in Fig. 5.

a solid solution of about (Ba0.8Pb0.2)TiO3. As shownin Fig. 6, a discontinuous change in Pb concentrationwas observed across the boundary between a growing(Ba0.8Pb0.2)TiO3 grain and a receding BaTiO3 grain.This implies that the growth of the BaTiO3 grain wasaccompanied by the alloying of Pb in the BaTiO3,occurring in the same way as DIGM.

During further heat-treatment of the BaTiO3 samplewith PbTiO3 particles at 1250◦C for 5 h, the grains grewto about 20µm in average size, but the scattered PbTiO3

particles still remained, as shown in Fig. 7. The Pb

Figure 7. Microstructure of the BaTiO3 sample sintered at 1350◦Cfor 1 h in air and then heat-treated with PbTiO3 particles at 1250◦Cfor 5 h in air.


concentrations in the scattered particles and grains wereabout 40 and 30%, respectively. The distribution of Pbin the grains was uniform and a discontinuous changein the Pb concentration across grain boundaries wasnot detected. This result implies that DIGM occurredfor all the grains. Comparing the microstructures of theBaTiO3 samples heat-treated at 1250◦C for 5 h without(Fig. 4) and with PbTiO3 (Fig. 7), it is evident that thePbTiO3 particles remarkably enhanced the growth ofthe BaTiO3 grains.

3.3. Coherency Strain Energyin the BaTiO3-PbTiO3 System

The above experimental results amply show that DIGMcan enhance grain growth. To quantitatively estimateits contribution, the coherency strain energy has to becalculated. A couple of rigorous equations are availablefor the calculation of the coherency strain energy, Ec,of a thin coherent diffusion layer on a crystal surface[18–24]. One of the equations proposed by Lee andKang [22] is expressed as:

Ec= (1/2)[Cijklεklεij − (σ1′1′ε1′1′

+ 2σ1′3′ε1′3′ + 2σ1′2′ε1′2′)]. (1)

Here,Cijkl is the elastic stiffness of the coherent dif-fusion zone,εij the elastic strain induced by alloying,εi ′ j ′ the relaxed elastic strain, andσi ′ j ′ the elastic stressreleased. The subscript 1′ indicates a direction perpen-dicular to the crystal surface. Sinceεi ′ j ′ , σi ′ j ′ , and thusEc are functions ofCijkl andεij , only the values ofCijkl

and εij are required to calculate Ec and its variationwith the directionn normal to the surface.

For the calculation of Ec of a (Ba,Pb)TiO3 diffu-sion layer on a BaTiO3 crystal surface the elastic con-stants of pure BaTiO3 measured at 423 K were usedbecause of the unavailability of data for (Ba,Pb)TiO3

solid solutions [25]. TheCijkl used areC1111= 1.7278,C1122 = 0.8196, andC2323= 1.0823 [103 N/m2]. Thestress-free strainsεij of BaTiO3 with cubic symmetrycan be calculated directly from the lattice misfit param-eterη1 and the solute mole fractionX:

ε11 = η1X (2)

To determine the lattice misfit parameterη1 betweencubic BaTiO3 and cubic PbTiO3, lattice parameter datameasured above 763 K are needed because of the phasetransformation of PbTiO3 from tetragonal to cubic at

763 K [26]. The lattice parameter of the cubic PbTiO3

at 773 K is reported to be 3.9600̊A, while that of thecubic BaTiO3 at 773 K is not as yet available. Thelattice parameter of cubic BaTiO3 at 773 K has beenestimated to be 4.0264̊A by linear extrapolation fromthe available data obtained between 392 and 493 K[27]. From these lattice parameter data at 773 K, thevalue of the lattice misfitη1 is calculated to be about−1.65× 10−2. Thus the value of the elastic strainε11

of the coherent (Ba0.8Pb0.2)TiO3 diffusion layer, whichis the composition of the migrated region in Figs. 3 and6, is estimated to be about−3.3× 10−3.

The calculated values of Ec with surface normalnare graphically represented as a coherency strain en-ergy map, Fig. 8, which depicts equi-coherency strainenergy lines and some specific crystallographic orien-tations [22]. In this map, the axes represent the inter-planar angles between the surface normal of the planein question and (100) plane (α), and between the sur-face normal of the plane in question and (010) plane(β). In (Ba0.8Pb0.2)TiO3 with ε=−3.3× 10−3, the cal-culation shows that a (111) plane has a maximum Ec(Ec(111)= 3.11 MJ m−3) and a (100) plane a minimumEc (Ec(100)= 1.94 MJ m−3). A (110) plane has Ec of2.90 MJ m−3.

Compared with the solute concentrations in othermaterials where DIGM occurs, the measured Pb

Figure 8. Calculated coherency strain energy (in MJ m−3) mapof (Ba0.8Pb0.2)TiO3 (with ε11 of −3.3× 10−3) coherent layer withBaTiO3. The axes represent the interplanar angles,α andβ. Crystal-lographic directions with minimum and maximum Ec are representedby• and¥, respectively.


concentrations of about 20 at% in the migrated re-gions are relatively high. For BaTiO3, because of itsrelatively low stiffness [25], a high degree of alloyingappears to be necessary to build up enough coherencystrain energy to induce its DIGM. On the other hand, formaterials with high stiffness, for example Al2O3, only5 at% Cr2O3 alloying in Al2O3 was enough to inducea coherency strain energy of about 2.5 MJ m−3 [28],comparable to the coherency strain energy induced bya Pb alloying of about 20 at% in BaTiO3.

3.4. Driving Force for Grain Growth During SoluteDiffusion Along Grain Boundaries

The driving force for grain growth without alloying orsolute diffusion along grain boundaries is the capillaryenergy difference between the adjacent grains com-ing from the grain boundary curvature. The capillaryenergy per unit material volume can be expressed as:

1GCP= CGBγGB, (3)

whereCGB is the grain boundary curvature andγGB

the grain boundary energy. The curvatureCGB is ex-pressed in terms of the principal radii of the curvature,ρ1 andρ2:

CGB = 1/ρ1+ 1/ρ2. (4)

When DIGM occurs, however, the coherency strainenergy1GCSE stored in a thin coherent diffusionallayer on a receding grain may be an additionaldriving force for grain growth.

1GCSE= Ec, (5)

where Ec is the coherency strain energy per unit volumeand expressed as Eq. (1). Therefore, the total drivingforce1GT for grain growth under chemical inequilib-rium is thought to be the sum of the capillary energyand the coherency strain energy.

1GT = CGBγGB+ Ec (6)

The relative contributions of the two componentscan be estimated in our case of BaTiO3–PbTiO3. Forthe calculation of the coherency strain energy of a co-herent diffusion layer on a BaTiO3 crystal surface, thechemical composition of the diffusion layer is assumedto be (Ba0.8Pb0.2)TiO3, because the composition of a

Figure 9. Comparison between capillary energy (2γGB/r, r: the av-erage radius of grain boundary curvature) and calculated coherencystrain energies stored in a (Ba0.8Pb0.2)TiO3 layer on the BaTiO3.Specific grain boundary energy is assumed to be 0.5 J m−2.

coherent diffusion layer on a receding grain is thoughtto be identical to that of the migrated region [16]. Thecoherency strain energy of a (Ba0.8Pb0.2)TiO3 layer onBaTiO3 is then read between 2 and 3 MJ m−3 depend-ing on the surface orientation, as shown in Fig. 8. Forthe calculation of the capillary energy, the grain bound-ary energy is assumed to be 0.5 J m−2, a typical valueof ceramic surface energies.

Figure 9 plots the estimated coherency strain energyand capillary energy in function of the radius of grainboundary curvature. Since the coherency strain energyis independent of the boundary curvature, the energyis plotted as a dotted horizontal line in the figure. Incontrast, the capillary energy is inversely proportionalto the radius of the grain boundary curvature. When theradius of the grain boundary curvaturer is 0.33µm,the capillary energy is about the same as the coherencystrain energy of 3 MJ m−3. Whenr = 0.5µm, the capil-lary energy is about 2 MJ m−3. This calculation meansthat for a radius of grain boundary curvature largerthan 0.3–0.5µm, the coherency strain energy domi-nates the capillary energy and thus becomes a majordriving force for grain growth. Since the radius of thegrain boundary curvature is much larger than the grainradius, the grain size must be much smaller than a fewtenths of a micrometer for the capillary energy to be-come comparable to the coherency strain energy. Sincethe average radius of the BaTiO3 grains in Fig. 4 is


about 2µm, the coherency strain energy should be themajor driving force for grain growth.

The grain boundary diffusion of the solute atomsin polycrystalline materials changes chemical com-position, structure, and mobility of the grain bound-aries. Grain growth rate in materials sometimes in-creases when the solute atoms diffuse into them. Theobserved increase in growth rate has, in general, beenthought to be due to the increase in grain boundarymobility. However, the present investigation shows thatthe grain growth enhancement during solute diffusionalong grain boundaries can be due to an additional driv-ing force for grain growth, i.e. the coherency strainenergy. In this case of enhanced grain growth underchemical inequilibrium, the coherency strain energymight be the major driving force for grains larger thana submicron.

4. Conclusions

When solute atoms diffuse along grain boundaries anda thin diffusion layer forms on grain interfaces, thecoherency strain energy is stored in the layer and of-ten induces grain boundary migration (DIGM). In suchcases, grain growth in the material may occur not onlyby the capillary energy coming from the grain bound-ary curvature but also by the coherency strain energyinduced by solute diffusion. In the present investiga-tion, we demonstrated enhanced grain growth of fine-grained BaTiO3 with the addition of PbTiO3. A calcula-tion of coherency strain energy in the BaTiO3-PbTiO3

system showed that the coherency strain energy storedin a diffusion layer was much higher than the capillaryenergy of the micron-size grains. It appears, therefore,that coherency strain energy can be the major drivingforce for the grain growth of polycrystalline materialsunder chemical inequilibrium.

During materials processing, compositional changesof solid grains often occur, which can result in grainboundary migration. In particular, during the sinteringof mixed powder compacts, alloying and sintering pro-ceed simultaneously. Since the usual sintering temper-atures of materials are much lower than their meltingtemperatures, alloying of the elemental powders canbe achieved by DIGM rather than by lattice diffusion.In such cases, the growth of solid particles may oc-cur extensively at the early stage of sintering, whichsuppresses the densification. The present investigation

suggests that any chemical inhomogeneity has to beminimized in order to prevent extensive grain growthand hence to improve sinterability in mixed powdercompacts.

Acknowledgment

This work was partially supported by the NationalResearch Laboratory (NRL) Program of the KoreanMinistry of Science and Technology.

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diffusion induced grain-boundary migration and enhanced grain growth in batio3

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