Hydrothermalsynthesisandsinteringofultrafineceo2powders.pdf

Hydrothermal synthesis and sintering of ultrafine CeO2 powders Y. C. Zhou Institute of Metals Research, Academia Sinica, 72 Wenhua Road, Shenyang 110015, People's Republic of China

M. N. Rahaman Ceramic Engineering Department, University of Missouri-Rolla, Rolla, Missouri 65401-0249 (Received 19 October 1992; accepted 5 March 1993)

Undoped CeO 2 and Y 2 O3 -doped CeO 2 powders, with particle sizes of —10-15 nm, were prepared under hydrothermal conditions of 10 MPa at 300 °C for 4 h. The compacted powders were sintered freely in air or in O 2 at constant heating rates of 1-10 °C/min up to 1350 °C. The undoped CeO 2 started to sinter at « 8 0 0 - 9 0 0 °C and reached a maximum density of 0.95 of the theoretical at 1200 °C, after which the density decreased slightly. Isothermal sintering at 1150 °C produced a sample with a relative density of =0.98 and an average grain size of =100 nm. The samples sintered above 1200 °C exhibited microcracking. The decrease in density and the microcracking above 1200 °C are attributed to a redox reaction leading to the formation of oxygen vacancies and the evolution of O2 gas. Doping with Y 2 O 3 produced an increase in the temperature at which measurable sintering commenced and an increase in the sintering rate, compared with the undoped CeO 2 . Sintered samples of the doped CeO 2 showed no microcracks. The CeO 2 doped with up to 3 mol % Y 2 O 3 was sintered to almost full density and with a grain size of =200 nm at 1400 °C.

I. INTRODUCTION Ultrafine-grained (or nanocrystalline) materials, characterized by a grain size of <=100 nm, have been the subject of much recent research. Experiments have shown that bodies formed by consolidating ultrafine powders have dramatically enhanced sintering rates and decreased sintering temperatures compared with their coarse-grained (micron-sized) counterparts.1"3 The properties of the ultrafine-grained materials are also significantly different and improved compared to coarsegrained materials.4"7 The successful fabrication, from powders, of ultrafine-grained materials involves the synthesis of powders with controlled characteristics, consolidation of the powders, and sintering procedures that can produce the required ultrafine microstructures. Most work on the fabrication of ultrafine-grained materials has utilized powders prepared by the inert gas condensation of Gleiter and co-workers. 8 Briefly, in this method, a metal (e.g., Ti) is first evaporated in a He atmosphere. Small particles condense from the vapor and are collected on a cold shroud. The chamber is then filled with O 2 which promotes rapid oxidation of the powder. Following the oxidation step, the powder is scraped off the cold shroud, lightly compacted, annealed again in O 2 , and finally compacted to provide samples for free sintering or pressure sintering. The gas condensation technique suffers from a number of limitations.9 These include strong particle agglomeration 1680

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during the oxidation step and difficulties in controlling the composition of the powder and in the incorporation of dopants. Hydrothermal synthesis is an attractive method for the preparation of crystalline ceramic oxide powders. It facilitates good control of the powder characteristics and the incorporation of dopants into the powder. 10'11 The process, involving precipitation from aqueous solutions under elevated temperature and pressure, has been used for decades for the preparation of ultrafine oxide powders. 12 While a variety of both simple and mixed oxides has been prepared, an in-depth study of the free sintering behavior of these powders has not been performed. The work described in this paper formed an initial study into the synthesis of CeO 2 powders and Y 2 O 3 doped CeO2 powders by hydrothermal processing and the free sintering of the consolidated powders. CeO 2 was chosen for the study because it has a relatively simple crystal structure, the cubic fluorite-type structure,13 and because it has good solid solubility for many cation dopants.14 II. EXPERIMENTAL A. Hydrothermal synthesis of CeO2 powders and Y2O3-doped CeO2 powders The starting materials for the preparation of CeO 2 powders and Y 2 O 3 -doped CeO 2 powders by hydrother1993 Materials Research Society

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Y. C. Zhou and M. N. Rahaman: Hydrothermal synthesis and sintering of ultrafine CeC>2 powders

mal processing were cerium (m) nitrate hexahydrate, Ce(NO 3 ) 3 • 6H 2 O, and yttrium chloride hexahydrate, YC13 • 6H 2 O. Both chemicals had a purity of 99.9% as stated by the manufacturer (Aldrich Chemical Company, Inc., Milwaukee, WI). The appropriate amounts of cerium nitrate (for the preparation of CeO 2) or cerium nitrate and yttrium chloride (for the doped CeO 2) were dissolved in distilled water and ammonium hydroxide solution was added, under vigorous stirring, to give a solution with a pH = 10. The precipitated material was filtered and washed with distilled water, after which a low viscosity gel was formed. The gel was transferred into a Teflon tube, which was sealed with a cap and placed in an autoclave. The system was kept under a pressure of =10 MPa at =300 °C for 4 h. After the system was quenched in water at 0 °C, the precipitated powder was washed with distilled water, dried in air at room temperature, then lightly ground in an agate mortar and pestle. Finally, the powder was dried at 200 °C for 2 h. B. Sintering of the consolidated powder Powder compacts (6 mm in diameter by 4 mm) were formed by pressing the powder uniaxially under a pressure of =50 MPa in a tungsten carbide die. Sintering was performed in a dilatometer that allowed continuous monitoring of the shrinkage kinetics. The samples were sintered either at constant rates of heating (1-10 °C/min to =1400 °C) or isothermally at =1150 °C. Most sintering runs were performed in air, but, for comparison, a few were performed in O 2 . The density of the sample at any time or temperature was measured from the initial density and the shrinkage. The final density was checked by the Archimedes method. C. Characterization of the powder and the sintered samples

where fim is the measured halfwidth, and /3S is the halfwidth of a standard CeO 2 sample with a crystal size greater than 100 nm. The reflection from the (422) plane, occurring at 88° 2 0 , was used for the crystallite size measurement. Specimens of the synthesized powder for TEM (Philips EM-300) were made by dispersing the powder in a liquid and then putting a drop of the suspension on a holey carbon film, followed by drying. TGA and DTA were performed at a heating rate of 10 °C/min. Microstructures of the sintered samples were observed using scanning electron microscopy of fracture surfaces. The grain size was measured by the line intercept method. Phase composition was analyzed by XRD of powder samples. III. RESULTS The x-ray diffraction pattern of the CeO 2 powder prepared by hydrothermal synthesis was identical, within the limits of detection by the diffractometer, to that for a standard CeO 2 with the cubic fluorite structure [Fig. l(a)]. For the Y 2 O3 -doped CeO 2 , free Y 2 O 3 was not detected by x-ray analysis for dopant concentrations of up to 3 mol % Y 2 O 3 used in this work [Fig. l(b)]. On the basis of the x-ray data, the Y 2 O 3 may be assumed to be incorporated into the CeO 2 solid solution during the hydrothermal synthesis. The crystallite sizes of the synthesized powders, calculated from the Scherer formula, were 12 nm for CeO 2 and for CeO 2 doped with 1 mol% Y 2 O 3 , and 9 nm for CeO2 doped with 3 mol % Y 2 O 3 . TEM showed that the synthesized powders (e.g., CeO 2 in Fig. 2) had faceted polyhedral morphologies. The particles also

o o

The ultrafine powders prepared by hydrothermal synthesis were characterized by x-ray diffraction (XRD), transmission electron microscopy (TEM), thermogravimetric analysis (TGA), and differential thermal analysis (DTA). Analysis of the crystal structure of the powder was performed in a diffractometer (Scintag XDS 2000) using CuK a radiation at a scan rate of 2° 20/min . Crystallite size was measured from XRD patterns at a scan rate of 0.5° 2 0 / m i n . The crystallite size, D, was calculated from the Scherer formula, k*jtfijf W\||UuwUUf \+

D = 0.9A/(y8 cos

(1)

where A is the wavelength of the x-rays, 0 is the diffraction angle, and ft is the corrected halfwidth given by: ?2

== Pi - Pi

(2)

25.0

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74.2

90.6

Degrees 20 FIG. 1. X-ray diffraction patterns of the ultrafine CeC>2 powders prepared by hydrothermal processing for (a) the undoped powder and (b) the powder doped with 3 mol% Y2O3.

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Y. C. Zhou and M. N. Rahaman: Hydrothermal synthesis and sintering of ultrafine CeC>2 powders

100 Ultra-fine powder

•

A Precipitated powder

Der isity

90-

01

D Commercial powder

80-

70-

(0 0)

rr

i

**•

60-

c 700

800

900

1000 1100 1200 1300 1400 1500 1600

Temperature

FIG. 4. Data for the relative density versus temperature for the ultrafine CeC>2 powder, compared with a coarser CeO2 powder prepared by chemical precipitation and with a commercial CeC>2 powder.

50 nm FIG. 2. TEM of the undoped CeC>2 powder prepared by hydrothermal processing.

appear to be fairly uniform in size. Particle size distribution calculated by measuring maximum diameters of more than 100 particles in TEM micrographs was fairly narrow, as shown in Fig. 3 for CeO2. The average particle size was calculated to be 14 ± 3 nm, which agrees very well with the average size (12 nm) obtained by x-ray line broadening. The sinterability of the ultrafine CeO2 powder is shown in Fig. 4, where the relative density, p, is plotted as a function of temperature, T. Also shown in Fig. 4 are data for a coarser CeO2 powder (particle size «100-200 nm) prepared by precipitation from solution

C

30-

ll

20-

(°C)

under nonhydrothermal conditions (i.e., room temperature and atmospheric pressure) and for a commercial high-purity CeO2 powder (purity 99.999%; Cerac, Inc., Milwaukee, WI). The samples were sintered at 5 0C/min in air. Each curve is the average of two runs under the same conditions, and the density at any T is reproducible to ±0.01. It is seen that the commencement of measurable sintering occurred at approximately the same T («800-900 °C) for all of the samples. However, after the commencement of sintering, the rate of densification increased significantly with the decrease in particle size of the powder. The densification rate, (\/p)dp/dt, was calculated by fitting smooth curves to the data of Fig. 4 and differentiating. Figure 5 shows (l/p)dp/dt vs T. It is seen that the curve becomes narrower and the maximum increases with decreasing particle size. •

Ultra-tine powder

A

Precipitated powder

D

Commercial powder

E

10

12

Size

14

16

18

20

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B O O 9 0 0 1000 1100

1200 1300

1400 1500

1600

Temperature (° C)

FIG. 3. Particle size distribution of the undoped CeO2 powder measured from TEM micrographs. 1682

700

Range (nm)

FIG. 5. Densification rate versus temperature calculated from the data of Fig. 4.

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Y. C. Zhou and M. N. Rahaman: Hydrothermal synthesis and sintering of ultrafine CeC>2 powders

100 T

100-

90-

D

CeO2

•

CeO2+1mol%Y2O3

f

tT

a

• CeO2+3mol%Y2O3

f

B0-

a

70-

/

/ 60"

a 50'

50'r 700

800

1 40-

900

1000

1100

Temperature

1200 1300 1400

700

8 0 0

9 0 0

1000 1100 1 2 0 0 1 3 0 0

Temperature

(°C)

(a)

1 4 0 0

(°C)

FIG. 7. Effect of Y2O3 dopant on the sintering of the ultrafine CeO2 powder.

too

90-

•

1 ° C/min

n

5° C/min

A 10° C/min >

80-

o

700

800

900

1000

1 100

1 200

Temperature (°C)

(b) FIG. 6. (a) Effect of atmosphere on the sintering of the ultrafine CeC>2 powder, (b) Effect of heating rate on the sintering of the ultrafine CeC>2 powder.

Figures 6(a) and 6(b) show the effect of atmosphere (air and O2) and heating rate (1, 5, and 10 °C/min) on the sintering behavior. The data show that the sinterability of the sample improves slightly in O 2 . The sintered density is almost independent of the heating rate and this means that the densification rate increases almost proportionally with increasing heat rate. Since the sintered density at any temperature was not affected significantly by the different atmospheres or heating rates, the experiments described in the rest of this section were performed in air at a fixed heated rate of 10 °C/min. The effect of Y 2 O 3 dopant on the sinterability of the ultrafine CeO 2 powder is shown in Fig. 7 for dopant concentrations of 0, 1, and 3 mol %. As described earlier, the average particle sizes (as measured by x-ray line broadening) were 12 nm for the undoped CeO 2 and for the CeO 2 doped with 1 mol% Y 2 O 3 , and 9 nm

for the CeO 2 doped with 3 mol% Y 2 O 3 . The powder doped with 3 mol % Y 2 O 3 could not be compacted to a green density > « 0 . 4 2 , which suggests that, apart from the difference in particle size, this powder may have somewhat different characteristics from the other two. These powders are currently being studied by TEM. However, it is seen that incorporation of the Y 2 O 3 produced a significant increase in the temperature at which measurable sintering is observed. After the commencement of sintering, the trajectory of the curve steepens with increase in dopant concentration. The densification rate, (l/p)dp/dt, as a function of T is shown in Fig. 8 for the undoped CeO 2 and the Y 2 O 3 doped CeO 2 . For the range of dopant concentration used in this work, the curve shifts to higher temperature, but the width of the peak narrows with increasing dopant concentration.

D

CeO2

A

CeO2+1mol%Y2O3

•

CeO2+3mol%Y2O3

50-

-10 800

900

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1100

1200

1300

1400

FIG. 8. Densification rate versus temperature calculated from the data of Fig. 7.

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Temperature (°C)

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Y. C. Zhou and M.N. Rahaman: Hydrothermal synthesis and sintering of ultrafine CeC>2 powders

As described earlier, the data of Fig. 7 show that the undoped CeO 2 sample reached a maximum density of =0.95 at =1200 °C, after which the density decreased slightly. The microstructure of the surface of an undoped CeO 2 sample sintered to 1350 °C [Fig. 9(a)] shows evidence of microcracking. An undoped CeO 2 sample sintered isothermally at 1150 °C for 2 h reached a density of =0.98 and shows no microcracking [Fig. 9(b)]. However, heating of this sample above =1200 °C (at 10 °C/min used for the constant heating rate sintering) produced microcracking. In contrast, a CeO 2 sample doped with 3 mol % Y 2 O 3 reached almost full density and, as seen from Fig. 9(c), had a fairly uniform grain size (=200 nm) after sintering to 1400 °C. Furthermore, there is no evidence of microcracking in the microstructure of the doped sample.

(a)

IV. DISCUSSION The experiments allowed an investigation into the sintering of ultrafine CeO 2 prepared by hydrothermal processing and into the effect of Y 2 O 3 dopant on the sintering of the CeO 2 powder. These effects are considered separately in the following sections.

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A. Sintering of ultrafine CeO2 powder The data of Fig. 4 show that the temperature (800-900 °C) at which measurable sintering of the ultrafine CeO2 powder commenced is approximately the same for coarser CeO 2 powders. However, after the commencement of sintering, the densification rate increases rapidly with decrease in particle size (Fig. 5). While the synthesis conditions for the commercial powder (average particle size =0.5 yam) are not available, the powder with the intermediate size (0.1-0.2 /xm) was prepared by precipitation, under nonhydrothermal conditions, from the same starting materials used for the hydrothermal process. Measurable shrinkage of compacts of CeO 2 powders prepared by spray pyrolysis 15 also commenced near the temperature observed in this work. The insensitivity of the temperature range for the commencement of measurable sintering to the particle size of the powder is somewhat surprising, but has also been observed for ZrO 2 powders.16 It is very different from the results of Hahn, Logas, and Averback,2 who found a drop of ==600 °C for ultrafine TiO 2 powder (particle size =14 nm) prepared by the inert gas condensation method, compared with coarse TiO 2 powder (particle size = 1 fim). The temperature at which measurable shrinkage occurs may depend on a number of factors, including particle size, degree of agglomeration of the powder, uniformity of packing in the green body, defect structure, and chemical composition. Further work is required for the understanding of the observed behavior. 1684

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FIG. 9. SEM of the sintered bodies formed from the ultrafine powders, showing (a) the top surface of an undoped CeC>2 compact sintered at 10 °C/min to 1350 °C, (b) a fracture surface of an undoped CeO 2 compact sintered isothermally at 1150 °C for 2 h, and (c) a CeO 2 compact doped with 3 mol % Y 2 O 3 after sintering at 10 °C/min to 1400 °C.

The sintering models 17'18 predict that the densification rate depends on temperature, T, and grain size, G,

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Y.C. Zhou and M.N. Rahaman: Hydrothermal synthesis and sintering of ultrafine CeO2 powders

according to: (l/p)dp/dt

= C exp[-e/(RD]/Gm,

(3)

where C is a constant, Q is the activation energy for the rate-controlling process, R is the gas constant, and m is an exponent that depends on the mechanism of sintering, e.g., m = 3 for lattice diffusion and m = 4 for grain boundary diffusion. At a fixed temperature, a double logarithmic plot of (l/p)dp/dt vs G would yield the exponent m. This plot, taken from the data of Fig. 5, is shown in Fig. 10 for various temperatures (850, 900, and 950 °C) during the early stages of sintering. In the plot, it was assumed that grain growth (or coarsening) prior to the attainment of these temperatures is negligible, so that the particle size of the powder was taken as the grain size. A least squares fit of the three data points for each temperature shows that the exponent m does not change significantly for these temperatures and is —0.5. This value is much smaller than the predicted values for solid-state sintering controlled by diffusional processes. Differences in agglomeration of the powders and in the packing of the green bodies may give rise to a significant deviation between the experimental values and the theoretical predictions. The sintering models assume geometrically similar packing.17 Another factor that may contribute to the low observed value for m is the assumption of negligible grain coarsening. Surface diffusion and evaporation/condensation produce coarsening but no densification during sintering, and their rates vary as 1/G4 and 1/G2, respectively. The rate of coarsening will therefore increase strongly with decrease in particle size. In this case, the exponent m will be reduced drastically. The sintering kinetics (Figs. 4 and 7) and the microstructural observations [Fig. 9(a)] showed that the

- 4

- 3

- 2

- 1

o

Ln(Grain Size/^m)

FIG. 10. Logarithmic plot of the densification rate versus grain size during the initial stage of sintering, plotted from the data of Fig. 5.

ultrafine CeO2 suffered from a decrease in density and microcracking when heated above =1200 °C in air or in O2. DTA analysis showed no evidence of a phase transformation for a powder heated in air at 10 °C/min to 1400 °C. XRD of samples sintered up to 1350 °C also showed no evidence for the formation of any new phase. However, a small weight loss beginning at ==1200 CC was detected by TGA. The decreased density at temperatures higher than 1200 °C is believed to be due to the reduction of CeO2 to Ce2O3 that occurs at high temperatures or in reducing atmosphere.19 This redox reaction is accompanied by the formation of an oxygen vacancy for each pair of Ce 4+ ions being reduced, i.e., 2CeO2 — Ce 2 O3 + l/2O 2

(4)

2e' + l/2O 2

(5)

or Oo

The decrease in density and the microcracking in the undoped CeO2 sintered above =1200 °C may be caused by the release of O 2 . As noted earlier [Fig. 9(b)], undoped CeO2 sintered at 10 °C/min to 1150 °C and held at this temperature for 2 h reached a density of 0.98 and showed no microcracking. However, microcracking occurred if the sample were then heated to temperatures above =1200 °C. The Ce2O3 cannot be detected by XRD performed at room temperature since it oxidizes back to CeO2 on cooling. B. Effect of Y 2 O 3 dopant on the sintering of ultrafine CeO2 powder

The incorporation of Y 2 O 3 dopant in the range of 0-3 mol % into the ultrafine CeO2 powder during hydrothermal synthesis produced an increase in the temperature at which measurable sintering commenced (Fig. 7). However, after the commencement of sintering, the curve increases more steeply with an increase in dopant concentration. Figure 8 shows that the densification rate curve becomes narrower and the height of the peak increases with an increase in dopant concentration. The delay in the onset of measurable sintering and the increase in sintering rate produced by the dopant have also been observed for a coarse, high-purity CeO2 powder obtained commercially.20'21 The presence of the Y 2 O 3 dopant essentially eliminated the microcracking problem observed for the undoped powder [Fig. 9(a)]. Doped powders can be sintered to nearly full density with a fine grain size (=200 nm), as seen in Fig. 9(c). The absence of microcracking in the doped samples can be explained in terms of the effect of the Y 2 O 3 dopant on the defect chemistry of the CeO2. Since the ionic crystal radius22 of Y 3+ (0.1097 nm) is close to that of Ce 4+ (0.097 nm),

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Y. C. Zhou and M.N. Rahaman: Hydrothermal synthesis and sintering of ultrafine CeC>2 powders

the yttrium ions would be expected to occupy normal cerium sites. The defect reaction for the incorporation of Y 2 O 3 may be written: Y2O3 —

2Y^ e + Vo + 2O O + 1/202.

(6)

The addition of Y 2 O 3 to CeO 2 influenced the redox reaction at high temperature because the balance between Ce 4+ and Ce 3+ can be compensated by the incorporation of Y 3 + in the normal Ce 4+ positions. For 1 mol% dopant, the redox reaction of ceria is not fully suppressed, as seen from the small kink at point A in the sintering curve (Fig. 7). The redox reaction appears to be fully suppressed in CeO 2 doped with 3 mol % Y 2 O 3 since no measurable change in the slope of the sintering curve was observed at =1200 °C. V. CONCLUSIONS Ultrafine powders of undoped CeO 2 and Y 2 O 3 doped CeO2 (particle size 10-15 nm) were synthesized under hydrothermal conditions, and the sinterability of the consolidated powders was investigated at constant rates of heating in air or O 2 . For the undoped powder, the temperature at which measurable shrinkage commenced (800-900 °C) was approximately the same as for coarser, micron-sized powder. This is very different from the published results for ultrafine TiO2 powders prepared by the inert gas condensation method, where a decrease of =600 °C was observed. For the initial stage of sintering, the densification rate showed a relatively weak dependence on the initial particle size of the powder, compared to the predictions of sintering models. The occurrence of a redox reaction at =1200 °C limited the sinterability of the undoped powder. The sintered density reached a maximum of 0.95 (of the theoretical) at this temperature and then decreased slightly. The undoped samples sintered above 1200 °C showed evidence of microcracking. Higher sintered densities can be obtained by sintering isothermally below 1200 GC or through the use of dopants. An undoped sample sintered isothermally at 1150 °C reached a density of 0.98 with a grain size of =100 nm. The incorporation of Y 2 O 3 dopant ( 0 - 3 mol %) into the CeO2 powder during hydrothermal synthesis pro-

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duced an increase in the temperature at which measurable sintering commenced and an increase in the sintering rate. The effects of the CeO 2 redox reaction on sintering were much less severe in the doped samples. A sample containing 3 mol % Y 2 O 3 was sintered to almost theoretical density with a grain size <200 nm and showed no microcracking. ACKNOWLEDGMENT This work was supported by a grant from the Weldon Spring Endowment, University of Missouri. REFERENCES 1. R. W. Siegel, S. Ramasamy, H. Hahn, L. Zongquan, L. Ting, and R. Gronsky, J. Mater. Res. 3, 1367 (1988). 2. H. Hahn, J. Logas, and R.S. Averback, J. Mater. Res. 5, 609 (1990). 3. W. Wagner, R. S. Averback, H. Hahn, W. Petry, and A. Wiedenmann, J. Mater. Res. 6, 2193 (1991). 4. H. Hahn and R.S. Averback, J. Am. Ceram. Soc. 74, 2918 (1991). 5. M.J. Mayo, R. W. Siegel, A. Narayanasamy and W. D. Nix, J. Mater. Res. 5, 1073 (1990). 6. G. W. Nieman, J. R. Weertman, and R. W. Siegel, J. Mater. Res. 6, 1012 (1991). 7. G. Skandan, H. Hahn, and J. C. Parker, Scripta Metall. 25, 2389 (1991). 8. R. Birringer, H. Gleiter, H-P. Klein, and P. Marquardt, Phys. Lett. 102A, 365 (1984). 9. H. Hahn and R.S. Averback, J. Appl. Phys. 67, 1113 (1990). 10. S. Somiya, M. Yoshimura, Z. Nakai, K. Hishinuma, and T. Kumaki, Adv. Ceram. 21, 43 (1987). 11. S. Komarneni, E. Fregeau, E. Breval, and R. Roy, J. Am. Ceram. Soc. 71, C26 (1988). 12. W.J. Dawson, Am. Ceram. Soc. Bull. 67, 1673 (1988). 13. R. N. Blumental, F. S. Brugner, and J.E. Garner, J. Electrochem. Soc. 120, 1230 (1973). 14. S.J. Wu and R.J. Brook, Solid State Ionics 12, 123 (1984). 15. B.R. Powell, R.L. Bloink, and C. C. Eickel, J. Am. Ceram. Soc. 71, C104 (1988). 16. C-L. Fan and M. N. Rahaman, unpublished work. 17. C. Herring, J. Appl. Phys. 21, 301 (1950). 18. R. L. Coble, J. Appl. Phys. 32, 787 (1961). 19. S. Meriani, Mater. Sci. Eng. A 109, 121 (1989). 20. M.N. Rahaman and C-L. Hu, in Synthesis and Processing of Ceramics: Scientific Issues, edited by W. E. Rhine, T. M. Shaw, R. J. Gottschall, and Y. Chen (Mater. Res. Soc. Symp. Proc. 249, Pittsburgh, PA, 1992), p. 427. 21. M.N. Rahaman and C-L. Hu, to be published in Materials & Manufacturing Processes 8 (1993). 22. R.D. Shannon, Acta Crystallogr. A 3232, 751 (1976).

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