stress induced by constrained sintering of 3ysz films measured by substrate creep

Stress Induced by Constrained Sintering of 3YSZ Films Measured bySubstrate Creep

Alan Atkinson,**,w Jung-Sik Kim,z Robert Rudkin, Samuel Taub, and Xin Wang

Department of Materials, Imperial College, London SW7 2AZ, U.K.

3YSZ green layers approximately 10 lm thick were screenprinted onto 3YSZ substrates up to 300 lm in thickness. Thestress induced by constrained sintering of the film (between11501 and 13501C) was measured by monitoring the bendingdisplacement of vertical strips of bilayers using a long-distancemicroscope. In order to deduce the stress it was first necessary tomeasure the creep properties of the substrates by monitoring thebending of horizontal beams under gravity. The creep strain rateof the 3YSZ substrates was linearly dependent on applied stressat the low stresses and strains involved in the present work. Thecreep viscosity appeared to increase with strain (time), whichmight be due to changes in grain-boundary composition, and hadhigher activation energy at temperatures above approximately12501C. The magnitudes of the creep viscosities are in reason-able agreement with other creep data in the literature for 3YSZ.

The in-plane stress induced during constrained sintering of the3YSZ films had a maximum value of approximately 3 MPaat 12001C. This behavior is consistent with literature resultsreported for constrained sintering of bulk alumina. The stressinduced by the constraint is of a similar order to the estimatedsintering potential.

I. Introduction

MANY planar solid oxide fuel cell (SOFC) concepts havea supported electrolyte with typical thickness in the range

5–20 mm in order to reduce the ohmic resistance loss associatedwith the electrolyte. Anode-supported electrolytes are usuallyfabricated by cosintering the electrolyte and the anode support.This imposes a constraint on the sintering of each component andleads to stress and distortion during the sintering process.1 How-ever, there is some common shrinkage in the plane of the com-posite and this relaxes the constraint somewhat. In other concepts(e.g., the integrated planar or segmented in series design of Rolls-Royce Fuel Cell Systems2) the substrate is effectively rigid. Insuch cases the film is fully constrained in its plane during sinteringand shrinkage can occur only perpendicular to its plane (exceptwithin a distance of a few layer thicknesses of the film edges).

This constrained sintering process is also a feature of manyother ceramic processing technologies and is a topic of consider-able fundamental interest. The constraint results in a tensile stressin the plane of the film that is just sufficient to oppose its tendencyto shrink in this plane, were it not for the constraint, and this hastwo main consequences. The first is that the densification isretarded,3,4 in comparison with an unconstrained case, and thesecond is that the in-plane tensile stress can cause crack-like defectsand porous channels in the constrained film or delamination fromthe substrate.5,6 Through-film defects are of particular concern in

SOFC applications because, if they penetrate the electrolyte, fuelcan then leak through the electrolyte and combust directly leadingnot only to loss of electrical efficiency, but also local hot-spots thatcan cause further mechanical damage.

Bang and Lu7 measured the stress induced when borosilicateglass films were sintered on silicon substrates at 7151C and foundvery low values of approximately 20 kPa. In contrast Calataet al.8 studied cordierite glass–ceramic films on silicon substrates at9001–10001C and found a stress of approximately 0.2 MPa beforecrystallization rising to almost 1 MPa after crystallization. Choeet al.9 also studied gold circuit paste sintering on silicon substratesand measured stresses of approximately 0.5 MPa at 7501C. Sim-ilarly Lin and Jean10 measured the stress induced when silver cir-cuit paste was sintered on a silicon substrate to be approximately1 MPa at 5501–7501C. In all these cases the substrate was siliconand deformed elastically in response to the induced stress and al-lowed the stress to be measured. The requirements for elastic de-formation and chemical inertness of the substrate have limitedthese experiments to relatively low temperatures when comparedwith processing temperatures used for many engineering ceramicssuch as yttria-stabilized zirconia (YSZ).

At these higher temperatures the substrates deform by creep andthe stresses can be deduced from specimen deformation assuminglinear viscosity as originally proposed by Bordia and Scherer.11 Forexample Chang and Jean12 used this approach to analyze thestresses induced on cofiring a bilayer of glass and a glass/ceramiccomposite between 5001 and 9001C by measuring the curvatureduring cofiring of horizontal bilayers. The uniaxial viscosities of thetwo layers were measured by thermo-mechanical analysis of bulkspecimens of the individual materials. The resulting mismatchstresses were deduced to be o30 kPa. However, no account wastaken of the effects of gravity in this horizontal configuration, whichwould have opposed the curvature caused by the mismatch stress.

As far as the authors are aware no studies have been reportedof stresses induced by constrained sintering of typical engineeringceramics on dense substrates at high temperatures by measuringcreep deformation of the bilayer. The objective of the currentwork is to apply this method to study the sintering-induced stressfor a typical fuel cell electrolyte film sintering on a dense substrate.

II. Experimental Procedure

The strategy adopted in the present study was to measure thestress in the sintering film by monitoring the resulting creepdeformation of the constraining substrate. Therefore it was firstnecessary to establish the creep properties (viscosity) of thesubstrate material, without the sintering film. Messing et al.13

described a method for measuring the viscosity of 8YSZ as afunction of relative density by observing the deformation of thinhorizontal simply supported beams deforming under gravity. Wehave used a similar method using in situ observation of an end-clamped horizontal cantilever of the dense substrate material,which creeps under gravity in this configuration and allows theviscosity of the substrate to be determined. For measurement ofthe stress induced by constrained sintering of the film, a bilayerbeam (the dense substrate with the sintering film applied on oneface) was clamped vertically to avoid the influence of gravity and

R. Bordia—contributing editor

**Fellow, The American Ceramic Society.wAuthor to whom correspondence should be addressed. e-mail: alan.atkinson@

imperial.ac.ukzPresent address: Department of Aeronautical and Automotive Engineering, Loughbor-

ough University, Loughborough LE11 3TU, U.K.

Manuscript No. 27920. Received April 30, 2010; approved September 5, 2010.

Journal

J. Am. Ceram. Soc., 94 [3] 717–724 (2011)

DOI: 10.1111/j.1551-2916.2010.04160.x

r 2010 The American Ceramic Society

717

mailto:[email protected]

mailto:[email protected]

the evolution of its curvature measured in situ. The stress was thenobtained from an analysis of the evolution of curvature of thebilayer and knowing the viscosity of the substrate obtained fromthe experiments with the horizontal substrate beams.

(1) Specimens

The system chosen for study was a 3YSZ (3 mol % Y2O3-dopedzirconia) film sintering on a dense 3YSZ substrate. This compo-sition was used because although it does not have the highestionic conductivity for an SOFC electrolyte in the YSZ system, ithas the best mechanical properties and is used as an electrolyte bysome SOFC developers.14 The film and substrate were chosen tobe the same composition to ensure that no stresses would begenerated by thermal expansion differences between the film andsubstrate. The substrates were commercially produced by KerafolGmbH (Eschenbach, Germany) and were received in the form of5 cm� 5 cm square plates with nominal thicknesses of 300 or 150mm. The material has a conductivity of 0.021 S/cm at 8501C andhas a low Si content to minimize grain-boundary resistance. Thesubstrate microstructure is illustrated in Fig. 1 and, from severalsuch areas, the mean grain size was measured to be 0.46 mm.

The 3YSZ powder used to prepare the screen-printing ink andother specimens was supplied by MEL Chemicals (Manchester,U.K.). Its particle size distribution was measured by light scat-tering and its parameters were: d105 0.38 mm; d505 0.50 mm; andd905 0.77 mm. The specific surface area was 6.9 m2/g.

YSZ layers were deposited on the substrates (in the as-received condition) by screen printing (165 mesh screen and2.5 mm gap) and the layers were oven dried at 1201C. At thisstage the films were in the thickness range 10–13 mm. Samples ofboth coated and uncoated substrates were then laser cut into 5-mm-wide strips for stress measurements. The film thicknesses inthe as-coated and dried condition and after the sintering exper-iment were measured using an interference microscope (ZYGO,Middlefield, CT). Similar 5-mm-wide strips without films de-posited were laser cut for the viscosity measurements.

(2) Substrate Viscosity

The substrate viscosity was measured by observing a horizontalcantilever beam of an uncoated substrate strip bend under itsown weight. The displacement of the tip of the beam relative to arigid reference (Fig. 2) was measured as a function of time dur-ing the heating cycle (3 K/min) followed by an isothermal holdof between 1 and 3 h at the test temperature (in the range 11001–13501C). At least two experiments using nominally identicalspecimens were carried out at each isothermal temperature.

The specimens were placed in a furnace, which had a windowthrough which they were observed using a long-distance micro-scope (Infinity K2, Boulder, CO) and camera. The ceramic fix-tures used to support the specimen were fabricated from 3YSZblocks with all mating surfaces ground flat and accurately at

right angles where critical. Digital images were recorded peri-odically and subsequently analyzed to quantify dimensionalchanges. A typical image is shown in Fig. 2. All images wereanalyzed automatically using the MATLAB code (The Math-Works) to identify best-fit straight lines to the specimen andreference block outlines. These are shown on the image togetherwith the dimension used to characterize the tip displacement,which is the difference in vertical coordinates between the pointat the intersection of the two faces of the reference block and thepoint at the end of the beam on its neutral axis (midline). Thetwo points are circled in Fig. 2. This maximizes the use of datapoints from the image and hence maximizes the resolution of thedisplacement measurement which was determined to be72 mm.

The results, in the region of constant temperature, were an-alyzed using a theory developed for similar experiments onporous ceramics13 and copper beams.15 This assumes a linearuniaxial viscous creep law of the form s ¼ Ep _e and thereforeanalogous solutions to small displacement elastic problems canbe used by replacing Young’s modulus by the viscosity, Ep. Thisform of creep law should apply to polycrystalline beams havinga constant grain size that is much smaller than the beam thick-ness, which was the case in this study.

The rate of vertical displacement of the horizontal beam, yh,as a function of distance, x, from its clamped end (beam oflength L and thickness hs) is given by15:

qyhqt¼ 6rg

Eph2s

L2x2

2� Lx3

3þ x4

12

� �(1)

where r is the density of the beam and g is the acceleration dueto gravity. Hence, the rate of deflection at the free tip of thebeam (x5L) is

qyhqt

� �x¼L¼ 3rgL4

2h2sEp(2)

The data for tip displacement as a function of time in theconstant temperature region thus allow the creep viscosity Ep tobe determined.Fig. 1. Substrate microstructure (secondary electron image of surface).

Fig. 2. (a) Schematic of the arrangement for measuring the substrateviscosity (the box indicates the region observed by the long distance mi-croscope and the arrow the measured beam tip displacement), and (b) anexample of an image. The broken lines show the fitted outlines of thebeam and the reference block and the center line of the beam. The circlesshow the points used for measuring the beam displacement.

718 Journal of the American Ceramic Society—Atkinson et al. Vol. 94, No. 3

(3) Sintering-Induced Stress

The deformation caused by the sintering-induced stress wasmeasured in a similar way, but with a sintering film applied tothe substrate and in vertical orientation so that there was littledeformation due to gravity (Fig. 3). The possible influence ofgravity as the bilayer beam bends away from the vertical is dis-cussed in the Appendix A. The distance between the tip of thespecimen and the reference block was measured as a function oftime as the sintering-induced stress caused the specimen to bendaway from the reference using the same heating cycle (afterbinder burn out at 5001C) as in the horizontal experiment andfor holding temperatures from 11001 to 13501C. A similar ap-proach to that described for analysis of images from the hori-zontal beam experiment was adopted for the vertical bilayerexperiments. After fitting straight lines to the beam and refer-ence outlines, the tip deflection was determined by measuringthe distance between the two circled points shown on the imagein Fig. 3. At least two experiments using nominally identicalspecimens were carried out at each isothermal temperature.

Hence, the beam curvature, k, was obtained from

k ¼ 2yv;LL2

(3)

where yv,L denotes the tip displacement in the vertical configu-ration. The sintering-induced stress was then calculated from the

curvature using the viscous analogue of Stoney’s formula forbeam bending (when the film is much thinner than the substrate)

s ¼ h2sEp

6hf

dkdt

� �(4)

where hf is the thickness of the film. In this expression the bend-ing rate is determined by the product of the stress and the filmthickness, which changes as the film shrinks. The film thicknessat the start of and end of the isothermal period was estimatedfrom a separate study of the constrained sintering kinetics ofthe films (J.-S. Kim, R. Rudkin, X. Wang and A. Atkinson,unpublished data) and the measured values of the initial andfinal film thicknesses.

III. Results and Discussion

(1) Substrate Viscosity

Figure 4 shows examples of the dense substrate deformation(beam tip displacement) under gravity in the horizontal config-uration during the isothermal hold period at 12501C and illus-trates two general points that were common in all theexperiments. The first is a gradual slowing of the deformationrate and the second is that there is considerable difference be-tween nominally identical specimens.

The slowing of the deformation rate implies an increase inviscosity with either time or strain, whereas the theory leading toEqs. (1) and (2) assumes a viscosity that is independent of timeand strain. Chokshi16 has summarized creep behavior in YSZpolycrystalline ceramics. Most experimental data were obtainedat higher temperatures than the current experiments, but inmany cases it has been reported that in the relationship _e / sn, nis often 41; being typically approximately 2 for low stress andapproaching 1 only for coarse-grained (B1 mm) material at highstress (30–200 MPa). In order to test the validity of the linearviscous assumption in the present study, we have compared thefinal profile of the beam at room temperature after the creepexperiment with that predicted for different values of stress ex-ponent n. The details of the analysis are given in the Appendix Aand the expression for the beam profile (Eq. (A-10)) agrees withEq. (1) for n5 1. A typical example of the measured profilecompared with the theoretical profile is shown in Fig. 5. In thefitting procedure the location of x5 0 was allowed to varybecause the exact location of the end of the clamped part ofthe beam could not be reliably identified by inspection after dis-mantling the experiment. The goodness of fit (from the sum ofresidual squares) to the measured profile for n5 1 is 0.9996,whereas for n5 2 it is only 0.995 (i.e., the fit for n5 1 is ap-proximately 10 times better than for n5 2). The better fit forn5 1 is also evident by inspection of Fig. 5 where it can be seenthat there is a systematic error in the fit for n5 2. This is because

Fig. 3. (a) Schematic of the arrangement for measuring sintering-in-duced stress (the box indicates the region observed by the long distancemicroscope and the arrow the measured beam tip displacement), and (b)an example of an image. The broken lines show the fitted outlines of thebeam and the reference block and the center line of the beam. The circlesshow the points used for measuring the beam displacement.

Fig. 4. Example of tip deflection as a function of time for two hori-zontal beam specimens (150 mm thickness and 40 mm length) during theisothermal hold period at 12501C.

March 2011 Stress Induced by Constrained Sintering of 3YSZ Films 719

the bending moment increases with distance from the free endof the beam and therefore the greater the value of n, the greaterthe beam curvature near the clamped end. The fit with n5 2 inFig. 5b clearly does not reproduce the curvature profile of thebeam, whereas the one with n5 1 in Fig. 5a does. Although thedifference in the profile between n5 1 and 2 is clear, thismethod is not sensitive enough to closely determine n. Thegoodness of fit was calculated as a function of n and found to be� 0.9995 for n in the range 0.6–1.1. It can therefore be con-cluded that although the value of n determined from the profileis not precise, linear creep (n5 1) gives a satisfactory descrip-tion of the creep behavior and there is no evidence for a sig-nificantly higher stress exponent in the substrate material. Thisappears to contradict the behavior reported in previous studiesof creep in 3YSZ. However, the previous studies concentratedon superplastic deformation with finer grained material athigher temperatures than in the present experiments. The su-perplastic experiments involve considerable grain growth dur-ing the creep deformation and Chokshi16 has shown that whengrain growth is taken into account the superplastic creep isconsistent with the Coble creep mechanism (grain-boundarydiffusion) for which n5 1. In the present study all the experi-ments were conducted at temperatures less than the substratesintering temperature and there was negligible grain growthduring the creep experiments. Therefore there is no incompat-ibility between the present results and those reported earlier.

However, the slowing of the creep rate (increasing viscosity)during the isothermal hold cannot be explained by grain growthbecause it was evident even at the lowest temperatures. In Coblecreep the viscosity is proportional to the cube of the grain size,but even with such a sensitive dependence this cannot accountfor the observed increase in viscosity because any change in grainsize is too small to be detected. The cause of the observed in-

crease in viscosity is thus unclear, but could be related to achange in grain-boundary diffusivity resulting from the creepdeformation (strain hardening). Consequently, the viscosity wascalculated as the average over the isothermal hold period and theresults are plotted in Arrhenius form in Fig. 6. The experimentsshowed some variation in creep viscosity from specimen to spec-imen and the error bars in Fig. 6 represent the percentage stan-dard deviation averaged over the whole set of specimens. TheArrhenius plot shows a distinct curvature with lower activationenergy at lower temperature (approximately 250 kJ/mol) andhigher activation energy at higher temperatures (approximately500 kJ/mol). This might be due to a change in creep mechanismfrom Coble creep (grain-boundary diffusion) at lower tempera-tures to Nabarro–Herring creep (lattice diffusion) at higher tem-peratures. The strains in the present experiments are of the orderof 10�4 and the strain rates of the order of 10�8 s�1. These areboth much lower than are typical for creep studies of this ma-terial which concentrate on its superplastic behavior and largestrains. Chokshi16 analyzed experimental creep data for 10001and 14001C and concluded that stress exponents greater thanunity can be explained by grain growth occurring during thecreep experiments. The results of that analysis have been usedto estimate the viscosity for tetragonal YSZ with a grain sizeequal to that of the substrates used in the present study.The resulting viscosities are shown in Fig. 6 compared withthe present results. Also shown are viscosities estimated fromcreep experiments assuming linear viscous creep at 1450117 and13501C.18 Despite the uncertainties, the present results are rea-sonably consistent with the existing literature in their approxi-mate order of magnitude and general trend. However, there arestill some significant differences. For example, Zapata-Solvas etal.18 claim a temperature-dependent threshold stress belowwhich there is no creep deformation, whereas in the presentstudy we observe creep at very low stresses. This difference couldbe due to differences in grain-boundary chemistry in the spec-imens used in the different studies. Therefore there are still sig-nificant unresolved fundamental issues concerning creep of YSZ.

(2) Sintering-Induced Stress

Figure 7 shows an example of tip displacement as a function oftime at the isothermal hold temperature for a vertical experimentwith a coated substrate beam. The bending profile was measuredafter each experiment and compared with the theoretical profilewhich is an arc of a circle. An example is shown in Fig. 8 fromwhich it can be seen that the profile is an excellent fit to the arc of

Fig. 5. Profile of 300-mm-thick beam after horizontal creep for 1.5 h at13501C; (a) fitted to theory for a stress exponent n5 1; and (b) fitted totheory for a stress exponent n52.

Fig. 6. Arrhenius plot of the viscosity of the 3YSZ substrates. Litera-ture data are from references.16–18


a circle as expected theoretically. (It should be noted that themaximum strain in the film caused by the bending is khs/2 and inthe experiments is typically 10�4. Because this is much smallerthan the unconstrained sintering strain, the bending has no effecton the sintering-induced stress in the constrained film.)

Therefore the specimen curvature can be calculated from thetip deflection in the vertical experiment, yn,L, using

k ¼ 2yv;LL2

(5)

It can be seen from Fig. 7 that the vertical experiments exhibittwo features in common with the horizontal ones. The first is avariability in the results of nominally identical experiments,which originates mainly from the variability in creep propertiesof the substrates. The second is that the rate of tip displacementslows down with time. This might be because the substrate vis-cosity is increasing with time (or strain) or because the sintering-induced stress is decreasing with time. Because the reason for theincreasing substrate viscosity is not currently understood, it isnot possible to deduce how the sintering-induced stress is chang-ing with time. Therefore we have analyzed the results by takingthe average deformation rate over the isothermal hold period.Similarly, we have used the average film thickness during theisothermal period when calculating the film stress from Eq. (4).The film density estimated from densification experiments (J.-S.Kim, R. Rudkin, X. Wang and A. Atkinson, unpublished data)at the start and end of the isothermal period of the correspond-ing vertical beam experiments is shown in Fig. 9 and the corre-sponding sintering-induced stresses averaged over the sameisothermal period in Fig. 10. The error bars are the percentagestandard deviation averaged over the whole set of experimentsand reflect the specimen-to-specimen variability in the results.The sintering-induced stress is of the order 1–3 MPa and peaksin the middle of the temperature range studied. However, this isnot necessarily an effect only of temperature because the averagedegree of densification during the isothermal period also in-creases with the holding temperature (Fig. 9).

It was noted earlier that the stress exponent in the creep equa-tion, n, could not be determined accurately from the profile of thebeam after the horizontal experiment and therefore the questionarises regarding the sensitivity of the deduced film stress to theassumed value of n. Equation (A-12) in the Appendix A allowsthis sensitivity to be examined. Because both the horizontal andvertical experiments were conducted with similar beams and withsimilar tip displacement rates in both orientations the factor in thebraces in Eq. (A-12) expresses the sensitivity to n. This factor is0.444 for n50.5; 0.5 for n51; and 0.577 for n52. Thus the stressis not sensitive to the assumed value of n. Even for these extremevalues the possible error is only approximately 715%, which issmaller than the observed specimen to specimen variability.

Using the linear viscous deformation model, the sintering-induced stress for a constrained film is given by19:

s ¼ Ep; f _ef1� np; f

(6)

where _ef is the unconstrained sintering rate and Ep,f and np,f arethe viscosity and Poisson ratio of the film. (It should be notedthat this equation is based on an assumption of isotropic filmproperties and it is known that constrained sintering gives rise toanisotropy in the film microstructure20 and hence to film prop-erties. Therefore Eq. (6) is applied with caution.) Both Ep,f and_ef are controlled by similar diffusion process (but in oppositesenses); namely by the slowest diffusing cation. Therefore theproduct is not particularly sensitive to either temperature ortime, but will change as the microstructure develops and/orchanges in temperature change the relative importance of differ-ent diffusion pathways. The relative activation energies for theavailable diffusion pathways are expected to be in the ordersurface diffusionograin-boundary diffusionolattice diffusion.At low temperatures surface diffusion is favoured which cancontribute to creep deformation (e.g., by transporting materialaround the necks that form between particles in the early stagesof sintering). Densification, on the other hand requires grain-boundary diffusion which is relatively more difficult at lowertemperatures. Thus the product Ep,f _ef tends to be low at lowtemperature and low relative density. Conversely at higher tem-peratures grain-boundary diffusion becomes faster relative to

Fig. 7. Example of tip deflection as a function of time for two vertical-coated beam specimens (150 mm thickness and 40 mm length) during theisothermal hold period at 12501C.

Fig. 8. Profile of 300-mm-thick beam coated with sintering film after thevertical experiment at 13501C fitted to the arc of a circle.

Fig. 9. The estimated relative density of the sintering film at the startand at the end of the isothermal hold at each temperature studied.


surface diffusion and, as the relative density increases, surfacediffusion becomes ineffective for creep deformation. Thus theproduct Ep,f _ef increases at first with increasing temperature.However, as the temperature is increased further the uncon-strained densification rate slows down relative to the creep ratebecause the distance between pores, which characterizes thescale of diffusion required for densification, becomes greaterthan the grain size, which is the characteristic scale of diffusionfor creep deformation. Hence, the product Ep,f _ef becomessmaller and, as the film approaches full density, the stress tendstoward zero even though the viscosity is high. The maximumstress therefore occurs around the mid-range of densificationand temperature as seen in Figs. 9 and 10. Typical microstruc-tures of the sintered films are shown in Fig. 11. The micro-graphs show that there is very little grain growth during dens-ification. The pores visible from the top of the film tend to beelongated and have a size of the order of 1 mm. Some localizeddefects were observed in the sintered films21 and these arethe subject of a more detailed investigation, but no crackswere observed.

The stresses measured in this study are significantly largerthan those measured in sintering of other constrained filmsmentioned in the introduction. However, those systems werenot crystalline ceramics sintering by solid-state diffusion. Zuoet al.22 have measured the uniaxial sintering stress using a hotforging technique for isostatically pressed bulk alumina speci-mens made from a powder with 150 nm grain size. (The uniaxialsintering stress is given by ss ¼ Ep _ef which differs from thebiaxial stress in Eq. (6) only in the term involving the viscousPoisson’s ratio.) They found the stress was of the order of a fewMPa and peaked at approximately 3.5 MPa at a relative densityof approximately 80%. Similarly, Ikegami et al.23 measured theuniaxial sintering stress in alumina rods using a tensile loadingmethod and found a maximum of approximately 3.6 MPa at12001C. Thus the present results are in qualitative agreementwith the studies of bulk alumina reported in the literature.

The constrained sintering-induced stress is also closely relatedto the thermodynamic sintering potential which is a function ofthe surface energy and the pore structure.24,25 From the princi-ple of virtual work the sintering potential (equivalent hydro-static stress on the porous unconstrained body) is given by

S ¼ gSVdS

dV(7)

where gSV is the surface energy, S is the internal surface area, andV is the external (bulk) volume and any contribution from thegrain boundaries is neglected. The relationship between S and V

is dependent on the details of the pore structure and how itevolves, but for a uniform population of spherical pores of radiusr, Eq. (7) gives S52gsn/r. Unfortunately, as far as the authors areaware there are no reliable experimental data for the surface en-ergy of zirconia. Redfern et al.26 calculated the surface energies ofvarious t-ZrO2 surfaces using atomistic simulation correspondingto 0 K and found typical values of 2 J/m2. (The surface energy athigher temperature is likely to be lower due to the greater entropyof surface atoms than bulk atoms.) Assuming this value for thesurface energy and a pore size of the order of 1 mm (r50.5 mm),then the sintering potential is estimated to be approximately 8MPa. This is reasonably consistent with the constrained sintering-induced stress measured here considering the approximationsinvolved in calculating the sintering potential.

IV. Conclusions

The creep rate of thin beams of 3YSZ is linearly dependent onstress at the low stresses and strains involved in the present work.The creep viscosity appeared to increase with strain (time) whichmight be due to changes in grain-boundary composition. The vis-cosity has a higher activation energy at temperatures above ap-proximately 12501C which could be due to a change from Cobleto Nabarro–Herring creep. The magnitudes of the creep viscositiesare in reasonable agreement with other data in the literature.

The in-plane stress induced during constrained sintering of a3YSZ film has a maximum value of approximately 3 MPaat 12001C. This behavior is consistent with literature resultsreported for constrained sintering of bulk alumina. The stressinduced by the constraint is of a similar order to the estimatedsintering potential.

Fig. 10. Sintering-induced stress in 3YSZ films averaged over theisothermal hold period at each temperature.

Fig. 11. Secondary electron micrographs of the top surfaces of con-strained 3YSZ films after sintering for 1 h at (a) 12001C; and (b) 13001C.


Appendix A

(A.1) Analysis of Beam Profile

Let the creep behavior be described by the relationship

s ¼ A�1e� 1=n

(A-1)

For the beam bent as a result of creep, the strain can be ex-pressed as e5 z/R(x), where R(x) is the local radius of curvatureat horizontal coordinate x (measured from the clamped end)and z5 0 is the neutral axis.

The moment on the rectangular cross section of the cantileverbeam is

M ¼Z h=2

�h=2swzdz (A-2)

where w is the width and h is the thickness of the beam.Hence,

M ¼ whð2þ1=nÞ

2ð1þ1=nÞð2þ 1=nÞAR�

ðxÞ1=n(A-3)

or

1

R� ¼

MnAn

In(A-4)

where

I ¼ whð2þ1=nÞ

2ð1þ1=nÞð2þ 1=nÞ (A-5)

For the cantilever beam the moment arising from the bodyweight is a function of x

MðxÞ ¼ ðL� xÞ2

2whgr (A-6)

with g the gravitational acceleration and r the density of thebeam material.

When the deflection of the beam from the initial horizontal,y(x), is small, the local curvature of the beam can be approxi-mated by:

kðxÞ ¼ 1

RðxÞ ¼d2y

dx2(A-7)

Therefore the beam profile can be expressed as:

yðxÞ ¼Z t

0

Z L

0

Z L

0

1

R� dxdxdt (A-8)

with the boundary conditions y5 0 at x5 0; dydx ¼ 0 at x5 0.

If the creep rate is assumed to be independent of the time thesolution is

yðxÞ ¼ tAn2ð2þ 1=nÞngnrnhnþ1

ðL� xÞ2nþ2

ð2nþ 1Þð2nþ 2Þ þL2nþ1x

2nþ 1� L2nþ2

ð2nþ 1Þð2nþ 2Þ

" #

(A-9)

(A.2) Sensitivity of Film Stress to Value on Creep Exponent

Using the creep relation in Eq. (A-1) we can calculate the influ-ence of the creep exponent, n, on the determination of the film

stress sf. Equation (A-9) gives the displacement rate of the tip ofthe beam in the horizontal experiment, _yH, which is used to givethe creep constant,A. In the vertical experiment the moment dueto the film stress is balanced by the moment caused by beambending. If the film thickness is small in comparison with thebeam thickness then the neutral axis of the beam is at its centerline. This is equivalent to the assumption leading to Stoney’sequation. The bending moment caused by the film stress is

M1 ¼ sfhfh

2(A-10)

And the opposing moment from beam bending

M2 ¼2 _k

1n

A

1

2þ 1n

h

2

� �2þ1n

(A-11)

Substituting k5 2yV/L2 (where yV is the tip displacement in

the vertical experiment), equating the moments and also substi-tuting for A from Eq. (A-9) (with x5L) finally gives the filmstress

sfhf ¼_yV_yH

� �1n

rgL2 2

2nþ 1� 2

ð2nþ 1Þð2nþ 2Þ

� �1n

(A-12)

(A.3) Influence of Gravity on Vertical Experiment

Considering a cross section of the beam at height x from theclamping point, it must support the body weight of a section ofbeam (L–x). The body weight provides a bending moment:

MG ¼rgbhsðL� xÞ2 sin y

2(A-13)

where y is the angle between the vertical and the local tangent tothe beam. Sin y is related to the curvature, ie., siny5 (3L1x)/4R, so the above equation can be rewritten as:

MG ¼rgbhsðL� xÞ2ð3Lþ xÞk

8(A-14)

where k is the curvature.Therefore for the vertical experiments, the total bending mo-

ment is provided not only by the sintering-induced stress, butalso by gravity to give the local rate of change of curvature:

_kðx; tÞ ¼ 3rgðL� xÞ2ð3Lþ xÞk2h2sEP

þ 6hfss

h2sEP(A-15)

Numerical simulation based on the above equation was car-ried out to assess the influence of gravity. Using parameterstaken from the experimental study, the gravitational contribu-tion to the deflection of the end of the beam was calculated for asintering-induced stress of 2 MPa. The contribution increaseswith time as the beam bends further away from the vertical.However, over a period of 2 h the contribution is only 3% and isnegligible on this time-scale.

An error due to gravity also arises if the beam is initially notperfectly vertical. If the misorientation angle is d then Eq. (A-15)is modified to give:

_kðx; tÞ ¼ 3rgðL� xÞ2½ð3Lþ xÞkþ 4d�2h2sEP

þ 6hfss

h2sEP(A-16)

The effect of a misalignment of 11 (which is easily detected) isshown in Fig. A1 for the same parameters used in the previouscalculation. The simulation shows that the tip deflection is notsensitive to misorientation of this order.


Acknowledgments

The authors are grateful to colleagues at Rolls-Royce Fuel Cell Systems forhelpful discussions and assistance with screen printing. This research was carriedout as part of the U.K. Supergen consortium project on ‘‘Fuel Cells: Powering aGreener Future.’’ The Energy Programme is an RCUK cross-council initiative ledby EPSRC and contributed to by ESRC, NERC, BBSRC, and STFC.

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Fig. A1. Calculated effect of gravity on the beam tip deflection (150 mmthickness, 13001C, film stress 2 MPa) for an initial misorientation angleof 711 from the vertical in a nominally vertical experiment.


stress induced by constrained sintering of 3ysz films measured by substrate creep

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