This work reports the elastic modulus and four-point flexural strength of a gelcast ceramic, cerium dioxide (ceria), with a microporosity of nominally 20% and a grain size of 11 μm from 23 to 1500 °C. The data augment the sparse data published for ceria and extend previous results by 150 °C. The ceria tested is representative of that constituting the ligaments of a reticulated porous ceramic. The elastic modulus decreases from 90 GPa at 23 °C to 16 GPa at 1500 °C. The flexural strength is 78 MPa below 900 °C and then decreases rapidly to 5 MPa at 1500 °C. These trends are consistent with data reported for other ceramics. Comparing the measured elastic modulus to prior data obtained for lower porosity shows the minimum solid area (MSA) model can be used to extend the modulus data to other porosities. Similarly, the flexural strength data agree with prior data when the effects of specimen size, porosity, and grain size are taken into account.

## Introduction

Cerium dioxide (ceria) is used in a variety of applications, including engine exhaust pollution control [1], wastewater treatment [2], thin film electrolytes within solid oxide fuel cells [3], and polishing materials [4]. Recently, ceria has been utilized in solar thermochemical reduction/oxidation (redox) cycles (for example, Refs. [57]). These cycles convert solar energy into chemical energy in the form of storable and transportable fuels. Ceria works well as a redox material because oxygen diffuses rapidly in its cubic fluorite crystal structure [8,9], it does not undergo structural phase transitions between its fully oxidized and partially reduced forms, and it maintains stable rates of oxidation over hundreds of cycles at temperatures as high as 1500 °C [1013]. The reduction step is endothermic and requires high temperatures (>1400 °C) and low-oxygen partial pressures. Thermodynamics shows the uptake of oxygen is improved in the exothermic oxidation step by performing it at temperatures 200 to 400 °C cooler [14]. Therefore, in many reactors, the ceria substrate must withstand substantial thermal stresses attributable to the temperature difference between reduction and oxidation.

Reticulated porous ceramic foams (RPCs) are good candidates for the ceria substrate in redox cycles because they combine structural integrity with reasonably high-specific surface area and accessibility required for rapid mass transport and fuel production [12,15,16]. Several alternatives to RPCs have been considered, with some morphologies exhibiting better stability than others. Ceria monolith structures thermochemically cycled between 1500 °C and 800 °C experienced a threefold increase in grain size [9]. Higher surface area materials such as a three-dimensionally ordered macroporous ceria lost a large fraction of the surface area and pore volume at temperatures as low as 825 °C [17], and the ordered structure was lost completely after 1 h at 1250 °C [16,18]. While electrospun ceria fibers cycled between 800 °C and 1500 °C maintained an open fibrous network up to 1400 °C [19], commercial ceria felts made with fibers exhibited significant sintering and shrinking in a prototype solar reactor [11]. Ceria particles made from fibers showed stable fuel production when isothermally cycled at 1500 °C [20,21]. RPCs made from other materials are also used in a variety of other applications, for example, filtration [22], radiant burners [23], and bone tissue engineering [24].

The motivation for the work described here is the desire to design a free-standing cylinder of ceria RPC for use in a solar reactor where the cylinder is subjected to a temperature differential of up to 500 °C [25]. Data on the structural properties of ceria (reviewed below) are sparse. The purpose of the present work is to fill a gap in the available data to enable analysis-driven design of ceria-based structures, especially those involving thermal stresses. While these data are essential for predicting the strength of ceria RPCs, the data are also necessary for modeling the strength of other ceria morphologies. The methods used to ensure that the specimens tested are representative of the ligaments of RPC are extensible to other ceramics commonly fabricated as RPCs, such as alumina.

Wachtel and Lubomirsky [26] summarize the state-of-the-art knowledge of the elastic modulus of pure and doped ceria. They comment on the difficulty of measuring the modulus at high temperatures and the limited availability of high-temperature data. The most complete data on elastic modulus as a function of temperature are reported by Wygant [27], where the data extend to 1300 °C. Unfortunately, the results are of limited usage, as the exact composition of the material, reported to be approximately 80% ceria, is not known. The elastic modulus at room temperature and 600 °C is reported by Sato et al. [28,29]. Additional data at room temperature and different porosities are reported by Lipińska-Chwałek et al. [30] and Wang et al. [31]. These previous results are compared to the results reported here in Sec. 6. Nakajo et al. [32] provide limited data on the elastic modulus of gadolinia-doped ceria, but doped ceria cannot be directly compared to the undoped ceria tested in the present study.

The availability of flexural strength data is more limited still. Akopov and Poluboyarinov [33] present flexural strength data for ceria up to 1350 °C, but the details of the experimental technique are not provided. The new data generated here are compared to available room temperature data of previous authors in Sec. 6.

Numerous studies have been performed to characterize the effect of nonstoichiometry on the mechanical properties of ceria-based oxides. Kossoy et al. [34] explore the effect of nonstoichiometry on the elastic modulus of thin film ceria, showing a decrease in modulus with an increase in chemical strain. Wang et al. [35] evaluate the effect of reduction on the room temperature flexural strength of ceria, observing a significant drop in strength with increasing nonstoichiometry. It should be noted that Wang et al. attribute some of the reduction in strength to internal stresses generated by having a highly reduced specimen surface with an unreduced interior below. Quantification of the impact of the gradient on the strength was not performed. In related work, Kaiser et al. [36] evaluated a doped ceria membrane for mechanical stability when subjected to a nonstoichiometry gradient across the thickness. Failure of the membrane occurred when relative chemical expansion exceeded a strain of 0.001. Various studies have investigated the effect of reduction on the chemical expansion of undoped and doped ceria [3739]. The present study is restricted to CeO2 and does not consider the impact of nonstoichiometry on mechanical properties.

This work reports the elastic modulus and four-point flexural strength of a gelcast ceria, with a microporosity of nominally 20% and a grain size of 11 μm from 23 to 1500 °C. The procedure used to fabricate test specimens with morphology similar to the material constituting ligaments of RPC is presented in Sec. 2. The experimental methods are described in Sec. 3. The results for the elastic modulus and flexural strength up to 1500 °C are presented in Sec. 4. Extending these results to RPC is discussed in Sec. 5. The results are compared to previous work in Sec. 6. Conclusions are drawn in Sec. 7.

## Test Specimen Fabrication

Flexural test specimens were fabricated which conform to size “B” specimens of ASTM Standard C1211 [40]. They were fabricated by gelcasting, a near-net-shape manufacturing process for ceramics which can yield strong, dense, and defect-free parts [41,42]. Gelcasting was chosen over alternative forming processes because the processing steps are the most similar to those used for RPCs. In particular, both methods utilize pressureless sintering. In contrast, alternatives such as extruding and pressing introduce high pressures that are likely to yield different porosities, pore shapes, and grain sizes. The pore shape and grain size of the specimens are further explored in Sec. 5.

The gelcast process used ceria powder (Alfa Aesar, 99.9% CeO2) with a D50 particle diameter of ∼5 μm measured using the SediGraph technique [43]. To reduce warping, the bars were fired in a tunnel kiln at a temperature of 1580 °C for 6 h, turned upside down, then fired for another 6 h at 1580 °C. The final firing temperature was set 80 °C above the highest test temperature to minimize any changes to the microstructure attributable to sintering. The nominal dimensions of the specimens, as fired, were 3.5 mm × 4.5 mm × 45 mm. The bars were then ground to their finished 3 mm × 4 mm cross-sectional dimensions with a parallelism tolerance of 0.015 mm on the longitudinal faces, as specified in the ASTM standard [40].

The porosity, P, of the test specimens was calculated as
$P=1−m/(hwL)ρ$
(1)

where m is the mass of the specimen, ρ is the density of ceria (7.65 g/cm3 [44]), and h, w, and L are the height, width, and length, respectively, of the specimens. The mass of each specimen was measured using a mass balance with a resolution of 0.001 g (Sartorius, GD-503-NTEP). The height, width, and length of each specimen were measured using calipers having a readout resolution of 0.01 mm (General, 147). The average porosity of the test specimens is 0.21 ± 0.02.

## Experimental Methods

All testing was done in accordance with ASTM Standard C1211 [40]. The test specimens described in Sec. 2 were loaded in a fully articulated, silicon carbide, four-point, 1/4-point, size B flexure fixture [40]. The fixture has a load span of 20 mm and a support span of 40 mm, with load point diameters of 4.8 mm.

A standard universal tester (Instron, Model 55R1123) was used to load the specimens with a constant displacement rate of 0.51 mm/min (0.020 in./min). A high-crosshead displacement rate was used to reduce any effects attributable to creep. The time, the crosshead displacement, and the load were recorded at a rate of 20 Hz.

The elastic modulus and the flexural strength are deduced from the recorded test data using beam bending equations, rearranged to solve for the value of interest [45]. The elastic modulus, E, is
$E=(Lo−Li)2(Lo+2Li)4wh3×ΔFΔδcrosshead$
(2)
where Lo is the support span of the test fixture, and Li is the load span. Only the linear portion of the applied force, F, versus the crosshead displacement, δcrosshead, curve is used. The flexural strength, σfs, is
$σfs=3Fmax(Lo−Li)2wh2$
(3)

where Fmax is the maximum total load applied to the specimen.

Multiple specimens were tested at several different temperatures between room temperature (nominally 23 °C) and 1500 °C. The number of specimens tested at each temperature is listed in Table 1. As the specimens exhibited a variation in porosity of ±0.02, they were distributed across the range of testing temperatures to equalize the porosity of the specimens in every range to the extent possible.

For tests above room temperature, the specimen and the fixture were heated at a rate of 10 °C/min in a clamshell furnace. Temperatures were measured using a Type R thermocouple positioned about 3 mm from the specimen. Once the temperature set point was reached, a 15 mins soak was employed. All tests were performed at ambient atmospheric conditions. Based on the data from Panlener et al. [14], the equilibrium nonstoichiometry of ceria in air at 1500 °C would be less than 0.001, and thus the chemical strain would be less than 0.0001 based on the chemical expansion versus nonstoichiometry relationship developed by Bishop et al. [37]. This chemical strain is equivalent to the thermal expansion of ceria with a rise in temperature of approximately 10 °C [46]. As such, the chemical strains of the ceria in this study are negligible compared to the thermal strains.

## Results

The measured elastic moduli of the gelcast ceria test specimens for temperatures from 23 to 1500 °C are plotted in Fig. 1. At room temperature, the elastic modulus of the gelcast ceria has a mean of 90 GPa with a standard deviation of ±4 GPa. The modulus decreases with increasing temperature to 16 ± 2 GPa at 1500 °C, with the reduction in modulus caused by a decrease in interatomic bonding force between ceria atoms. The measured moduli exhibit larger scatter in the data at intermediate temperatures, with a standard deviation of ±10 GPa at 800 and 1000 °C, and ±7 GPa at 500 °C.

A curve having the following form was fit to the experimental data for elastic modulus using a least squares regression:
$E=−1.92×10−5T2−0.0191T+89.0 (GPa)$
(4)

where T is the temperature in  °C. The large scatter of the data in the 500–1000 °C range leads to a coefficient of determination for the curve fit, R2, of 0.948. The result is shown as a solid curve in Fig. 1.

Some nonlinear effects were observed in the 1400 and 1500 °C data sets just prior to failure, where the failure mode changed from brittle to plastic. Creep does not impact the data. Based on ceria creep characterizations performed by Lipińska-Chwałek et al. [47], the creep rate is less than 1% strain per year when calculated at conditions of 1500 °C and a stress of 5 MPa. An example of the nonlinear effect is shown in Fig. 2. The slope of the linear portion of the curve was used to determine the elastic modulus, as indicated in the figure.

The measured flexural strength of the gelcast ceria from 23 to 1500 °C is shown in Fig. 3. The mean flexural strength is 78 ± 6 MPa at 23 °C. The strength is relatively stable until 900 °C. Thereafter, it decreases dramatically with increasing temperature. The flexural strength is 5 ± 4 MPa at 1500 °C. Like the elastic modulus, the measured flexural strength has more scatter at the intermediate temperatures. The standard deviation is ±14 MPa at 500 and 1000 °C, and ±19 MPa at 800 °C. At low temperatures, the feature limiting the strength of the specimen is expected to be the size of the critical flaw, namely, an existing pore in the specimen. At elevated temperatures, failure is attributed to subcritical cracks propagating into a critical flaw [48]. This change in failure mechanism dictates the shape of the strength curve.

A curve having the following form was fit to the data for flexural strength using a least squares regression:
$σfs=80.1−82.61+4.33×104e−0.00845T(MPa)$
(5)

The form of the curve is that used by Munro for the flexural strength of alumina [49]. The coefficient of determination for the curve fit is 0.808. The resulting curve is shown superimposed on the experimental data in Fig. 3.

## Extension of the Results to RPCs

The results reported in Sec. 4 are general, i.e., they are not limited to RPCs. Nevertheless, the immediate motivation for obtaining these results was to predict the mechanical properties of RPCs. Extending these results to RPCs is discussed here.

The mechanical properties of RPCs relevant to this study are described in Sec. 5.1. The methods applied to ensure that the flexural test specimens are representative of the material constituting the ligaments of a ceria RPC are described in Sec. 5.2. These methods are extensible to RPCs fabricated from other materials.

### Porosity, Elastic Modulus, and Flexural Strength of RPCs.

The image in Fig. 4 is of an RPC fabricated using the replication method [50,51], where a polyurethane foam is immersed in a ceramic slurry, purged of excess slurry, and fired at high temperatures to volatilize the foam and sinter the ceramic. The result of the process is an open-cell structure made of a series of interconnected ceramic ligaments, as illustrated in Fig. 4. Depending on the slurry constituents, firing temperature, heating rate, and processing steps, varying degrees of porosity can be obtained both at the macroscopic and microscopic levels [5153]. The dual-scale porosity of the RPC is clarified in Fig. 5.

RPCs are generally described in terms of their macroscopic porosity, PRPC, and linear macroscale pore density, λP, generally given as pores per inch (ppi) (e.g., Ref. [55])
$PRPC=1−ρ*ρs$
(6)

$λP=NLP$
(7)
where ρ* is the effective density of the RPC, ρs is the density of the material from which the ligaments are constructed, and N is the number of macroscopic pores within the RPC that are intersected by a straight line of a known length, LP. The macroscale pore density of the RPC illustrated in Fig. 4 is determined by the 30 ppi polyurethane foam used to fabricate it.
Analytical and semi-empirical models have been developed that relate the macroscale properties of foam materials to the microscale properties of the constituent materials [5660]. Gibson and Ashby [56] developed a power law relationship between the modulus and the density based on a staggered cubic structure of ligaments
$E*Es=C(ρ*ρs)n$
(8)

where E* is the effective elastic modulus of the foam, Es is the elastic modulus of the material constituting the ligaments, and C and n are empirically fit constants. Hagiwara and Green [61] found that this model provided a good fit for the elastic modulus of alumina RPC using constants of C = 0.3 and n = 1.93. The elastic modulus reported in Sec. 4 constitutes Es. The relative flexural strength of ceramic foams can also be predicted by substituting the strengths for the moduli in Eq. (8) [62,63]. Therefore, Eq. (8) can be used to extend the results reported here to ceria RPC.

### Comparison of Test Specimens With RPC Ligaments.

Several attributes of the morphology of the test specimens were analyzed to demonstrate that they are representative of those in ceria RPC ligaments. These attributes include area porosity fraction, pore density, pore area on a cross section, pore roundness, and grain size. These attributes were also measured in a test specimen taken to 1500 °C then cooled to room temperature, to confirm that exposure to the maximum temperature considered in this study did not change the morphology of the specimen.

The material morphology was quantified by photographing the microstructure of specimens, then processing the resulting images. One specimen was selected for each sample type: RPC ligament, room temperature test specimen, and a test specimen that had been taken to 1500 °C. Each specimen was potted in a cylindrical mold of epoxy (Buehler, EpoThin2). As the porosity of the test specimens varied slightly between specimens, those that were potted were selected to have porosity as close as possible to the average value. The potted specimens were ground and polished on one face. A series of images of the polished surface of each specimen was obtained using a visual microscope (Nikon, Optiphot) and digital camera (Canon, SL1), thereby giving a visual representation of one cross section through each specimen. One such image is shown in Fig. 6(a), where the light areas of the image are ceria and the dark areas the pore space. Thresholding was performed on the cross-sectional specimen images using an image processing program (NIH, ImageJ) to highlight the pores within the specimens. A sample result of this process is illustrated in Fig. 6(b).

The area porosity fraction was determined by dividing the total highlighted pore area on a designated area by the enclosing area. If one assumes that the pores are uniformly distributed in the material, then the area porosity fraction is the same as the porosity. Based on an analysis of 31 images of the test specimen tested at 23 °C, the average porosity is 0.20, with a standard deviation of 0.01. This value is in agreement with the measured porosity of 0.21 ± 0.02 (see Sec. 2). The area porosity fraction of the gelcast sample is close to the target value measured for the RPC (based on eight images): 0.23 ± 0.02.

The pore density was estimated by counting the number of pores appearing in the imaged areas. The pore density of the gelcast material (8100 pores/mm2) is about 40% higher than the RPC (5800 pores/mm2). However, as noted below, the pores in the gelcast specimen are smaller than those in the RPC, yielding similar overall porosity.

Pore area on a cross section examines the area of each pore on the polished face of each specimen. The pore area is only an approximate measure, as the pores are actually three-dimensional ellipsoids having irregular shapes. The area produced by each pore on an intersecting plane will depend on the position and orientation of the plane relative to each pore. Nevertheless, the overall pore area distributions are expected to follow the same general trends for similar materials.

A histogram of the pore areas on the imaged surface of each specimen type is shown in Fig. 7. The area of the room temperature gelcast specimen inspected included 12,989 pores, while the areas of the 1500 °C gelcast specimen and the RPC specimen included 10,318 and 1999 pores, respectively. The area distributions for the two gelcast specimens are very similar. The pore areas for the RPC specimen are somewhat larger but cover an overall range similar to the gelcast specimens. The average pore area for the RPC specimen is 40 μm2, about 1.6 times the average pore area for the gelcast specimens of 25 μm2.

Determining the roundness of each pore was initiated by best fitting an ellipse to each pore using the image processing software. “Best fit” is defined as matching the area and centroid of the ellipse with the area of the dark pixels defining the pore, respectively, then solving for the orientation, major axis length, and minor axis length using a Newton-based Pratt fit [64]. Examples of how ellipses are fit to the pores on the imaged surfaces are illustrated in Fig. 8. Once the ellipse is fit, the roundness of the pore is calculated as

$Roundness=βα$
(9)

where α and β are the length of the major and minor axes, respectively, of the best fit ellipse. Roundness values of unity indicate perfectly circular pore areas with decreasing roundness values indicating elongated pore areas. As with pore area on a cross section, pore roundness is an approximate measure, since the roundness of the pores will vary depending on the position and orientation of the intersecting plane.

A histogram of the pore roundness corresponding to the imaged surface of each specimen type is shown in Fig. 9. The same pores used to calculate the area porosity fraction were used here. All specimens show a similar trend: the frequency of pores with improving roundness increases to roundness of 0.85–0.90, then drops in the final bins. We conclude that the pore shapes are similar between all specimens.

The average grain size of each test specimen was determined visually using the linear intercept method [65]. The polished face of each specimen was chemically etched at room temperature in a solution of 50% H2O, 45% HNO3 (70%), and 5% HF (49%) for 1 h to expose the grain boundaries. The faces were then rephotographed in the same manner as was done for the pore analysis. An example of the grain boundaries in the room temperature test specimens is provided in Fig. 10. The grains have approximately the same average size of 11 μm in all samples.

The morphological attributes of the test specimen taken to 1500 °C are all very close to the values measured for the test specimen that was not heated beyond room temperature following firing. From this result, we conclude that the morphology of the gelcast specimens did not change as a result of heating to 1500 °C. Therefore, it appears that the gelcast material did not densify at high temperatures.

The results of comparing the gelcast ceria test specimens with the material comprising the ligaments of a ceria RPC are summarized in Table 2. The gelcast ceria test specimens are representative of the material comprising the ligaments of the ceria RPC. The grain size is approximately the same between materials. While the pores in the RPC are somewhat larger, the RPC also has fewer pores, so that the overall porosity of the test specimens and RPC samples agree to within about ±0.02. The average pore roundness of the test samples are also similar, with similar roundness distributions. Raising a test sample to 1500 °C did not appear to change its morphology.

## Discussion

The results for elastic moduli and flexural strength are demonstrated to be reasonably consistent with those in the literature. While data are limited for pure ceria at high temperatures, baseline comparisons are made to available room temperature data.

The effect of porosity must be considered to compare the data measured in this work with those in the literature. The minimum solid area (MSA) model is a widely accepted approach for accommodating the effects of both porosity and pore shape [66]
$XX0=e−bP$
(10)
where X is the porous material property of interest, X0 is the corresponding property of the nonporous material, and b is an empirical constant related to the property of interest and pore shape. For porous media manufactured using pressed and colloidal processing techniques and having random pore stacking, the b-values are typically between 3 (representative of cubic stacking of spherical pores) and 5 (representative of cubic stacking of spherical solids) [6668]. Similarly, two materials, A and B, of different porosity can be compared as
$XAXB=e−b(PA−PB)$
(11)

Figure 11 illustrates published elastic modulus data for less porous [2729,31] and more porous [30] ceria, formed by pressing, superimposed on the new data for gelcast ceria. All values from the literature are adjusted to an equivalent 0.20 porosity by applying Eq. (11). Using a b-value of 3.0 in Eq. (11) produces good agreement with the data obtained in the present study and the literature values obtained using bulk measurement methods (see Fig. 11).

The elastic moduli at room temperature nominally agree between this study, Sato et al. [28,29], and Wygant [27]. However, the moduli of Lipińska-Chwałek et al. [30] and Wang et al. [31] are approximately 50% higher. The difference is attributed to the methods used to measure the moduli. The first group of authors used “bulk measurement” techniques: flexural tests or the “small punch” method [28,29]. The second group used indentation techniques. This discrepancy is consistent with variability observed by Radovic et al. [69] and Wachtel and Lubomirsky [26] when comparing different elastic modulus measurement techniques.

The probability of failure of a ceria specimen is assumed to follow a Weibull distribution [70]
$Pf=1−e−(σ/σ0)m$
(12)
where Pf is the probability of failure, σ is the stress at failure of an individual specimen, σ0 is the characteristic strength, and m is the Weibull modulus. Note from Fig. 3 that the strength of the ceria specimens tested here stays nominally constant from room temperature through 800 °C. Therefore, the Weibull modulus is estimated using all of the available data points in that range. The nominal Weibull constants are m = 4.9 and σ0 = 88.7 MPa. Note that the Weibull modulus indicates high variability in the data. The limits of the 90% confidence interval on the Weibull modulus corresponding to the 11 available data points [71] are
$2.8≤m≤6.5$
(13)

The probability of failure may be related to either an effective volume, if the predominant flaw type is volume-distributed, or an effective area, if the predominant flaw type is surface-distributed [72]. Fractographic analysis was performed on the failed test specimens. For specimens tested up to 1300 °C, fractures originated at both surface and subsurface pores. (The fracture origin could not be determined for the samples tested above 1300 °C.) An image of a fracture surface with a subsurface fracture origin is shown in Fig. 12. As a result of some failures originating at subsurface pores, the strength is assumed to scale with effective volume [73]

$σ1σ2=(VE,2VE,1)1/m$
(14)

Equation (14) enables comparing the strengths of specimens having different sizes and loading geometries. In addition, specimen strength is expected to depend on porosity and grain size. Porosity is accounted for using Eq. (11). The Hall–Petch relationship [74,75] can account for grain size; the general trend of the relationship shows decreasing strength with increasing grain size.

The room temperature flexural strength from previous studies of undoped ceria is summarized in Table 3. It is not possible to directly compare the results obtained here with Akopov and Poluboyarinov [33] nor Maschio et al. [76] because of missing information about the specimens or test procedures. Nevertheless, Akopov and Poluboyarinov provide the only published data on the high-temperature strength of ceria. A qualitative comparison between our results and theirs is performed by normalizing the flexural strength versus temperature data of each study, dividing the data by the average room temperature strength (see Fig. 13). Comparison is difficult due to the gap in their data between room temperature and 800 °C. However, their strength data appear to start decreasing at a lower temperature than was observed in our tests.

The remaining strength data were obtained at room temperature. The flexural strength of Wang et al. [35] shows substantial scatter. Equation (11) is applied to adjust to an equivalent porosity of 0.20 using a b-value of 3.0. A lower bound of 58 MPa is obtained by applying a porosity of 0.04 to the lower bound of measured flexural strength, and an upper bound of 87 MPa is obtained by applying a porosity of 0.06 to the upper bound of measured flexural strength. The effective volume for their 1/3-point tests is obtained from [73]
$VE,W=hWwWLo,W6 m+3(m+1)2$
(15)
where VE is the effective volume, and the “W” subscript indicates the first author, Wang. The effective volume for the 1/4-point tests performed in our study is
$VE,S=hSwSLo,S4 m+2(m+1)2$
(16)
where the “S” subscript indicates the first author in the present study. The flexural strength can be adjusted to account for effective volume by substituting Eq. (15), Eq. (16), and the specimen dimensions (from Table 3) into Eq. (14)
$σSσW=[13(m+3)30(m+2)]1/m$
(17)
The data of Wang et al. can then be used to predict the strength expected in this study. Note that Eq. (17) indicates that our failure strength is expected to be lower than that of Wang et al. for all positive values of m. A lower bound is estimated by using their minimum strength, after adjustment to 0.20 porosity, and the minimum Weibull modulus of 2.8 in Eq. (17). A parallel procedure is used to estimate an upper bound, yielding
$46 MPa≤σS,predicted≤78 MPa$
(18)

While the measured data from our study are at the upper limit of the estimated range, the effect of grain size has not yet been considered. Unfortunately, no available experimental studies quantify the constants in the Hall–Petch relationship for ceria. Therefore, we can only utilize the qualitative relation that a smaller grain size is expected to produce a higher strength. Because the grain size of the specimens tested here is smaller than that reported by Wang et al., a higher strength is expected from this study, which is consistent with the strength of our specimens being at the upper limit of the range in Eq. (18).

Comparing our data with that of Cutler and Meixner [77] is more challenging still, as the porosity of their specimens varied over a wide range and their grain size is unknown. However, their specimen geometry is identical to ours, so compensation for effective volume effects is not required. A coarse lower bound for the expected strength for our specimens is made by adjusting their lower bound for flexural strength from their minimum porosity of 0.03 to 0.20. A coarse upper bound is estimated in a parallel manner, yielding
$85 MPa≤σS,predicted≤142 MPa$
(19)

While our data fall below the lower bound of Cutler and Meixner, this trend could occur if their specimens had a smaller grain size. Note that the minimum strength predicted from Cutler and Meixner's data is above the maximum predicted strength resulting from the data of Wang et al.

The trends observed for elastic modulus and flexural strength are similar to those seen in comparable ceramics. For example, Munro [49] provides equivalent data for dense alumina. For comparison, our data are corrected to zero porosity. The elastic modulus of alumina exhibits a near-linear decrease from 0 to 1600 °C. This trend is similar to our elastic modulus, except the alumina is approximately 4.5 times stiffer at room temperature, and the ceria data are better described by a quadratic fit. The rate of decrease of modulus with temperature is about the same for ceria and alumina. Dense alumina is approximately 2.8 times stronger than dense ceria, but both exhibit a precipitous dropoff in strength beginning around 900 °C. The dropoff in strength of ceria appears to be more linear than that of alumina. The trend of the strength data for alumina is consistent with our data in Fig. 13, but different than that of Akopov and Poluboyarinov [33].

## Conclusion

This paper presents new data on the temperature dependency of the elastic modulus and flexural strength of ceria, specifically gelcast ceria, to 1500 °C. The elastic modulus decreases monotonically from about 90 GPa at room temperature to about 16 GPa at 1500 °C. The flexural strength remains approximately constant at about 78 MPa to a temperature to about 900 °C. At higher temperatures, its strength drops roughly linearly with increasing temperature to about 5 MPa at 1500 °C.

The data add to the very limited data available in the scientific literature for ceria and extend the upper temperature range of previous results by 150 °C. Overall, the trends of decreasing strength and elastic modulus with increasing temperature are consistent with previously reported trends in the literature.

When using the data presented in this study, care should be taken to consider the impact of material morphology. While the data were obtained for gelcast specimens with a porosity of 0.20, the results can be extended to other low porosities using the minimum solid area model. Adjusting the previously published moduli to equivalent porosity using the MSA model produced good agreement with the new experiments.

Only one previous source for high-temperature flexural strength data is published [33]. Some discrepancy appears to exist between the previous and current flexural strength data after correcting for porosity. However, little is known about the testing details for the earlier data; the discrepancy is possibly attributable to differences in test fixture or grain size in the earlier test specimens. Reasonable agreement was found with available room temperature flexural strength data.

The present study was motivated by the desire to utilize ceria RPC in solar thermochemical reactors as self supporting catalytic structures. A gelcast ceria was used as the surrogate because of manufacturing similarities with the replication method used to manufacture RPC. Great care was taken to ensure that the morphology of the gelcast material tested here is representative of that in RPC ligaments. Nevertheless, the results are applicable to other ceria morphologies in addition to RPC. The enhanced data presented here facilitate exploiting ceria in new high-temperature applications.

## Acknowledgment

This research was supported by the National Science Foundation (EFRI-1038307) and the University of Minnesota Initiative for Renewable Energy and the Environment. Parts of this work were carried out in the Characterization Facility, University of Minnesota, which receives partial support from NSF through the MRSEC program. The authors appreciate the assistance of Steven Goodrich, University of Dayton Research Institute, in performing the flexural tests at elevated temperature, and Rudolph Olson, SELEE Corp., in preparing the samples. Micrographs of RPC are courtesy of Peter Krenzke.

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