Size dependent compressibility of nano-ceria: Minimum near 33 nm

It is well-established that nanomaterials often have physical and chemical properties that differ from their bulk counterparts.¹ It is common, however, to treat bulk modulus as an attribute that does not change with crystallite size,² particularly when explaining size-dependent lattice parameter expansions or contractions in nanomaterials.^3–5 Bulk modulus refers to the reciprocal of a material's relative volume decrease under hydrostatic pressure and can be measured independently in both bulk- and nano-scale materials to check for constancy. The term compressibility, which here specifically refers to the reciprocal of bulk modulus, is also often employed in these types of studies. Metal oxides, whose lattices often expand with decreasing crystallite size, are useful as microelectronic circuits, sensors, piezoelectric devices, fuel cells, and catalysts.⁶ Good evidence has been mounting that bulk modulus of nano metal oxides is different from the same material at larger grain size,^7,8 but few studies test over a range of different crystallite sizes, which is key to detailing trends.⁹ We present here experimental evidence that bulk modulus of ceria increases from the bulk to reach a maximum near a crystallite diameter of 33 nm and then decreases for smaller crystallite sizes. We examine several potential explanations for this behavior, which together may lead to a better understanding of the physics of nano metal oxides.

Nanoceria was synthesized with the hexamethylenetetramine (HMT) method.^5,10 Equal volumes of aqueous solutions of 0.0375M Ce(NO₃)₃·6H₂O and 0.5M HMT were combined at room temperature with stirring, which resulted in the precipitation of ceria nanoparticles. The aqueous behavior of cerium ions is complex.¹¹ Ceria precipitates because of the gradual increase in basicity from the slow decomposition of HMT to ammonia and formaldehyde. Details of homogeneous nucleation and subsequent uniform growth of nanoceria has been previously captured by synchrotron in-situ small angle x-ray scattering with flow cell and temperature control.¹² Initial nanoparticle sizes are typically in the range of 6–12 nm, depending on aqueous mixing time. It is thought that the cage-like structure of HMT may play a role in directing the formation of particularly small and uniform nanoparticles.¹² Some of the resulting nanoceria was annealed in air (typically for two hours at temperatures ranging from 500 to 700 °C with fast heating and cooling times) to coarsen the particles, and in this way moderately uniform nanocrystallite sizes from 15–100 nm were realized. X-ray diffraction (XRD) scans were performed on an Inel XRG 3000 diffractometer with a curved detector using Cu-K_α1 radiation. XRD data work-up was done with MDI JADE software version 2010. The standard reference for lattice parameter was taken from the Crystallography Open Database (COD ID: 9009008),¹³ and the standard reference pattern for bulk size (i.e., greater than 1 μm) was a scan of the micron-sized Alfa Aesar CeO₂, which agree within experimental error. Figure 1 show the lattice parameter measured under ambient conditions as a function of crystallite diameter, including the experimental bulk measurements shown as a dotted line in the plot. These measurements are all consistent with earlier reports.^4,5,10 The expansion of lattice with decreasing particle size is discussed elsewhere,⁴ but reiterated in brief here.

There are several potentially overlapping models that may explain expansion of lattice parameter with decreasing particle size, including a Madelung pressure model,³ arguments from surface stress,⁴ and larger Ce³⁺ ions from the reduction of cerium cations on the surface of nanoparticles.¹⁴ Indeed it has been demonstrated experimentally that for smaller ceria nanoparticles, an increased occurrence of Ce³⁺ ions and corresponding oxygen vacancies are observed by XANES at the surface.¹⁴ Up to 6% of the cerium atoms are reduced Ce³⁺ state for 6 nm ceria. It is thought that the larger, reduced Ce³⁺ may account for 40% of the lattice expansion, whereas a negative Madelung pressure model³ may account for the other 60% of the lattice expansion in 6 nm ceria. Recent improvements in the sophistication of density functional theory calculations have also helped illuminate the experimentally observed expansion of metal oxides lattices via negative (i.e., compressive) surface stress arguments.⁴ 

Transmission electron microscopy (TEM) images were recorded on a JEOL-TEM instrument, to confirm the uniformity of the nanoceria. Suspensions of ceria were sonicated in ethanol and drop-cast onto carbon film on 300-mesh copper grid from Electron Microscopy Sciences. Measurements of particle diameter were accomplished with Image-J software. Samples with small particle diameters (6–20 nm) were uniform in size, but limited polydispersity was observed in some of the larger nanoceria (20 nm+). Figure 2 shows representative TEM images of small and large nanoceria particles.

For the bulk modulus measurements, high pressure x-ray diffraction measurements were performed at ambient temperature using a standard symmetric diamond anvil cell which consists of two 300 μm culet diamonds with a rhenium gasket indented to ∼50 μm thickness near the sample chamber. Ceria powder mixed with a pressure marker (ruby chips) was loaded in an 110 μm hole drilled at the center of the indentation. Helium, neon, or argon was used as the pressure transmitting fluid. X-ray diffraction data were collected on beamline 12.2.2 of the Advanced Light Source (LBL-ALS).¹⁵ A Si(111) double-crystal monochromator was used for wavelength selection, and diffraction data were collected using a MAR345 image plate at an x-ray energy of 30 keV (0.4133 Å) and a sample-to-detector distance of 280 mm determined with a LaB₆ standard. Diffraction data were integrated using Fit2d¹⁶ and unit cell parameters and volumes were determined with Celref software.¹⁷ 

For the pressure calibration in these measurements, the ruby fluorescence spectra showed a sharp doublet throughout the measurements, with no measurable broadening of the peaks. Small ruby chips were placed in the sample chamber. Pressure was measured from each of these chips, and differences in the measured pressure between these chips were found to be within the experimental precision of the pressure scale used. The sharpness of the doublets and the uniformity of the pressures recorded indicate that any non-hydrostaticity that might be present in our measurements was, for the most part, below a measurable level over the entire range of pressures investigated in this study.

The collected high-pressure XRD data is plotted in Figure 3. The data, although neither perfectly linear nor a perfect fit with a Birch-Murnaghan (BM) equation of states (EoS), are free from drastic discontinuities and sharp slope changes sometimes seen in other reports on nanoceria.⁸ We attribute this to selection of the pressure-transmitting medium (PTM) as helium, neon, or argon. Experimenters who have chosen a mixture of methanol and ethanol as a PTM sometimes see significant and sharp deviations from linearity (especially above 15 GPa), which may come from the non-hydrostaticity of the PTM solidifying.

The bulk modulus, B, was calculated several different ways. First, it was calculated as $−V0dPdV$⁠, over the linear range of data. Then, it was calculated using the BM EoS, with software EoSFit7 as developed by RJ Angel. Finally, the bulk modulus was calculated with f-F plots. For the BM EoS, some of the fits were not perfect. It was decided that the best representation of the bulk modulus came from using the parameters detailed in the supplemental information section.¹⁸ We have decided that these parameters are appropriate because they fit the experimental data best while still adhering to non-negative Bo's and reasonable Vos.

All three of these methods for calculating the bulk modulus produced the same general trend: as crystallite size decreased, bulk modulus first increased, then reached a peak at 33 nm, and then reached the lowest value from our set at 6 nm. This gives us confidence that the trend we report is real. The differences between the bulk modulus values using different calculation methods, however, are concerning. Even within the BM EoS method, values can change substantially through the tradeoff between Bo and Bo'. In the end, we have the most confidence in the BM EoS values as reported here, and those are the values plotted in Figure 4. It is evident that with decreasing crystallite size, bulk modulus first increases, and then decreases, reaching its maximum near 33 nm.

There are several different explanations that have been proposed for changes in bulk modulus on the nanoscale. Wang and colleagues in 2001¹⁹ reported that the structure of their commercial nanoceria (9–15 nm in diameter) started to change from fluorite to orthorhombic α-PbCl₂ at 22 GPa. Our experiments did not reproduce such a phase change. We do note that our maximum pressure (∼25 GPa) was much lower than Wang's maximum pressure in this case (∼38 GPa). Their 2001 studies also indicated that the bulk modulus of their nanoceria was enhanced relative to the bulk, micron-sized material. Specifically, their nanoceria had a bulk modulus of 328 GPa, whereas their bulk ceria had a bulk modulus of 230 GPa. In a follow-up study in 2004, Wang and colleagues²⁰ reported noticeable discontinuities in their P vs V data for their nanoceria, starting at 20 GPa. They used no PTM for these experiments. Their conclusions are complicated by the proposal of in-situ particle coarsening, as evidenced through XRD peak sharpening, which was not evident in our experiments. Such coarsening may be related to synthesis method or absence of PTM. In continued studies reported in 2007,²¹ Wang and colleagues experimented with very small nanoparticles (3 nm) and found their bulk modulus to be enhanced relative to the bulk. They also postulated (through arguments based on homogenous stress field distributions and unusual behavior of defects and vacancies) that there may be a critical size where bulk modulus of nanoceria may suddenly become very enhanced; they estimate the critical size to be 7–18 nm.

Ge and colleagues⁷ reported sudden changes in compressibility of their nanoceria at 10 GPa using methanol-ethanol-water as their PTM. Specifically, they saw their volumes become less responsive to pressure, indicating an increase in bulk modulus. Such apparent changes may be due to solidifying media. They followed their methanol-ethanol-water studies with N₂ as their PTM, obtaining similar results. Their nanoparticles (4.7 and 5.6 nm diameter) were smaller than the ones used here, so it is possible that they are observing a separate phenomenon, but the PTM of helium or argon was not tested. Their explanation for the sudden bulk modulus increase has to do with increasing surface energy for smaller nanocrystals, which leads to hardening at small sizes. Ge's interpretation is insufficient to explain the results obtained here, as it accounts only for an increased bulk modulus as size decreases and not for a peak in bulk modulus at intermediate size.

Wang⁸ and colleagues reported in 2014 that the compression curve for their nanoceria was irregular above 12 GPa using silicon oil as the PTM. They supposed that good PTMs like helium or neon would be too soft to hinder the grains from touching each other and thus would be functionally equivalent to no PTM. We suggest that this may also depend on how much sample there is in the chamber compared to PTM. Our results, using small volume percentage of sample in the diamond anvil cell, suggest that the selection of helium, neon, or argon as PTM can avoid such irregularities. Their core-shell interpretation of their data is focused on explaining sharp slope sign changes, which are not seen in our data. Their core-shell interpretation is distinct from the core-shell model of Bian,²² which we will return to shortly.

None of these nanoceria works consider crystallites over a wide range of diameters, whereas Chen and colleagues⁹ reported the bulk modulus of anatase titania nanoparticles over a wide range of diameters, and their results are similar to our nanoceria. They measured 12 different-sized samples: the smallest was 3 nm and the largest was 45 nm (and then bulk). As their crystallite size decreases, they see an initial increase in bulk modulus, followed by a decrease, with bulk modulus peaking near 15 nm particle diameter. They use hardening from dislocation networks to explain the increase in bulk modulus, and then they propose that the particles become too small to sustain such dislocations, which results in a decreasing bulk modulus after 15 nm. This explanation supposes that dislocations (traditionally thought to influence plastic properties) can influence elastic properties. Further studies on anatase titania nanoparticles raise concerns²³ over experimental details, but Chen's work covered a comprehensive size range of nano oxides.

A similar trend was observed by Bian and colleagues for a wide size-range of PbS nanoparticles.²² They describe a core-shell model. Following this analysis, the synthesis methods for our nanoceria, as well as Bian's PbS, yield high-quality crystallites that are defect-free, as evidenced by high-resolution TEM,⁵ and therefore Chen's explanation of bulk modulus trends depending on dislocations does not apply. Explanations that rely on diameter-dependent lattice expansion also do not apply here, because the lattice of ceria only expands at small diameters, which does not fit bimodal bulk modulus behavior. For our ceria, lattice parameter changes only occur at crystallite diameters under 20 nm, while bulk modulus peaks at 33 nm. Furthermore, Bian et al. have shown that the parallel effect seen in the expanding lattice spacings of nano-PbS with smaller particle size is quantitatively inadequate to explain the magnitude of the changes in bulk modulus.

We use the core-shell model to understand the trend observed in our nanoceria. Under this model, the ceria has a shell region and a core region. The core region has a constant bulk modulus, equal to that of the bulk, while the shell region has a size-dependent bulk modulus. In a large crystallite, the bulk modulus of the shell is higher than that of the core, because of surface reconstruction and higher packing density at the surface. The concept of shorter and stronger Ce-O bonds at the surface of ceria is supported by first-principle electronic calculations.⁸ But at large crystallite sizes, this counts little overall because of the small surface area to volume ratio. As crystallite size decreases, the shell becomes important, as the surface area to volume ratio increases, and the consequence is an increased effective bulk modulus up to a peak near 33 nm. At diameters smaller than 33 nm, the size-dependency of the bulk modulus of the shell begins to matter. The shell bulk modulus is expected to decrease with smaller nanocrystal sizes because of curvature effects. The bulk modulus of a surface shell is greatest in a flat slab and decreases as surface curvature becomes more severe and radius decreases. The surface ions have fewer partners to bond with even under the most favorable reconstruction. The shell is also expected to have a decreased bulk modulus at small crystallite diameters because of an increased occurrence of oxygen vacancies and larger Ce³⁺ cations at small crystallite diameters, as measured by XANES.¹⁴ Overall, this core-shell empirical qualitative model satisfactorily explains the peaking of bulk modulus at an intermediate crystallite diameter in all three inorganic crystallites considered here: nanoceria, nanotitania, and nano lead sulfide, where considerable ionicity in bonding is observed. Any theoretical derivation beyond the rationalization under the core-shell model is beyond the scope of this letter.

We have presented experimental evidence of the size-dependence of the bulk modulus of nanoceria, illustrated the importance of testing nanomaterials over a range of sizes, and emphasized the importance of using a suitable PTM for high-pressure XRD studies. Our results are consistent with a core-shell theoretical explanation for the observed experimental behavior. This finding of bulk modulus extremum in nanoceria will shed light on their bonding characteristics, which affects their use in applied technologies such as heterogeneous catalysis.

S.W.C., P.P.R., and J.S. acknowledge the support from National Science Foundation-DMR 1206764 and the special beamtime granted by the ALS-director for the final data collection to complete this study. We acknowledge S.W.C.'s former students Sungjoo Kim, Joan Raitano, and Feng Zhang for their initial investigations. We thank Alastair MacDowell, Jinyuan Yan, and Andrew Doran for their help and guidance at LBL-ALS 12.2.2. Support for B.K. was provided by The Scientific and Technological Research Council of Turkey fellowship under Contract No. 114C120. The Advanced Light Source was supported by the Director, Office of Science, Office of Basic Energy Sciences of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This research was partially supported by COMPRES, the Consortium for Materials Properties Research in Earth Sciences under NSF Cooperative Agreement No. EAR 11-57758.