Axial DAC XRD

Spinel represents one of the most important structure groups in ceramics. Its rheological properties have a wide range of implications to multiple fields, from materials science to Earth science [Reference Dal Maschio, Fabbri and Fiori22, Reference Burnley and Green23, Reference Merkel, McNamara, Kubo, Speziale, Miyagi, Meng, Duffy and Wenk24]. We choose the archetypal end-member magnesium aluminate (MgAl₂O₄) spinel mainly because its normal, ordered spinel structure ([Mg]₈^IV [Al]₁₆^VI O₃₂) is stable over a wide pressure range and the lack of pre–phase-transition disordering allows us to focus on pressure-induced microdeformation mechanisms. 25-nm MgAl₂O₄ was compressed to ∼43 GPa in a DAC at room temperature (Fig. 1). At ambient conditions, the lattice parameter a was measured to be 8.096 ± 0.003 Å, in excellent agreement with previous results [Reference Kruger, Nguyen, Caldwell and Jeanloz25, Reference Levy, Pavese and Hanfland26]. The lattice strain initially changes approximately linearly with pressure between 0 and ∼20 GPa [Fig. 2(a)]. At higher pressures, the slope of the pressure versus lattice strain trend increases, implying that a change in the deformation mechanism has occurred. In contrast, for bulk MgAl₂O₄ crystals, the stress–strain relationship is nearly linear over the pressure range to 40 GPa [Reference Kruger, Nguyen, Caldwell and Jeanloz25], implying that the deformation mechanism does not change in the pressure range of the study.

Figure 1:

Axial DAC XRD patterns of MgAl₂O₄ nanocrystals at various pressures.

Figure 2:

(a) The lattice deformation behavior of MgAl₂O₄ nanocrystals. Lattice strain is the fractional change of the lattice parameter. (b) Equation of state of MgAl₂O₄. The closed/open circles represent data taken in this study from nanocrystalline samples on compression/decompression. Square symbols are data on bulk material [Reference Kruger, Nguyen, Caldwell and Jeanloz25].

The bulk modulus (the inverse of compressibility), a measure of how the volume of a material changes during elastic deformation, can be determined from the changes in the lattice parameters with pressure. The unit cell volume as a function of pressure was fitted with the Birch–Murnaghan equation of state [Fig. 2(b)].

We determined the lattice parameter of MgAl₂O₄ at each pressure with MAUD, a Rietveld refinement program [Reference Lutterotti, Matthies, Wenk, Schultz and Richardson27], and analyzed the data in terms of the Birch–Murnaghan equation of state [Reference Birch28],

(1)$$P = 1.5K\left[ {{{\left( {{{{V_0}} \over V}} \right)}^{{7 \over 3}}} - {{\left( {{{{V_0}} \over V}} \right)}^{{5 \over 3}}}} \right]\left\{ {1 - {3 \over {4\left( {4 - K'} \right)\left[ {{{\left( {{{{V_0}} \over V}} \right)}^{{2 \over 3}}} - 1} \right]}}} \right\}\quad ,$$

where V/V₀ is the ratio of unit cell volume at pressure P and ambient pressure. K is the zero-pressure bulk modulus, and K′ is the pressure derivative of the bulk modulus.

We obtained the zero-pressure bulk modulus (K) and its pressure derivative (K′) to be 148 ± 6.3 GPa and 15.8 ± 1.0, respectively. Rather than indication of the second-order phase transition, the extremely large value of K′ is possibly related with a pressure-induced shift in compressional mechanism, which is identified as occurring near 20 GPa based on the deviation between the nanocrystal data and the bulk data (Fig. 2). Therefore, we separate the dataset into two discrete pressure regimes: low-pressure and high-pressure, and apply a Birch–Murnaghan equation of state to each regime. From the equation-of-state fits to the low- and high-pressure regions of our data, we obtained a K of 181 ± 7.1 GPa and K′ of 7.3 ± 0.5 for the pressure range 0–20 GPa, and 285 ± 9.3 GPa and 4.4 ± 0.3 for 20–43 GPa. Compared with previous results for bulk MgAl₂O₄ (K of 196 ± 1.0 GPa and K′ of 4.7 ± 0.3) (Reference Kruger, Nguyen, Caldwell and JeanlozRefs. 25 and Reference Levy, Pavese and Hanfland26), the nanoceramic sample has a close compressibility to that of its bulk counterpart at low pressures, but becomes markedly less compressible at higher pressures.

Some well-known models for nanomechanics, such as strain gradient plasticity [Reference Nix and Gao29] and dislocation source starvation [Reference Budiman, Han, Greer, Tamura, Patel and Nix30], seem not applicable to this case, because no strain gradients can be well examined, and we actually tend to suspect that dislocations are more favored in nanoceramics than bulk counterparts. The observed lattice stiffening indicates a deformation mechanism transition, which suggests that crystal defects could be induced by stress in the nanoceramic sample. To evaluate the hypothesis that a defect-mediated mechanism generates discontinuity in the pressure dependence of strain [Fig. 2(a)], the linewidths of several representative diffraction peaks were analyzed (Fig. 3). Peak broadening is generated by several factors, including instrumental broadening, grain size effects, and stress (microstrain)-induced broadening [Reference Groma and Borbely31]. A standard material, LaB₆, was used to characterize the instrumental broadening. The peak broadening of unstressed samples was used to confirm the particle size that was measured with electron microscopy. Thus, the observed peak broadening can be ascribed to local stresses produced by inhomogeneous strain. Based on previous studies [Reference Groma and Borbely31], we infer that dislocation effects are reflected in the peak profiles.

Figure 3:

(a) Representative XRD patterns of nanocrystalline spinel under compression, showing the shifting and broadening of peaks under pressure. Peaks labeled with asterisks are from the gasket. Peak profile fits: (b) The full width at half maximum (FWHM in terms of two theta) of the Gaussian and Lorentzian components of the diffraction peaks. (c) The relative intensities of several representative diffraction lines with respect to the strongest (311) diffraction peak. For clarity, the FWHM of the (220) peak is offset by 0.9, whereas that of the (111) peak is offset by −1.4. Blue and yellow symbols represent the Gaussian and Lorentzian components, respectively. Closed symbols correspond to data taken on compression, whereas open symbols correspond to data taken on decompression. The pressures at which a shift in deformation takes place are labeled with arrows. The pressure dependences of the Gaussian and Lorentzian components diverge at high pressures, indicating that the stress/strain inhomogeneity increases with pressure.

The peak profiles were fitted with a pseudo-Voigt function [Reference Wertheim, Butler, West and Buchanan32]. The pseudo-Voigt function is an approximation of the Voigt line shape by the sum of a Gaussian and a Lorentzian function. It can fit both symmetric and asymmetric peaks. To reflect the contribution from the inhomogeneously strained lattices of materials, the peak centers and widths of the Gaussian and Lorentzian components are allowed to vary independently within the fit. The fit function is written as

(2)$$\eqalign{f\left( x \right) = {y_0} + A\left\{ {\left( {1 - \eta } \right)\exp \left[ { - {{\left( {{{x - {x_{\rm{G}}}} \over {0.600561 \cdot {\rm{FWH}}{{\rm{M}}_{\rm{G}}}}}} \right)}^2}} \right] \cr+ {\eta \over {1 + {{\left[ {{{\left( {x - {x_{\rm{L}}}} \right)} / {\left( {0.5{\rm{FWH}}{{\rm{M}}_{\rm{L}}}} \right)}}} \right]}^2}}}} \right\},$$

where (x_G, x_L) and (FWHM_G, FWHM_L) are the peak centers and widths of Gaussian and Lorentzian components, respectively. A, η, and y₀ are fit coefficients.

The Gaussian centers and linewidths vary smoothly with pressure, and so the Gaussian component reasonably represents the homogeneous strain in the nanocrystals and the effect of pressure gradients across the nanocrystalline aggregate. Larger variations are seen in the Lorentzian components, as would be expected for pressure-induced inhomogeneous strain fields. The shift in the pressure dependence of the peak positions and widths indicates the onset of plastic deformation (Fig. 3). The combination of the peak broadening of the Lorentzian components and the decreased compressibility of the nanocrystals implies that defects within the nanocrystals have shifted the local strain fields. Hysteresis in the inhomogeneous strain fields on decompression also indicates that pressure-induced defects are incompletely removed during decompression (Fig. 3). Peak symmetry is recovered upon complete quenching, and this observation further shows that strain inhomogeneity causes the asymmetry of diffraction peaks at high pressure. Reversible peak broadening also indicates that no permanent change in crystallite size and shape resulted from compression; hence, pressure-induced coarsening/fracturing of particles appears not to take place in MgAl₂O₄ nanocrystals. However, the radial diamond-anvil cell (rDAC) XRD (Fig. 4) results presented in the next section capture the dislocation-mediated texturing in the nanocrystal samples, which may suggest that the shear stresses in these aDAC XRD (Fig. 5) experiments are not large enough because of the use of the pressure medium. In a recently reported study, the increase in defect intensity in the deformed indium pillar was probed with Laue diffraction peak broadening. This technique could be useful for further exploring the defect evolution of nanoceramics [Reference Ali, Tamura and Budiman34].

Figure 4:

The rDAC XRD setup [Reference Chen, Lutker, Raju, Yan, Kanitpanyacharoen, Lei, Yang, Wenk, Mao and Williams33].

Figure 5:

The axial DAC XRD setup.

The pressure-induced shifts and broadening of the Gaussian components of different peaks are quite similar, indicating that they reflect homogeneous and isotropic deformation (Fig. 3). In contrast, the Lorentzian components of the different diffraction peaks behave differently, indicating the presence of anisotropic inhomogeneous deformation.

The asymmetry of the different diffraction peaks increases with pressure (Fig. 3). Up to ∼20 GPa, the Gaussian and Lorentzian components have both the same centers and linewidths [Fig. 3(b)], showing that the peak profiles are symmetric and that the strain distribution in each of the nanocrystals is homogeneous. On further compression, substantial deviations are seen between the peak centers and the linewidths of the Gaussian and Lorentzian components of the (400) peak. Similar deviations are also seen at higher pressures for the (220), (311), and (111) peaks. Thus, strain inhomogeneity in these nanocrystals appears to increase greatly between 23 GPa and the highest pressure.

The onset of asymmetric peak broadening occurs for the (100) peak at >23.5 GPa, followed by (110) at >27.1 GPa, (311) at >31.4 GPa, and (111) at >35.0 GPa. The abruptly increased peak broadening above these pressures is not likely to be due to pressure gradients across the nanocrystal aggregates, nor is it readily attributed to a bulk crystallographic distortion (such as a transition to lower symmetry) because the onset of such a crystallographic distortion is usually coherent and is unlikely to span a >12 GPa pressure range. Instead, we infer that the variable onset of peak broadening is due to strain fields generated as defects with differing preferred orientations are activated at varying critical stresses. Crystals usually contain a mixture of edge and screw dislocations. Weak-beam dark-field microscopy reveals that in MgAl₂O₄ full-edge dislocations can dissociate into partial dislocations [Reference Mitchell35]. The dissociation spacing can be as small as 10 nm. Dislocations move through the lattice and create regions of compressive, tensile, and shear stresses. The enhanced broadening of the (111) peak is seen at much higher pressure than for the other peaks. Thus, the critical stresses associated with pressures of less than 35 GPa mainly generate dislocations with preferred slip within the {111} plane.

The relative intensity of diffraction lines also varies with pressure [Fig. 3(c)], as is anticipated from the generation of defects during compression causing a shift in structural order. The Burgers vectors and dislocation lines lie preferentially on {111}, so dislocation-associated disordering affects the (111) diffraction line the least. The intensities of the other diffraction lines are more affected by compression. As a consequence, the relative intensity of the (111) peak increases with pressure with respect to the other peaks [Fig. 3(c)].

rDAC XRD

To have experimental evidence for the dislocation activity in the nanoceramics, we conducted high-pressure texturing measurements with rDAC XRD techniques (Fig. 4). Deformation texturing is known to originate from dislocations [Reference Merkel, McNamara, Kubo, Speziale, Miyagi, Meng, Duffy and Wenk24, Reference Kocks, Tomé, Wenk and Beaudoin36]. Plastic deformation by dislocation glide results in crystallite rotations, generating lattice preferred orientation or texture. We deformed MgAl₂O₄ samples nonhydrostatically in a panoramic diamond-anvil cell. The particle sizes of the MgAl₂O₄ samples are 25 ± 7 nm for nano-sized samples and 3 ± 0.8 µm for bulk-sized samples. No systematic variations in diffraction intensity are seen for the bulk-sized sample. In contrast, modest but resolvable intensity variations are observed in the nanoceramic sample above 4.0 GPa (Fig. 6).

Figure 6:

(a) Azimuthally (0 to 360°) unrolled diffraction images of MgAl₂O₄ nanocrystals (left) and microcrystals (right) from radial DAC diffraction measurements. The long thick arrows represent the maximum compression direction and the short thin arrows the minimum compression direction. The curvature within the diffraction lines indicates that the sample is stressed. For MgAl₂O₄ nanocrystals at >4 GPa, texture is manifested by the systematic intensity variations of the diffraction peaks along the azimuthal direction. The narrow white line (horizontal) is due to blocking by the beam stop arm. The unlabeled weak diffraction lines are due to the partial crystallinity of the amorphous boron-epoxy gaskets. (b) The inverse pole figures (IPF) of MgAl₂O₄ nanocrystals (left) and microcrystals (right) along the compression direction (normal direction). Equal area projection and a linear scale are used. Texture strength is expressed as multiples of random distribution (m.r.d.), where m.r.d. = 1 denotes a random distribution and a higher m.r.d. number represents stronger texture. The IPF maximum of stressed MgAl₂O₄ nanocrystals appears at the 〈110〉 corner, which is compatible with the slip system {111}〈110〉.

Rietveld refinement, implemented in the MAUD software [Reference Lutterotti, Matthies, Wenk, Schultz and Richardson27], was used to analyze the differential stress, microstructure, and texture of the samples at each pressure. Texture is represented with inverse pole figures of the compression direction (Fig. 6); these show the probability of finding the poles (normal) to lattice planes in the compression direction. Texture strength, the degree of lattice preferred orientation, remains unchanged at about 1 multiple of random distribution (m.r.d.) over the entire pressure range. Hence, no resolvable texture in the bulk ceramics evolves at pressures up to 37 GPa. In the experiment with the nanoceramic particles, texture starts to develop at 4.0 GPa with 1.27 m.r.d. On further compression to 36 GPa, the highest pressure of the measurements, the texture strength of the nanoceramics increases to 2.22 m.r.d. The observed texturing indicates that dislocation-mediated plastic deformation is active in the nanoceramics when high external pressures are applied. It is known that dislocation glide on preferred slip systems gives rise to crystallographic texture, whereas grain rotation by grain boundary (GB) sliding alone randomizes the grain orientation distribution [Reference Weissmüller and Markmann17]. Previous studies have established that the deformation texturing of nanocrystals arises from dislocations, not from GB processes.

The physics governing the observed pressure-promoted stiffening and texturing in nanoceramics can be understood from the effect of pressure on dislocations. According to dislocation theory [Reference Hirth and Lothe37], stress can promote the retention of dislocations by decreasing their equilibrium distances and the distances between dislocations and the interfaces (Fig. 7). An external shear stress of 0.44 GPa (which is equivalent to an external pressure of 18 GPa in our axial DAC XRD experiment) could counteract the repulsion between dislocations and allow them to move within 10 nm of each other, as long as other effects do not impede dislocation motions and interactions. The estimation of image force and dislocation interaction includes straight edge dislocations chosen for the estimation of dislocation–dislocation interaction, for simplicity. The force per unit length of two edge dislocations separated by a distance of l_dd is $f = {{G{b^2}} \over {2\pi \left( {1 - \nu } \right){l_{{\rm{dd}}}}}}$ (Reference Hirth and LotheRefs. 37 and Reference Nieh and Wadsworth38). In the estimation of the interaction force, published values, 62.9 GPa (Ref. Reference Anderson and Isaak39), 0.35 (Ref. Reference Wdowik, Parliński and Siegel40), and 0.286 nm (Ref. Reference Doukhan, Duclos and Escaig41), are used for the shear modulus (G), Poisson’s ratio (ν), and Burgers vector (b) of MgAl₂O₄, respectively. The image force (the interaction between a dislocation and a surface/interface) on per unit length of an edge dislocation at a distance of l_ds from the particle surface is $f = {{G{b^2}} \over {4\pi \left( {1 - \nu } \right){l_{{\rm{ds}}}}}}$ (Reference Hirth and LotheRefs. 37 and Reference Nieh and Wadsworth38). The applied stress needed for canceling out the dislocation interaction force or image force is σ_ext = f/b (Ref. Reference Nieh and Wadsworth38). Although more complicated dislocations may exist, the simplified calculation above is useful for a rough estimation of the dependence of dislocation concentration on compression.

Figure 7:

External stress needed to achieve dislocation separations in the nanoceramic samples. External stresses push dislocations closer to one another, so dislocations become less controlled by particle size in stressed nanocrystals. The measured pressure gradients across the sample give an estimation of the shear stress induced by the nonhydrostatic pressure.

Because of nonhydrostaticity and different elastic strengths of nanocrystal cores and surface regions, external pressure induces shear stress within nanocrystals [Reference Ashby, Gelles and Tanner42, Reference Meade and Jeanloz43]. At each compression, the pressure was measured at multiple spots across the sample. Shear stress was estimated by using the empirical formula, ${\rm{shear}}\;{\rm{stress}} = {h \over 2}{{{\rm{d}}P} \over {{\rm{d}}r}}$ (Ref. Reference Meade and Jeanloz43). The gasket thickness was assumed to be 10–30 µm. These values are underestimated because the elastic strength difference of nanocrystal cores and grain boundaries was not included in the estimation.

The dislocation density of MgAl₂O₄ at ambient conditions was reported to be of the order of 10¹² m⁻² (Ref. Reference Doukhan, Duclos and Escaig41). So, at room pressure, an MgAl₂O₄ particle of size 20 nm contains 10¹² m⁻² × π(10 nm)² ≈ 0 dislocations. Under an external shear stress of 0.44 GPa, dislocations can be pushed as close as 10 nm to one another. Assuming there are only full-edge dislocations for simplicity, the dislocation density can be 4/[π(10 nm)²] ≈ 10¹⁶ m⁻².

Because of the solidification of the pressure medium, the pressure in the axial DAC experiments is nonhydrostatic above 10 GPa. In the current study, the sample was compressed to 43 GPa, which should be sufficient to generate the shear stress needed to produce dislocation microstructures even within the nanocrystals (Fig. 7). Because no pressure medium was used in the texturing experiments, shear stress was generated even at low pressures. According to the MAUD calculations, at 4 GPa (the external pressure), the shear stress in the nanoceramic sample is ∼0.92 GPa (extracted from the differential strain of the crystal lattices) in this radial DAC XRD experiment.

Previous studies suggested that nanoceramics are expected to exhibit enhanced diffusional creep because of their greatly increased creep rate associated with grain boundary diffusion [Reference Karch, Birringer and Gleiter2]. Therefore, the conventional belief is that the enhanced diffusional activities would make nanoceramics less brittle. The contribution of diffusion cannot be assessed in this study, but the crystallographic dependence of texturing suggests that the deformation texturing is not arising from diffusion. The observed texturing in the nanoceramic sample indicates that dislocation mechanisms provide a significant mechanism of plastic deformation in nanoceramics. Although the textures formed at high pressure are quenchable, dislocations usually cannot remain in nanocrystals when no external shear stresses sustain them, so postdeformation measurements may not constrain dislocation-associated deformation in nanoceramics. In contrast, although dislocations can exist in bulk-sized ceramics, they tend to not move as rapidly in ceramics as they do in metals. This limits the amount of energy that can be dissipated ahead of a growing crack by plastic flow, resulting in brittle failure [Reference Clegg1]. The stress needed for the nucleation and propagation of cracks is lower than that for dislocations, so fracturing processes are more favorable, and dislocation mechanisms are not operative. Although grain fracturing may occur under pressure, the compression behavior of bulk ceramics appears to involve purely elastic deformation over a relatively large pressure range (Fig. 2).

In the nanoceramic sample, dislocations are not suppressed by cracks, as indicated by the observed texturing. The frictional-wear mechanism–associated crack growth was reported in grain-bridging ceramics [Reference Dauskardt44]. However, no cracks seem nucleating in the MgAl₂O₄ nanocrystals [Fig. 3(b)] in our experiments, although stepwise compression could be treated as escalating cyclic loading. Finite element analysis, which has been successfully applied to the crack behaviors in accumulative roll-bonded Cu/Nb nanolaminates [Reference Radchenko, Anwarali, Tippabhotla and Budiman45], could be useful for understanding the crack–dislocation transition in ceramics. Dislocations usually change the stress and strain profiles of a crystal and result in inhomogeneous deformation. The dislocationless-to-dislocation–mediated deformation transition accounts for the observed stiffening in the compression behaviors of MgAl₂O₄ nanocrystals. The stiffness of lattice is primarily determined by atomic bonding strength, and distorted bonds due to dislocations or other defects usually exhibit enhanced strength [Reference Gilbert, Huang, Zhang, Waychunas and Banfield46]. Even in the same sample, multiple deformation mechanisms could coexist and result in different mechanical behaviors in the different parts of the sample [Reference Lee, Kim, Burek, Greer and Tsui47, Reference Razorenov48]. The deformation mechanism transition pressure obtained in the rDAC XRD experiments, i.e., the texturing pressure (Fig. 6), is lower than that obtained in the axial DAC experiments (Figs. 2 and 3), which is expected from the lack of a pressure medium, and hence larger nonhydrostatic stresses, in the texturing measurements. Once immobilized, dislocations always tend to move from glide to climb configurations, forming a 3D dislocation network. Dislocation networks resist compression more than normal regions because the structure is already internally strained. We propose that the enhanced overlap of dislocation strain fields in smaller particles accounts for pressure-induced stiffening, as we observed in MgAl₂O₄ nanocrystals at high pressures. The differential stress analysis [Reference Chung, Weinberger, Levine, Kavner, Yang, Tolbert and Kaner49] indicates that the strength of the nano-sized and bulk-sized MgAl₂O₄ is comparable over the pressure range of this study. Thus, MgAl₂O₄ nanoceramics are stiff, strong, and ductile.