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Below the Hall−Petch Limit in Nanocrystalline Ceramics Heonjune Ryou,† John W. Drazin,† Kathryn J. Wahl,‡ Syed B. Qadri,§ Edward P. Gorzkowski,§ Boris N. Feigelson,∥ and James A. Wollmershauser*,§ †
American Society for Engineering Education Postdoctoral Research Fellow sited at the U.S. Naval Research Laboratory, Washington, D.C. 20375, United States ‡ Chemistry Division, U.S. Naval Research Laboratory, Washington, D.C. 20375, United States § Material Science & Technology Division, U.S. Naval Research Laboratory, Washington, D.C. 20375, United States ∥ Electronics Science & Technology Division, U.S. Naval Research Laboratory, Washington, D.C. 20375, United States S Supporting Information *
ABSTRACT: Reducing the grain size of metals and ceramics can significantly increase strength and hardness, a phenomenon described by the Hall−Petch relationship. The many studies on the Hall−Petch relationship in metals reveal that when the grain size is reduced to tens of nanometers, this relationship breaks down. However, experimental data for nanocrystalline ceramics are scarce, and the existence of a breakdown is controversial. Here we show the Hall−Petch breakdown in nanocrystalline ceramics by performing indentation studies on fully dense nanocrystalline ceramics fabricated with grain sizes ranging from 3.6 to 37.5 nm. A maximum hardness occurs at a grain size of 18.4 nm, and a negative (or inverse) Hall−Petch relationship reduces the hardness as the grain size is decreased to around 5 nm. At the smallest grain sizes, the hardness plateaus and becomes insensitive to grain size change. Strain rate studies show that the primary mechanism behind the breakdown, negative, and plateau behavior is not diffusion-based. We find that a decrease in density and an increase in dissipative energy below the breakdown correlate with increasing grain boundary volume fraction as the grain size is reduced. The behavior below the breakdown is consistent with structural changes, such as increasing triple-junction volume fraction. Grain- and indent-size-dependent fracture behavior further supports local structural changes that corroborate current theories of nanocrack formation at triple junctions. The synergistic grain size dependencies of hardness, elasticity, energy dissipation, and nanostructure of nanocrystalline ceramics point to an opportunity to use the grain size to tune the strength and dissipative properties. KEYWORDS: nanocrystalline, ceramic, MgAl2O4, spinel, Hall−Petch, energy dissipation, indentation inverse5 Hall−Petch relationship). Hall−Petch breakdown is generally described to occur when dislocation-mediated strain accommodation requires a higher stress than the activation stress of other deformation mechanisms.5,8,13 While most Hall− Petch breakdown models do employ dislocation activity,5,6 solely considering the limit of dislocation activities in nanocrystalline materials without including a secondary mechanism typically does not show a decrease in strength and hardness, i.e., a negative Hall−Petch slope.5 Several models covering the whole range of behavior of metals (increasing and decreasing strength) often attribute the reduction in strength
H
all−Petch strengthening and its breakdown in nanocrystalline materials have great engineering and scientific relevance because they describe mechanical property limits and the onset of other plastic deformation mechanisms. Numerous experimental studies in metals show that the strength increases as the grain size decreases and relate the mechanisms behind the Hall−Petch strengthening to interactions of dislocations within the grain interior and grain boundary.1−7 Importantly, there is a size limit where the Hall− Petch relationship breaks down5,6,8−12 and the strength and hardness follow a different grain size dependence. This phenomenon often leads to a slope change and is called Hall−Petch breakdown. When a negative slope appears after the breakdown, where the strength and hardness decrease as the grain size is reduced, the phenomenon is known as a negative Hall−Petch relationship5 (also called a reverse8 or © XXXX American Chemical Society
Received: October 17, 2017 Accepted: February 26, 2018
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DOI: 10.1021/acsnano.7b07380 ACS Nano XXXX, XXX, XXX−XXX
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Figure 1. Nanostructure of nanocrystalline MgAl2O4 ceramic samples. (a) Low-magnification bright-field TEM image of nanocrystalline MgAl2O4 ceramic prepared by ion milling into a wedge shape, showing a uniform nanoscale polycrystalline structure. (b) Highermagnification TEM image of the sample. The inset shows the fast Fourier transform diffraction pattern representing a polycrystalline and randomly oriented crystallographic texture. (c) High-resolution TEM image of the edge of the wedged TEM sample at high magnification, showing no porosity and a thin grain boundary region even at the edge of the wedge, which has extreme beam damage, coarsening the structure. (d) Stacked X-ray diffraction patterns of all samples studied with grain sizes ranging from 37.5 nm down to 3.6 nm. The increasing broadening of the peaks indicates smaller grain size. Some diffraction patterns include additional peaks that originated from sample packaging used in the synthesis process.
Petch breakdown,17 but another work observed a Hall−Petch breakdown at 30 nm and a negative Hall−Petch relationship below 30 nm.8 While the reason for such discrepancies in ceramics has not been debated in the literature, it is accepted14−18 that to measure the intrinsic mechanical response, the material must be free of porosity (fully dense) and the measurement technique must be, at least, selfconsistent to allow a qualitative comparison.5 Theories that describe the negative Hall−Petch relationship in ceramics generally treat the material as a fully dense composite comprising a crystalline phase and distinct grain boundary phase.16,19,20 In these composite models, the crystalline phase accommodates plastic deformation through dislocation activity and diffusion creep, while the grain boundary phase accommodates strain by diffusional flow of material within the grain boundary. The models rely on the assumption that the total volume fraction of the grain boundary increases significantly as the grain size decreases to the
and hardness to diffusion-based mechanisms that allow grain boundary activities (e.g., grain boundary sliding and grain boundary diffusion) and grain activity (e.g., grain rotation) in conjunction with dislocation mechanisms.5,10,12 Conventional Hall−Petch relationships have been observed in ceramics,14,15 but the limited number of studies exploring the breakdown of the Hall−Petch relationship in nanocrystalline ceramics contain contradicting results.16−18 Attempts to quantify the full extent of the Hall−Petch relationship have been undertaken only for nanocrystalline MgO16 and MgAl2O4.15,17,18 Unlike in nanocrystalline metals, where the breakdown occurs over a narrow grain size range (from 10 to 20 nm),5,12 a negative Hall−Petch relationship in nanocrystalline ceramics is observed over an order of magnitude (from ∼130 nm16 to 30 nm18). Furthermore, the existence of the breakdown itself is in question even for the same material (MgAl2O4).17,18 For example, one study showed MgAl2O4 ceramics with grain sizes as small as 7.1 nm with no Hall− B
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Figure 2. Mechanical properties of nanocrystalline MgAl2O4 ceramic samples using an indentation force of 100 mN. (a) Hardness as a function of grain size showing three different regions (I, II, III). Region I represents the conventional Hall−Petch relationship, where the hardness increases with decreasing grain size. There is a hardness limit to the conventional Hall−Petch relationship at the estimated critical grain size of 18.4 nm (inset), below which the hardness decreases with decreasing grain size (region II). Below a grain size of ∼5 nm, the hardness becomes insensitive to the grain size (region III). (b) Elastic modulus as a function of grain size, showing a decreasing elastic modulus with decreasing grain size. Within region II, the elastic modulus starts to deviate from the bulk elastic modulus reported in the literature. Similar to the hardness, the elastic modulus becomes insensitive to the grain size below dplateau ≈ 5 nm. The inset shows representative indentation force−displacement curves at the limit of the conventional Hall−Petch relationship (18 nm grain size sample), within region II (10.5 nm grain size sample), and within region III (3.6 nm grain size sample).
nanoscale. It should be noted that assuming diffusional flow as the only mechanism for grain boundary deformation is an oversimplification since other mechanisms can also accommodate grain boundary deformation. The models predict both the increase and decrease in strength, but the onset grain size of the Hall−Petch breakdown can be significantly affected by the strain rate as a result of the diffusional flow in the grain boundary phase.20 The predicted high strain rate sensitivity is also not consistent with observations at room temperature, limiting the utility of diffusion-based models in explaining negative Hall−Petch behavior in nanoceramics.21 Furthermore, the diffusion-based mechanisms utilized in the models to accommodate strain are generally not readily active in ceramic materials at room temperature.14 To explore the full range of behavior of ceramics and determine the intrinsic mechanical response (i.e., is there a Hall−Petch breakdown?), one must first produce fully dense ceramics with grain sizes in the range of tens of nanometers down to single-digit nanometers. Thus, here we investigate the extent of the Hall−Petch relationship and breakdown by performing instrumented indentation studies for fully dense nanocrystalline MgAl2O4 ceramics fabricated via an environmentally controlled pressure-assisted sintering (EC-PAS) technique with average grain sizes ranging from 37.5 nm down to 3.6 nm. While we show that a Hall−Petch breakdown is present in ceramics, our work also contends that a distinct underlying structure of nanocrystalline ceramics drives changes in hardness, elastic modulus, energy dissipation, and fracture behavior.
ranging from 2.5 to 22.5 nm. Batches were calcined at various temperatures to achieve different average particle sizes (see Methods and Figure S1 and Table S1 in the Supporting Information). After calcination and removal of all adsorbents, the nanopowders, without exposure to air, were pressed into green compacts and sealed in an evacuated metal capsule to prevent exposure to atmosphere. The sealed capsules were placed in a high-pressure assembly and loaded into a multianvil pressless split-sphere apparatus22 and pressed to 2 GPa at room temperature. The samples were sintered at various temperatures from 640 to 850 °C and cooled to room temperature before the pressure was released (see Methods and the Supporting Information for details). Transmission electron microscopy (TEM) of the nanocrystalline MgAl2O4 samples evaluated over a 2 μm × 2 μm area showed a uniformly distributed equiaxed polycrystalline nanostructure without porosity (Figures 1a,b and S2). The high-resolution TEM image in Figure 1c, which was taken at the edge of a wedgeshaped sample, shows no porosity and narrow grain boundaries at sub-10 nm grain sizes. The average grain sizes of the nanopowder and nanocrystalline MgAl2O4 ceramics were determined using a Halder−Wagner23 crystallite size analysis from X-ray diffraction (XRD) peaks (Figure 1d). All of the ceramic samples produced had a Halder−Wagner average residual strain measurement of zero. A more focused residual stress measurement on the 3.7 nm sample utilizing the sin2 ψ method24 also found an average residual stress near zero (see the Supporting Information for details). Relationship between Grain Size and Mechanical Behavior. The hardness measurement of the nanocrystalline ceramics reveals that a conventional Hall−Petch relationship holds for larger grain sizes but breaks down at a critical grain size (dcritical). Below dcritical, two distinct regions are observed: a slope change to negative, suggesting a negative Hall−Petch
RESULTS AND DISCUSSION Microstructure of Nanocrystalline MgAl2O4. Fully dense nanocrystalline MgAl2O4 ceramic samples were prepared by EC-PAS from nanoparticles with average particle sizes C
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Figure 3. Mechanical properties of nanocrystalline MgAl2O4 ceramic samples measured at different strain rates using constant-strain-rate indentation studies. (a) Hardness as a function of indentation strain rate for samples from each region (I, II, III). The inset shows the indentation displacement profiles to achieve different constant strain rates. The hardness is insensitive to the strain rate ranging across 3 orders of magnitude, from 0.05 to 35 s−1. This suggests that diffusion mechanisms do not play a significant role in the mechanical behavior of nanocrystalline ceramics at room temperature. (b) Elastic modulus as a function of indentation strain rate for samples from each region (I, II, III), showing that the elastic modulus is independent of the strain rate. This is expected since the elastic modulus is intrinsically insensitive to the strain rate, as it is related to the interatomic potentials.
region, and another distinct “plateau” region where the hardness is not influenced by the grain size. Thus, for the grain sizes studied, there are three distinct regions of interest: conventional Hall−Petch (I), negative Hall−Petch (II), and plateau (III). Figure 2a shows the hardness of the nanocrystalline MgAl2O4 as a function of grain size. The hardness of the 37.5 nm grain size ceramic is 20.9 ± 0.4 GPa, and the hardness increases to 22.1 ± 0.6 GPa as the sample grain size decreases to 18.0 nm. Within this region (region I), a conventional Hall− Petch relationship is valid with the standard linear expression, H = H0 + k / d , where the initial hardness H0 and slope k are 17.9 ± 1.4 GPa and 21.2 ± 7.1 GPa·nm1/2, respectively. Further reduction of the sample grain size from 18.0 to 4.6 nm results in a hardness decrease from 22.1 ± 0.6 GPa to 19.9 ± 0.7 GPa (region II). The linear expression constants H0 and k for the negative Hall−Petch relationship in region II are 25.2 ± 0.4 GPa and −11.4 ± 1.2 GPa·nm1/2, respectively. The intersection of the conventional and negative Hall−Petch linear expressions reveals that the maximum theoretical hardness of the nanocrystalline MgAl2O4 ceramics is 22.5 GPa at an estimated critical grain size (dcritical) of 18.4 nm (Figure 2a inset). Below a grain size of 4.6 nm, the hardness remains constant at approximately 19.9 ± 0.6 GPa (region III). Neither the conventional nor the negative Hall−Petch relationship holds below the plateau grain size (dplateau) of around 5 nm. We find that the grain size also contributes to the elastic modulus in our nanocrystalline ceramics. Figure 2b shows the elastic modulus of the nanocrystalline ceramics as a function of grain size. By instrumented indentation methods, the reduced modulus can be determined from the slope of the unloading curve (Figure 2b inset) and converted to the elastic modulus. The reduced modulus (Er) was converted to the elastic modulus (Esample) by means of a contact mechanics relationship, 1 Er
=
2 (1 − vindenter ) E indenter
+
2 (1 − vsample ) 24 , Esample
ratio (vsample) of 0.276.25,26 For the largest nanocrystalline ceramic grain size, the elastic modulus is 292 ± 2 GPa, which is near the reported value for fully dense microcrystalline MgAl2O4 (289 ± 10 GPa).27 As the grain size decreases, the elastic modulus decreases linearly from 292 ± 2 GPa (37.5 nm grain size) to 264 ± 3 GPa (4.6 nm average grain size) and remains constant at approximately 260 GPa below dplateau. Unlike the hardness profile, the elastic modulus gradually decreases until the grain size reaches the second critical grain size (dplateau ∼ 5 nm). A reduction in elastic modulus has also been observed in highly oriented nanocrystalline metallic thin films.28 In metals, negative Hall−Petch slopes are typically related to a change in plastic deformation mechanisms involving diffusion, such as Coble creep and diffusion-activated grain boundary sliding.5,6,12 Increases in grain boundary volume fraction promote diffusion-based plastic flow in nanocrystalline metals even at room temperature.29 However, no such behavior has been proposed or observed in hard ceramics at room temperature,30,31 even though prevailing models describing the mechanical response of nanocrystalline ceramics employ diffusion-based mechanisms.16,19 If the deformation behavior of nanocrystalline ceramics is associated with diffusion, one would expect the hardness to exhibit strain rate sensitivity since diffusion-based plastic deformation mechanisms are sensitive to changes in strain rate.31−34 Influence of Diffusion-Based Mechanisms. To test whether diffusion-based mechanisms are active in nanocrystalline ceramics at room temperature, constant-strain-rate indentations were performed at various indentation strain rates on a ceramic from each region: 37.5 nm (region I, conventional Hall−Petch), 8.0 nm (region II, negative Hall− Petch), and 3.7 nm (region III, plateau). For each sample, indentations were performed at strain rates of 0.05, 0.1, 1, and 34 s−1 (the maximum of our indenter). Figure 3 shows the hardness (Figure 3a) and elastic modulus (Figure 3b) as functions of strain rate. The loading profiles of the different strain rates are shown in the Figure 3 inset. While the average
with a diamond indenter elastic
modulus (Eindenter) and Poisson’s ratio (vindenter) of 1141 GPa and 0.07, respectively, and a polycrystalline MgAl2O4 Poisson’s D
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Figure 4. Influence of the crystalline and grain boundary volume fractions on the density. (a) Density of nanocrystalline MgAl2O4 ceramic samples as a function of grain size. The measured density is maintained through region I, which indicates that the density is independent of grain size. Below dcritical, density starts to decrease through regions II and III. The inset show a TEM micrograph of fully dense MgAl2O4 obtained from a sample in region III. (b) Representative illustration of a tetrakaidecahedron-shaped ideal grain with grain-face boundaries and triple junctions in region II (18 nm grain size). (c) Ideal total grain boundary, grain-face boundary, and triple-junction volume fractions as functions of grain size calculated for a grain boundary thickness of 1 nm. (d) Representative illustration of an ideal grain in region III (3.6 nm grain size). The density starts to decrease within region II when the grain-face boundary volume fraction reaches an inflection point at about 8 nm, signifying a shift in grain boundary characteristics from the grain-face boundary to the triple junction.
region (region III). The density starts to decrease at a grain size of approximately 8 nm (within region II), but the grain size does not coincide with dcritical or dplateau (Figure S7). To understand why the density starts to decrease at a grain size of about 8 nm, it is necessary to estimate how the total grain boundary volume fraction and the volume fractions of different types of grain interfaces (Figure 4b) change as functions of grain size in these nanocrystalline ceramics. For our analysis, we define the grain boundary volume as comprising two distinct regions, as shown in Figure 4b: the grain boundary volume where two adjacent faces meet is the grain-face boundary volume fraction (Figure 4b, colored blue), and the portion of the grain boundary where the edges from three adjacent grains meet is the triple-junction volume fraction (Figure 4b, colored white). The grain boundary, grain-face boundary, and triple-junction volume fractions were estimated as functions of grain size (Figure 4c) assuming that the grains
hardness of the different grain sizes follows the same trend shown in Figure 2 (showing three distinct regions of hardness), neither the hardness nor the elastic modulus is influenced as the strain rate is changed by over 3 orders of magnitude. The strain-rate-independent hardness measured for 8.0 and 3.7 nm grain size samples indicates that a diffusion-based mechanism does not dominate the negative Hall−Petch behavior within the tested strain rate regime. Therefore, the Hall−Petch breakdown and plateau of hardness presented in Figure 2a likely are not due to diffusion in nanocrystalline ceramics. Crystalline and Grain Boundary Volumes. Density measurements for the fully dense nanocrystalline ceramics show that the grain boundary has lower density than the crystalline grain interior. As shown in Figure 4a, the density of nanocrystalline ceramics in the conventional Hall−Petch region (region I) remains constant at 3.56 g/cm3 but decreases in the negative Hall−Petch region (region II) and through the plateau E
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Figure 5. Mechanical properties as functions of the grain boundary and triple-junction volume fractions: (a) hardness as a function of grain boundary volume fraction; (b) hardness as a function of triple-junction volume fraction; (c) elastic modulus as a function of grain boundary volume fraction; (d) elastic modulus as a function of triple-junction volume fraction. While the hardness (a) did not show a good correlation with the grain boundary volume fraction, the elastic modulus (c) decreased proportionally as the grain boundary volume fraction decreased. The hardness (b) and elastic modulus (d) sharply decreased as the triple-junction volume increased. These results indicate that the triplejunction volume has a more profound influence on the measured hardness and elastic modulus.
have an ideal tetrakaidecahedron shape35,36 with a grain boundary thickness (t) of 1.0 nm, as estimated from the high-resolution TEM image in Figure 1. The structural character determinations show that the ceramics have a total grain boundary volume fraction of approximately 0.2 at dcritical (18.4 nm); further reduction of the grain size to dplateau increases the total grain boundary volume fraction to approximately 0.5. While these are significant volume fractions, the strictly increasing trend of the total grain boundary volume fraction does not suggest a specific change in density at a grain size of ∼8 nm. The grain-face boundary volume fraction, however, initially increases in region I but reaches an inflection point in region II at a grain size of ∼8 nm. The triple-junction volume fraction is maintained at a minimal value in region I but begins to increase quickly in region II (near the grain-face boundary volume fraction inflection point) and throughout region III. Importantly, both the grain-face boundary and triple-
junction volume fractions are at their transition points at a grain size of ∼8 nm. The inflection point in the grain-face boundary volume fraction (Figure 4b) represents a shift in grain boundary character toward one more dominated by triple-junction volume (Figure 4d), since the two constitute the total grain boundary volume fraction. Our results suggest that the decrease in overall density with smaller grain size is strongly related to the grain boundary volume fraction, especially the triplejunction volume fraction. Prior work on thick nanocrystalline metallic coatings provides evidence that a lower density is linked to structural features.37 Our current work extends the understanding to ceramic-based nanocrystalline materials while also providing evidence for a definitive grain-size-dependent trend of density in bulk nanocrystalline materials. On the basis of the findings from the density measurements, dense nanocrystalline ceramics with grain sizes below ∼20 nm can be described as composite materials19,38 with two F
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Figure 6. Elastic (red ●), plastic (red ▲), and total energy (black ■) as determined from indentation force−displacement curves using an indentation force of 100 mN for nanocrystalline MgAl2O4 ceramic samples. (a) The energies are plotted as a function of grain size. Since the energy measurements were subjected to the maximum load and displacement of individual indents, the stored and dissipative energies were normalized by the total energy, as shown in (b). In region I, the stored energy increases (indicating a higher yield strength) while the dissipative and total energies decrease (indicating lower ductility), which is consistent with the conventional Hall−Petch relationship. Below the conventional Hall−Petch relationship limit (dcritical ≈ 18.4 nm), the normalized stored energy decreases while the normalized dissipative energy increases.
constituents: a crystalline grain interior and a very thin neighboring grain boundary, which includes grain-face boundaries and triple junctions that have lower hardness, modulus, and density.19 At large grain sizes, the crystalline grain structure dominates the microstructure, and deformation mechanisms associated with the crystalline grain structure dictate the hardness and elastic modulus. As the grain size becomes smaller, the grain boundary volume fraction increases, and mechanisms associated with the grain boundary begin to affect the overall deformation process and mechanical properties (Figure 5). When the grain boundary volume fraction is parsed, the hardness is seen to be more strongly influenced by the triplejunction volume fraction than the grain boundary volume
fraction. As shown in Figure 5a, the hardness slowly increases in region I, where the grain boundary volume fraction increases to approximately 0.16 (16%) and decreases only in region II and III. The region II−III transition occurs where the grain boundary volume fraction is greater than 0.5 (50%). In contrast, when the hardness is plotted as a function of triplejunction volume fraction, region I is very small, and the hardness starts to decrease when the triple-junction volume fraction is
