Large increase in fracture resistance of stishovite with crack extension less than one micrometer

The development of strong, tough, and damage-tolerant ceramics requires nano/microstructure design to utilize toughening mechanisms operating at different length scales. The toughening mechanisms so far known are effective in micro-scale, then, they require the crack extension of more than a few micrometers to increase the fracture resistance. Here, we developed a micro-mechanical test method using micro-cantilever beam specimens to determine the very early part of resistance-curve of nanocrystalline SiO2 stishovite, which exhibited fracture-induced amorphization. We revealed that this novel toughening mechanism was effective even at length scale of nanometer due to narrow transformation zone width of a few tens of nanometers and large dilatational strain (from 60 to 95%) associated with the transition of crystal to amorphous state. This testing method will be a powerful tool to search for toughening mechanisms that may operate at nanoscale for attaining both reliability and strength of structural materials.

due to fracture-induced amorphization 14,15 . Stishovite is a high-pressure phase of silicon dioxide (SiO 2 ) stable at pressures above 9 GPa, and metastable at ambient conditions 16 . While low-pressure phases, e.g., quartz and coesite, consist of networks of corner-sharing SiO 4 tetrahedra, stishovite possesses the rutile structure and is described as chains of edge-sharing SiO 6 octahedra. Stishovite is 60% more compact than quartz, and has the highest hardness (33 GPa) of any stable/metastable oxide under ambient temperatures 17 . Only diamond and cubic boron nitride are industrially used materials that are harder than stishovite. Hard materials with limited ability of plastic deformation tend to be brittle, then, the fracture toughness K 1C of a single crystal is 1.6 ± 0.3 MPa m 1/2 by the indentation method 18 . Recently, Nishiyama 14 found that a polycrystalline stishovite with submicron grain size had a fracture toughness of 7.9 MPa m 1/2 by the indentation fracture method (IF method). Furthermore, a nanocrystalline stishovite with the average grain size of 127 nm had fracture toughness of 13 ± 3 MPa m 1/2 . The IF method is highly questionable for estimating the accurate fracture toughness value 19 , but, the results indicate the fracture toughness increases with decreasing grain size. This discovery shows that the nanocrystalline stishovite attains both excellent hardness and toughness which are vital requirements for structural materials.
Transformation of metastable stishovite with six-fold coordination to the stable phase with four-fold coordination often occurs through intermediate amorphous state, because the Gibbs free energy of stishovite is higher than that of amorphous SiO 2 at any temperature above 0 K at ambient pressure 20 . The direct transition of crystal to amorphous state, which is analogous to melting, is triggered by heating at 1 bar, or decompression at room temperature. The terms, amorphization/vitrification and melting are used for the transition below and above the glass transition temperature. The amorphization accompanies a volume expansion up to about 100%. The amorphization is induced by fracture since the tensile stress, which is equivalent to negative pressure, can be very large at the crack tip. The formation of amorphous phase with the thickness of a few tens of nanometers was observed on the fracture surface by X-ray absorption near edge structure (XANES) spectroscopy 15 . In analogy to the transformation toughening of zirconia where metastable tetragonal phase transforms to stable monoclinic phase at crack tip 3,4 , the fracture-induced amorphization can be a toughening mechanism of stishovite. In order to understand why the nanometer-thick amorphous layer can increase the fracture resistance significantly, we have investigated R-curve behavior by using micro-cantilever beam specimens.

Results
The nanocrystalline stishovite with the average grain size of 128 nm was synthesized at 1473 K and at a pressure of 15 GPa by using a pure bulk silica glass as the starting material 15 . For comparison, we have also studied a pure silica (SiO 2 ) glass, a submicron alumina (Al 2 O 3 ), and 3 mol%-Y 2 O 3 stabilized tetragonal zirconia polycrystals (ZrO 2 , 3Y-TZP). The full detail of materials and specimen preparations are included in the supplementary information. The micro-cantilever beam specimens with notch were used to measure R-curve behavior in microscopic scale (Fig. 1a). The resulting load (P) -displacement (u) curves are plotted in Fig. 1b. Unstable fracture occurs when the material has a flat R-curve. The principal mechanism of rising crack resistance in alumina is grain-localized bridging, which is more pronounced for coarse-grained materials than for fine-grained materials 21 , then, the R-curve of fine-grained alumina is almost flat. As expected, pure silica glass and the alumina with submicrometer grain size showed unstable fracture. The stress intensity factor K 0 at the crack initiation was 0.8 MPa m 1/2 for pure silica glass, and 2.8 MPa m 1/2 for the alumina. The measured values were in good agreement with the fracture toughness reported by macroscopic tests. In the materials which have rising R-curves, the crack starts propagation at K 0 , and grows stably, and then, unstable fracture occurs at a critical crack extension 5 . The stable fracture was observed in 3Y-TZP and stishovite. The occurrence of unstable fracture is indicated by the "x" mark. The R-curves were calculated by the compliance method from P-u curves, and plotted in Fig. 2a. The dimensions of each specimen and elastic constants used in the analysis are summarized in Supplementary table S1. The crack growth resistance of stishovite increased from K 0 = 4 to 8 MPa m 1/2 with crack extension about one micrometer. The initial slope Δ K/Δ a was 7 MPa m 1/2 /μ m at K 0 . In 3Y-TZP, the crack growth resistance increased from K 0 = 2 MPa m 1/2 with the initial slope of 0.2 MPa m 1/2 /μ m. The initial steep rise in toughness has been found also in the silicon nitride (Si 3 N 4 ) with MgO and Y 2 O 3 as sintering aids by correcting the effect of notch root radius of macroscopic specimens 8 . The initial slope of stishovite was more than three times steeper than that of the silicon nitride toughened by bridging mechanisms 9 . The R-curves measured by micro-cantilever beam specimens show only the early part of rising R-curve, and cannot be used to determine the plateau value. But, they are advantageous to determine K 0 and the initial slope accurately. R-curve behaviors with crack extension of several tens of micrometers are shown for some ceramics in Fig. 2b. The extrapolation of R-curve of stishovite indicates that it can reach the stress intensity factor higher than 10 MPa m 1/2 , which is close to the fracture toughness measured by IF method. The plateau value of nanocrystalline stishovite is higher than that of the silicon nitride 8 and 2 mol%-Y2O3 stabilized tetragonal zirconia polycrystals (2Y-TZP) 22 . The fracture toughness of nanocrystalline Y-TZP materials increases with increasing grain size and decreasing dopant content. The fracture resistance of 3Y-TZP measured in air was lower than that of 2Y-TZP measured in vacuum due to the susceptibility to subcritical crack growth under humid atmosphere. Figure 3 shows SEM micrographs of fracture surfaces of nanocrystalline stishovite (Fig. 3b) and Y-TZP (Fig. 3c). The fracture surface of the nanocrystalline stishovite showed a rough morphology accompanied with many worm-like textures scattered over the whole surface. The dimensions of a "worm" are several tens and hundreds of nanometers in diameter and length, respectively. This feature of nanocrystalline stishovite was quite different from that of the intergranular fracture of Y-TZP. The worm-like textures observed on a micro-cantilever beam specimen of nanocrystalline stishovite had been also observed on a surface of mechanically crushed sample 15 . The selected-area electron diffraction patters of the worm-like textures had a halo, then, it showed the presence of amorphous phase on the fracture surface 15 .

Discussion
The effect of extrinsic toughening mechanism is described by a stress intensity change Δ K between the applied stress intensity factor and the stress intensity factor at the crack tip, which is assumed to be equal to the intrinsic toughness K 0 of the material. In the transformation toughening, the transformation zone develops upon crack extension by stress-induced transformation as illustrated in Fig. 4 3,4 . The transformation zone width h depends on the critical stress σ c for the phase transformation: The transformation zone width increases with crack extension Δ a, and then, saturates at Δ a* ≈ 5h. The extent of Δ K is proportional to h and the dilatational strain ε T accompanying transformation (i.e. the volume increase by the transformation): where V f is the volume fraction of transformed phase in the transformation zone. From equations (1) and (2), the toughness is expressed as: The sharply rising R-curve indicates that the transformation zone width h of stishovite is much narrower than that of Y-TZP. This is the evidence that the toughening by fracture-induced amorphization acts in nano-and submicron-scales. The width h of Y-TZP is estimated from the monoclinic phase content measured by Raman microscopy: 4 μ m for the grain size of 900 nm, and 2 μ m for the grain size of 140 nm 22 . The critical stress σ c for transformation determines the transformation zone width h. σ c is estimated to be about 1.3 GPa for tetragonal-monoclinic transition in 2Y-TZP 23 .
As well as stishovite, dense high pressure crystalline silicates, where silicon is in octahedral coordination, transform under tension to a low-density glass composed of tetrahedral silicon. The toughening by fracture-induced amorphization can occur in a number of such materials, for example, CaSiO 3 and MgSiO 3 perovskites. In the molecular dynamic simulation, high-pressure phases of CaSiO 3 and MgSiO 3 started amorphization at tensile stress of 18 GPa at 300 K 24 . This result suggests that the critical transformation stress of stishovite is more than 10 times higher than that of Y-TZP. From equation (1), the transformation zone width of stishovite is estimated to be a few tens of nanometers. This estimate agrees with the experimental observation by the surface-sensitive Si-K XANES measurement 15 . The initial slope of R-curve Δ K/Δ a is a function of parameter α ε σ = / EV K f T c 0 , and increases linearly with α approximately 25 . While the fracture toughness of stishovite was about two times larger than that of Y-TZP, Δ K/Δ a of stishovite was 30 times larger than that of Y-TZP. The slope of stishovite is very steep, because it is proportional to the product of ε T and σ c , both of which are about 10 times larger in stishovite than in Y-TZP. Steep initial slopes are desirable for R-curves because materials that achieve high toughness at small crack length also achieve high strength.
The increase of fracture toughness with decreasing grain size of stishovite is opposite to the behavior expected from the crack bridging effect and the results observed in transformation toughening of Y-TZP 26 . But, the grain size effect on fracture toughness cannot be separated from other factors such as internal microstrain and dislocation densities 14 . The microstrain arose during synthesis of stishovite  which crystallized from bulk glass at 15 GPa. As the synthesis temperature decreases, the grain size decreases, and the microstrain increases.
The amorphization of stishovite is dominantly a heterogeneous nucleation and growth controlled process 27,28 . The transformation proceeds by nucleation of amorphous at defects, such as grain boundaries, surfaces, and dislocations 27 . Many dislocations were observed in fine stishovite grains by TEM observation 11 . The origin of dislocations is unknown. One possible explanation is that local deformation occurred during the phase transformation from glass to stishovite at the synthesis pressure 29 . Dislocation density decreased with increasing the synthesis temperature, and then, grain size. It is possible that the fine grain size promotes the amorphization process due to high defect density. Indeed the amount of transformed phase on the fracture surface and the fracture toughness increased with decreasing grain size 15 . The high density of nucleation sites increases the volume fraction of amorphous phase in the transformation zone, which in turn, enhances the crack shielding effect Δ K in equation (3).
The relation between fracture strength and the critical transformation stress is illustrated in Fig. 5. When the strength is limited by failure from pre-existing flaws, the strength is proportional to fracture toughness. From equation (3), the toughness, and then, strength increase as σ c decreases. However, as the critical stress decreases, the strength is limited by σ c due to the transformation induced deformation 4,30 . The achievement of peak strength by optimizing σ c is the target in the material design of zirconia. In fracture-induced amorphization of stishovite, the volumetric transformation strain ε T is from 60 to 95%, while it is about 4% in the tetragonal-monoclinic transition of zirconia. From equation (3), the increase of ε T enhances the fracture toughness at a given σ c as illustrated in Fig. 5. This effect shows that the peak strength of stishovite can be higher than that of zirconia ceramics. The very narrow transformation width and high critical stress are effective in toughening of stishovite due to large ε T .
In summary, the fracture resistance measurement by using micro-cantilever beam specimens revealed that there exists a toughening mechanism which can operate at nano-scale. The narrow transformation zone in fracture-induced amorphization of nanocrystalline stishovite was very effective to increase the fracture resistance by a crack extension of less than one micrometer. This finding will provide a motivation to search for other nano-scale toughening mechanisms in many structural materials, such as ceramics, composites, nanocrystalline materials, and also, synthetic materials with hierarchical architecture.

Methods
Material. Nanocrystalline stishovite samples were synthesized using a Kawai-type multi anvil apparatus 15 . The dimensions of the starting material (pure SiO 2 glass) are 2.5 mm in diameter and 1.4 mm in height. Load was applied first up to 15 GPa under room temperature, and the temperature was increased rapidly from ∼ 450 °C to 1473 K within 30 s at constant load. The temperature was held constant for 30 min. The decompression (3 h) was started when the temperature was decreased to ∼ 450 °C. The nanocrystalline stishovite contained numerous dislocations.
Mechanical testing. The loading of the cantilever beams is carried out in air using a mechanical testing machine designed for micro-sized specimen. The basic concept of this machine is described in ref. 10 . The displacement was applied by a piezoelectric device at a displacement rate of 10 nm/sec. The load was applied by using a spherical diamond indenter with radius of 2.5 μ m. The machine compliance was 0.023 μ m/mN. Load and displacement data were acquired with the sampling interval of 0.1 second.
Stress intensity factor and compliance of micro-cantilever beam specimen. The stress intensity factor (mode I) K I of a side-grooved cantilever beam specimen in Fig. 1a was calculated by the extended finite element method (XFEM) using the commercial solver Abaqus 6.13. The stress intensity factors were determined by path-independent contour integral. Side grooves were introduced for straight crack propagation under the presence of K II component in cantilever beam test. Three-dimensional isotropic elastic model was employed to simulate the cantilever beam specimens with side grooves (H: L: W: B = 1:3:1:1, B N = 0.8B), where H was the distance between the notch and the root of the cantilever beam. The ratio K II /K I was less than 5%. The critical specimen thickness limit for plane-strain fracture toughness is given as 12 The yield stress σ y is estimated from Vickers hardness H V according to Tabor's approximation The critical specimen thickness is 1.25 μ m for stishovite. We used specimens with the thickness B of 20 μ m for reliable fracture toughness tests. Furthermore, side-grooves increases the level of constraint, then, promotes a more uniform stress state according to ASTM E399-90.
The stress intensity factor of cantilever beam specimen with side-grooves was calculated, and expressed in the following equation: where P is the applied force, L is the distance between the notch and the loading point, a is the crack length, W is the cantilever width, B is the thickness, and B N is the minimum thickness measured at the root of the side grooves. The shape factor can be used for 0.2 ≤ a/W ≤ 0.7: The compliance C ≡ u/P is expressed as: where E is Young's modulus and v is Poisson's ratio. The dimensionless shape factor for the compliance f c is given as: The micro-cantilever beam test is carried out by controlling the displacement u′ of the actuator: where u = CP is the displacement of the specimen, and C machine is the machine compliance. The crack length is determined by the compliance method using equations (7) and (10).