Correlation between the local stress and the grain misorientation in the polycrystalline Al₂O₃ measured by near-field luminescence spectroscopy

For the proper operation management of structural components, the local residual stress in ceramics has been widely assessed by optical spectroscopy due to its capability to sense stress with high spatial resolution. Although Raman spectroscopy is a versatile tool for stress measurement available for various materials,^1–3) the photoluminescence from Al₂O₃ offers a much sensitive stress detection owing to its strong signals, which has been applied to not only the monolithic bulk Al₂O₃,^4–7) but also thermally grown Al₂O₃ films,^8–12) and composite materials.^13–16) The luminescence technique makes use of the linear relationship between stress and the shift of the luminescence R-lines of Al₂O₃. In addition, polarization characteristics of the R-lines have also been studied for their possible use for assessing the crystallographic orientation of Al₂O₃.^17,18)

In conventional luminescence spectroscopy using lens-based optical systems, the spatial resolution is still restricted to ∼1 µm owing to the diffraction limit of light. On the other hand, a near-field technique enables spectroscopic investigation with a spatial resolution of 30–500 nm.^19–23) There is thus significant interest in the application of this method to sensing stress in a narrower region, such as stress within grains of polycrystalline Al₂O₃. Recent technological improvements of lens-based luminescence detection system also enable the detailed analysis of each stress component inside Al₂O₃.^24,25) Attempts using such a system, however, were made on the polycrystalline Al₂O₃ with an average grain size larger than 10 µm. Moreover, in many luminescence studies of stress inside grains of Al₂O₃, the properties of emission anisotropy have not been examined despite the fact that the local polarization characteristics relevant to the individual grain orientation are important for the effect of anisotropic thermal expansion on stress, which causes fracture.^26–28) To date, the evaluation of grain-anisotropy-induced stress has been limited to numerical stress simulation using grain-orientation-limited polycrystalline ceramics as a model.^{26,27,29–31)} Simultaneous high-resolution imaging of stress and grain orientation by near-field luminescence spectroscopy will provide important insights for further advancements of local stress simulation.

In previous studies, luminescence imaging of Cr³⁺-doped polycrystalline Al₂O₃ was performed under near-field excitation with the resolutions of 300–500 nm.^22,23) We obtained typical features of grain-dependent luminescence anisotropy as well as a smooth spatial distribution of a stress-induced peak shift. Here, using the obtained luminescence images, we statistically investigate the contribution of grain misorientation to stress. The grain misorientation is evaluated from the deviation of data of the luminescence anisotropy calculated within a certain spatially limited area. We demonstrate the analysis of correlation between two images of the misorientation parameter and peak shift.

The properties of spectra of near-field excited luminescence from Al₂O₃ and its spatial images are depicted in Fig. 1.²²⁾ The imaging was conducted for a single-crystal α-Al₂O₃ with a Cr content of 0.17 mass % and polycrystalline α-Al₂O₃ with a Cr content of 0.7 mass % (average grain size: ∼5 µm). Figure 1(a) shows examples of luminescence spectra coming from the $(11\bar{2}0)$ plane of the single-crystal Al₂O₃ and polycrystalline Al₂O₃. The intensity ratio of R₁ to R₂ lines, I_R2/I_R1, varies with sample rotation angle. I_R2/I_R1 reflects the angular relationship between the polarization direction of spectrometer system sensitivity and the crystallographic orientation of Al₂O₃.¹⁷⁾ R-lines have two polarization components, the electric field parallel to the [0001] direction (π polarization) and that perpendicular to the [0001] direction (σ polarization). I_R2/I_R1 differs between two components, which is responsible for the angular dependence of I_R2/I_R1 from the [0001] direction. Therefore, the present result [Fig. 1(a)] shows the existence of an angular dependence on the detection sensitivity of the near-field system.

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Indeed, in the image of polycrystalline Al₂O₃ [Fig. 1(b)], I_R2/I_R1 varies from grain to grain owing to the heterogeneity of the [0001] direction. The location where I_R2/I_R1 sharply changes coincides well with the grain boundary. This suggests that the rough tendency of grain anisotropy and resultant misorientation between grains can be detected on the basis of I_R2/I_R1 using our near-field system. I_R2/I_R1 is insensitive to the variation of Cr concentration, when compared with the absolute intensities of the R₁ and R₂ lines.²²⁾ The variation of I_R2/I_R1 values within a grain is ∼0.02 at maximum, which may come from the spatial variation of Cr concentration. This Cr-induced I_R2/I_R1 variation is smaller than the typical values of grain-anisotropy-induced variation (0.02–0.07) in the present measurement, suggesting that I_R2/I_R1 is an effective parameter for examining grain anisotropy. For a sample with greater Cr concentration fluctuation, however, a large error of orientation estimation occurs; hence, a systematic investigation of I_R2/I_R1 for different Cr concentrations is necessary to discriminate between the contributions of Cr concentration and crystal orientation. Since I_R2/I_R1 correlates with the angle between the near-field polarization, E_NF, and the [0001] direction, it is possible to determine the [0001] direction by measuring I_R2/I_R1 for different E_NF fields in more than two directions.

The R₂ peak shift distribution is shown in Fig. 1(c). When the luminescence-detected region can be assumed to be an untextured polycrystalline Al₂O₃, under the plane stress condition, the relationship between the peak shift of R-lines and the stress on the surface is expressed as

$$\begin{equation} \Delta \nu = \frac{2}{3}\Pi_{ii}\bar{\sigma}, \end{equation} \tag{ 1 }$$

where Δν is a shift of R₂ peak frequency from its stress free value, which is assumed to be the average peak frequency over the entire mapped area, Π_ii is the piezospectroscopic constant of polycrystalline Al₂O₃, and $\bar{\sigma }$ is the average stress of two in-plane stress components, σ₁₁ and σ₂₂, in the orthogonal coordinate systems on the surface [i.e., $\bar{\sigma } = (\sigma _{11} + \sigma _{22})/2$]. The coefficient Π_ii for the R₂ line is Π_ii = 7.61 cm⁻¹ GPa⁻¹.⁷⁾

The luminescence-detected region for the near-field illumination is 300 nm, which is smaller than the average grain size of polycrystalline Al₂O₃. As a result, the detected luminescence strongly reflects the anisotropy of a single or few grains of Al₂O₃; therefore, the effect of anisotropy of piezospectroscopic constant should be taken into account when estimating the stress from Δν. However, for convenience, by assuming that the anisotropy of the piezospectroscopic constant is negligible, the variation of $\bar{\sigma }$ is roughly estimated to be ∼320 MPa using Eq. (1), which is comparable to those derived by stress simulations.^27,29,30)

It can be seen that Δν changes sharply at some grain boundaries. Detailed profiles of I_R2/I_R1 and Δν for lines A and B in Fig. 1(c) are shown in Figs. 2(a) and 2(b), respectively. In the line A, a small difference in I_R2/I_R1, Δ(I_R2/I_R1) ∼ 0.01, is found between grains, where the change in Δν is slight, ∼0.05 cm⁻¹. On the other hand, at the grain boundary in line B, I_R2/I_R1 shows a marked variation of more than Δ(I_R2/I_R1) = 0.04 and the change in Δν is also large, ∼0.2 cm⁻¹. These results imply that the degree of difference in I_R2/I_R1 at the grain boundary is related to the magnitude of Δν change, although some exceptional places showing the poor relationship of I_R2/I_R1 and Δν exist.

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From the viewpoint of the effect of grain anisotropy on stress, we carried out correlation analysis between Δν and I_R2/I_R1. The stresses concentrated at the boundary between two grains are related to the difference in the thermal expansion coefficient mismatch due to the difference in crystallographic orientation, i.e., misorientation. Near the triple junction of a grain boundary, the magnitude of variation in orientations for three grains has a relevant effect on stress. As a parameter indicating such nature of misorientation, we employ the standard deviation, s, of I_R2/I_R1 within a certain area of a circle with the radius r given by

$$\begin{align} s & = \frac{1}{r\sqrt{\pi}}\left\{\iint_{i}\left[\frac{I_{\text{R2}}(x,y)}{I_{\text{R1}}(x,y)} - \mu\right]\,dx\,dy\right\}^{0.5}, \end{align} \tag{ 2 }$$

$$\begin{align*} \mu & = \frac{1}{r^{2}\pi}\int_{i}\frac{I_{\text{R2}}(x,y)}{I_{\text{R1}}(x,y)}\,dx\,dy, \end{align*}$$

where i is the calculation area of s and μ is the average I_R2/I_R1 inside the area i.

Figure 3 shows a schematic representation of s at different positions. In the area inside the grain (area I), s remains low because of the constant I_R2/I_R1, whereas s increases when the calculation area includes a grain boundary with a different I_R2/I_R1 (area II). We evaluate the correlation between s determined in the circle area centered at the position $(x_{i},y_{i})$ and Δν at the same position $(x_{i},y_{i})$. Figure 4 shows Δν–s plots for all elements in the luminescence-detected area in the case of r = 0.5 µm. The normalized s values, 〈s〉, are distributed in the range of ∼0.8. This 〈s〉 varying range is extended, when increasing r, because the I_R2/I_R1 variation affects s only under the condition that the characteristic length of the spatial fluctuation of I_R2/I_R1 is equal to or less than r. Therefore, for a smoothly varied I_R2/I_R1, s is higher for a large r than for a small r. Many of the elements are highly concentrated in the lower 〈s〉 region (〈s〉 ≦ 0.4), because the s circles are more likely to be taken within a grain than in the area including the grain boundary. In the higher 〈s〉 region, the data are concentrated in the higher range of |Δν| (|Δν| ∼ 0.4 cm⁻¹). We classify the Δν groups according to the magnitude of 〈s〉 to assess the Δν histograms, as shown in Fig. 5. In lower 〈s〉 regions, Δν of ∼0 cm⁻¹ accounts for a large fraction of data, indicating that the stress is reduced at the area with less misorientation away from the grain boundary. On the other hand, the area corresponding to Δν of ∼0 cm⁻¹ is explicitly reduced with increase in 〈s〉 and the dispersion of Δν is pronounced. At highest 〈s〉 region [Fig. 5(e)], Δν distribution is eventually split into two peaks corresponding to high tensile and compressive stresses. This tendency is qualitatively consistent with the stress generation in the region at greater misorientation, demonstrating that the stress can be discriminated by 〈s〉.

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The correlation of |Δν|–〈s〉 is also highlighted clearly in the two-dimensional (2D) histogram of |Δν| and 〈s〉 (Fig. 6). By deducing the modal center by linear least-square regression analysis of the 2D histogram, we determined coefficients as indexes for the magnitude of the correlation: the slope (d|Δν|/d〈s〉), the determination coefficient (R), and the variable range of modal center (δ|Δν|_max). Table I shows the 3 coefficients at different r values. All the coefficients increase with the decrease in r and they are maximum at r = 0.15 µm. This parameter r provides the spatial distance from the grain boundary where the effect of grain misorientation on stress is prominent. r is crucially affected by the average grain size of Al₂O₃ and the spatial resolution of luminescence measurement.

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In this study, the s calculation is based on the I_R2/I_R1 variation reflecting the angle difference between the near-field polarization direction E_NF and the [0001] orientation of Al₂O₃. It is expected that the s calculation based on the direct difference in crystallographic orientation between neighboring grains would considerably improve the correlation properties, i.e., increases in d|Δν|_max and R. Future works including a more accurate grain orientation analysis by measuring I_R2/I_R1 at different E_NF fields are required for advanced correlation analysis.

Despite the wide use of optical spectroscopy as a sensing tool for stress, the crystallographic orientation and crystallinity in ceramics, and their relationships, which are particularly important in a local area, have rarely been examined by this method, owing to its poor spatial resolution or detection sensitivity. In this study, on the basis of the highly resolved images obtained by near-field spectroscopy, we have investigated the relationship between stress and crystallographic orientation by focusing on two characteristics of the luminescence spectra of polycrystalline Al₂O₃. At the grain boundaries where the mismatch of R-line intensity ratio is large, a large R₂ peak shift corresponding to the stress localization is found. By incorporating the concept of the crystal misorientation parameter using the data for intensity ratio, the stressed and unstressed regions are successfully discriminated. The apparent correlation between the magnitude of stress and the misorientation parameter indicates that the grain-anisotropy-induced stresses are successfully detected by our image analysis method combined with near-field luminescence measurement. Our methodology has the further potential applications in local stress analysis of fractured surfaces in ceramics where the crystal structure is damaged.