A Subgrain‐Size Piezometer Calibrated for EBSD

We calibrate a subgrain‐size piezometer using electron backscatter diffraction (EBSD) data collected from experimentally deformed samples of olivine and quartz. Systematic analyses of angular and spatial resolution test the suitability of each data set for inclusion in calibration of the subgrain‐size piezometer. To identify subgrain boundaries, we consider a range of critical misorientation angles and conclude that a 1° threshold provides the optimal piezometric calibration. The mean line‐intercept length, equivalent to the subgrain‐size, is found to be inversely proportional to the von Mises equivalent stress for data sets both with and without the correction of Holyoke and Kronenberg (2010, https://doi.org/10.1016/j.tecto.2010.08.001). These new piezometers provide stress estimates from EBSD analyses of polymineralic rocks without the need to discriminate between relict and recrystallized grains and therefore greatly increase the range of rocks that may be used to constrain geodynamic models.


Introduction
Quantitative constraints on the stresses associated with past deformation events in the ductile portion of the lithosphere are key to developing and testing geodynamic models. One method of estimating past stress magnitudes is to measure microstructural elements that can be related to stress through experimental calibrations, a technique known as piezometry. Paleopiezometry has provided key estimates of the strength of continental (e.g., Kohlstedt & Weathers, 1980;Stipp et al., 2002;Weathers et al., 1979) and oceanic (e.g., Hansen et al., 2013;Jaroslow et al., 1996;Speckbacher et al., 2011;Warren & Hirth, 2006) fault zones, yielded insight into the mechanisms of localization in outcrop-scale shear zones (e.g., Austin et al., 2008;Gueydan et al., 2005;Haertel & Herwegh, 2014;Linckens et al., 2011;Skemer et al., 2010;Skemer et al., 2013), and enabled tests of the extrapolation of laboratory-derived rheological laws to geological conditions (e.g., Behr & Platt, 2011;Hansen & Warren, 2015;Hirth et al., 2001;Stipp et al., 2002;Wex et al., 2019). Thus, paleopiezometry is an essential tool for field-based quantitative investigations of the mechanical behavior of the lithosphere.
Like grain size, subgrain size varies systematically with differential stress during steady-state deformation (Luton & Sellars, 1969;Twiss, 1986). Piezometers based on subgrain size are advantageous for two reasons: (1) subgrain sizes are not modified by grain-boundary pinning (Hansen & Warren, 2015;White, 1979), and (2) subgrains can form over relatively small strain intervals (≤10%, Biberger & Blum, 1992;Ross et al., 1980) in both relict and recrystallized grains. Therefore, subgrain-size piezometry can be applied to samples subject to small strains and obviates the need to identify grains that formed by dynamic recrystallization. Subgrain-size piezometry therefore provides a tool to evaluate past stress experienced by a broader range of rocks, including polymineralic rocks (e.g., Hansen & Warren, 2015).
Relationships between applied stress and subgrain size measured using optical microscopy or transmission electron microscopy (TEM) have been established for quartz (Mercier et al., 1977), olivine (Durham & Goetze, 1977;Goetze, 1975;Karato et al., 1980;Toriumi, 1979), and calcite (Friedman & Higgs, 1981;Platt & de Bresser, 2017). In addition, subgrain-size piezometers applicable to multiple minerals have previously been proposed (Shimizu, 1998;Twiss, 1986). However, different methods for measuring microstructural features have different detection limits, which can lead to systematic offsets between piezometric relationships for the same mineral (Cross et al., 2017;Hansen et al., 2011). Currently, no subgrain-size piezometers have been calibrated for data collected by electron backscatter diffraction (EBSD), despite the relatively simple sample preparation, rapid data acquisition, and precision in misorientation angles of better than ±0.3°( Figure 4 in  associated with this technique.
Here, we present EBSD measurements of subgrain size in experimentally deformed olivine and quartz samples and derive a single piezometric relationship for both minerals. We explore the sensitivity of the piezometric calibration to the lower cutoff of subgrain-boundary misorientation angles and provide tests for the number of intercept lines, the number of grains, and the step size required to accurately capture the subgrain size. The resulting piezometer greatly extends the range of rocks for which EBSD data can provide quantitative stress estimates.

Sample Description
Samples from three sets of experiments conducted on quartz and olivine were analyzed. The experimental conditions associated with each sample can be found in the supporting information (Table S1). Each stress reported in Table S1 is the von Mises equivalent stress (hereafter referred to solely as the equivalent stress, σ) experienced by the sample, which is assumed to be associated with the microstructures observed at the end of the experiment. All samples were deformed in a regime in which deformation is rate limited by the motion of dislocations.

Quartz Experiments
Quartz samples prepared from Black Hills Quartzite were deformed in two sets of experiments, denoted Qz-1 and Qz-2. Qz-1 samples were deformed in a Griggs apparatus at Brown University (Stipp & Tullis, 2003). Qz-2 consists of samples deformed in a Griggs apparatus either at Texas A&M University or at Brown University (Holyoke & Kronenberg, 2013;Holyoke & Tullis, 2006). Both sets of experiments were conducted at a confining pressure of 1.50-1.56 GPa and temperatures in the range 800-1,100°C. Samples included in the calibration lay within Recrystallization Regimes 2 and 3 as defined by Hirth and Tullis (1992), as the relationship between subgrain size and stress appears to change in Regime 1 (cf. Stipp & Tullis, 2003).
Qz-2 samples were subjected to either general shear or axial compression at constant equivalent strain rates of 1.6 × 10 −6 to 1.15 × 10 −5 s −1 . Final shear strains ranged from 50% to 610%. The only experiment deformed in axial compression in Qz-2, TMQ-7, reached a final axial strain of 15.6%. All Qz-2 samples, excluding W-1105, were deformed to their yield point. The ability of subgrains to keep pace with the stress after very little strain means the lack of steady-state conditions should not influence the piezometric calibration. Therefore, the final flow stress, converted into the equivalent stress, was taken as the stress associated with the final microstructures.
Correction of the mechanical data for the friction on the σ 1 piston was performed in Qz-1 according to the procedures of Gleason and Tullis (1995) and Stipp and Tullis (2003) and in Qz-2 according to the procedures of Holyoke and Tullis (2006). We then compare two sets of stresses to subgrain size, either with or without a second friction correction described by Kronenberg (2010, 2013) to account for increased friction due to a Poisson effect on the load column during loading of the sample.

Olivine Experiments
Data from olivine samples deformed in a number of different axial compression and torsion experiments were compiled to form the "Ol" data set (Hansen et al., 2011(Hansen et al., , 2012Pommier et al., 2015;Tasaka et al., 2016;Tasaka, Zimmerman, Kohlstedt, Stünitz, et al., 2017). Polycrystalline Fo 90 and Fo 50 samples were fabricated from San Carlos olivine (Hansen et al., 2011) or from a combination of oxide powers and San Carlos olivine , respectively. Experiments were conducted in a servo-controlled, internally heated, gas-medium apparatus (Paterson, 1990) at the University of Minnesota at a confining pressure of 300 MPa and temperatures of 1,150-1,250°C. Equivalent strain rates were 9 × 10 −6 to 1.48 × 10 −3 s −1 . Stress was measured with an internal load cell and controlled to ±1 MPa for axial compression experiments (Hansen et al., 2011) and ±2 MPa for torsion experiments (Tasaka et al., 2016). Stresses measured during torsion experiments were converted to equivalent stress following Paterson and Olgaard (2000). In axial compression experiments, maximum axial strains reached up to 20%. In torsion experiments, except PT-0966, strain was recorded as the shear strain at the outer radius and reached up to 880%. For sample PT-0966 the shear strain, recorded as 500%, was calculated from an originally vertical crease in the jacket formed during the initial pressurization of the sample.

Acquisition of Microstructural Data
For EBSD analyses, samples deformed in axial compression were cut parallel to the cylindrical axis (Hansen et al., 2011;Holyoke & Kronenberg, 2013;Stipp & Tullis, 2003). For samples deformed in torsion, tangential sections were chosen for analysis (Hansen et al., 2012;Tasaka et al., 2016;Tasaka, Zimmerman, Kohlstedt, Stünitz, et al., 2017). In the direct shear geometry, sections parallel to the shear direction and perpendicular to the shear plane were chosen (Holyoke & Tullis, 2006). The sectioned surfaces were prepared by polishing with diamond lapping films or suspensions with decreasing grit sizes from 30 to 0.05 μm and most samples were finished by polishing with 0.03 or 0.04 μm colloidal silica.
EBSD data were collected with an FEI Quanta 650 FEG E-SEM in the Department of Earth Sciences, University of Oxford, equipped with Oxford Instruments AZtec (Version 3.3) acquisition software and a NordlysNano EBSD camera. Samples were tilted at 70°and mapped in low vacuum (50-60 Pa, H 2 O) at accelerating voltages of 20-30 kV with step sizes of 0.1-1.0 μm. Noise reduction was performed using Oxford Instruments Channel5 software. Nonindexed points with ≥6 indexed neighbors within the same grain were assigned the average orientation of their neighbors. In addition, pixels in images of quartz that were systematically misindexed due to pseudosymmetry were corrected by applying a rotation of 60°around [0001] (Trimby et al., 2002).
For our piezometer, subgrains were measured using the line-intercept method. This method compares adjacent pixels on evenly spaced lines transecting the sample (Figure 1a). Along each line, misorientations greater than a specified angle, the critical misorientation angle, are detected, and the intercept length between misorientations is recorded. As such, both subgrain boundaries and grain boundaries are captured in this measurement. The number of intercept lines is increased until the measured mean line-intercept length stabilizes (±2.5%), thus ensuring the number of intercepts is sufficient to accurately estimate the mean value.
The line-intercept method is preferred over other grain-size measurement techniques because it includes nonclosing subgrain boundaries. When analyzing subgrain boundaries with small misorientation angles, it is important to include such primitive structures because subgrain boundaries do not always fully enclose an isolated region (e.g., Figures 1a and 2). Additional benefits compared to alternative area-based subgrain-size measurement include lower sensitivity to changes in step size (Mingard et al., 2007;Valcke et al., 2006), faster processing times (Humphreys, 2001), and lower sensitivity to anomalous clusters of data points missed during postprocessing (Hansen et al., 2011).
To obtain sample averages, we use the arithmetic mean of the line-intercept lengths. This approach provides the mean spacing between all boundaries with misorientation angles above the chosen critical value. An alternative average, the geometric mean, is more sensitive to misindexed pixels and variations in step size (Mingard et al., 2007). In previous studies, a stereological correction was commonly applied to convert the mean line-intercept length from a 2-D section to the mean grain diameter in 3-D (Hansen et al., 2011;Hansen & Warren, 2015;Underwood, 1970, pp. 80-93;Valcke et al., 2006). As the choice of 3-D correction adds an additional layer of uncertainty to the data processing, we avoid any correction and simply use the mean line-intercept length, λ, as the microstructural length scale in our piezometric calibration.

Sensitivity Tests
As EBSD maps only examine a portion of a deformed material, we devised two tests to assess whether a mapped area is representative of the bulk sample. These tests examine the effect of the size and spatial resolution of the map on the measured values of mean line-intercept length. MATLAB® scripts to perform these tests are presented in the supporting information (Text S1).
As increasing the step size of an EBSD map leads to overestimates of grain size (Cross et al., 2017;Humphreys, 2001), we tested the influence of step size on measured subgrain size. We evaluated the spatial resolution by implementing the same step-size analysis test as Cross et al. (2017). In this procedure, the resolution of the map is artificially reduced (i.e., the effective step size increased) by selecting subsets of points in the map on a regular grid. The mean line-intercept length is then measured and compared to the mean line-intercept length associated with the smallest step size, λ best . We refer to the ratio of λ to λ best as the intercept variation factor. The spatial resolution is deemed sufficient if the measured mean line-intercept length is not sensitive to the effective step size. For each sample in Figures 1d and S2, decreasing the pixels per intercept length (defined as the λ best divided by the effective step size) had an insignificant effect on the mean line-intercept length. The presence of an asymptote at an intercept variation factor of 1 is evidence that that step size is small enough to capture the mean line-intercept length.
In the second test, we evaluated the size of the map relative to the mean line-intercept length, that is, whether a sufficient number of intercepts were measured for their mean value to be representative of the sample mean. We propose that, rather than being fixed (Humphreys, 2001(Humphreys, , 2004Valcke et al., 2006), the number of required subgrains (or intercept lengths) is likely to depend on the variance of the true subgrain-size distribution. Therefore, for each map, we tested the effect of map area on the mean line-intercept length by measuring the mean line-intercept length from a centered subarea of the original map. Initially, this subarea was 1% of the size of the original map. We then measured the mean line-intercept length as the subarea was progressively increased in size. As the size of the subarea increases, the mean line-intercept length should asymptotically approach the mean for the entire map, indicating the full map area is sufficiently large. Examples of the area-analysis technique are presented in Figures 1e and S3. Samples that demonstrated a strong dependence of mean line-intercept length on the size of the subarea (e.g., Figure S4) were mapped multiple times. The mean line-intercept length was then calculated as the average intercept length across all the maps to ensure representative measurement.

Results
To identify a general piezometric relationship applicable to both quartz and olivine, the equivalent stress and the mean line-intercept length were normalized by the shear modulus μ and the Burgers vector b, respectively. We then explored the effect of the choice of critical misorientation angle on mean line-intercept length, testing angles ranging from 1°to 10°(e.g., Figure 1c). We did not evaluate critical misorientation angles below 1°, the minimum angle that has previously been used to define a subgrain boundary (e.g., Ambrose et al., 2018;Hansen & Warren, 2015). Reanalysis of a number of our maps (e.g., Figure S6) with high-angular resolution EBSD (Wallis, Parson, & Hansen, 2019;Text S3) found the resolution adequate for subgrain-size piezometry. We did not evaluate critical misorientation angles above 10°as we consider this value to be the lower bound of the definition for a high-angle grain boundary (e.g., Karato, 2012, page 243;Trimby et al., 1998). For all values of the critical misorientation angle, in all three data sets, the mean line-intercept length exhibits a power-law relationship with the equivalent stress ( Figure S5). The three data sets coincide best at lower critical misorientation angles. The discrepancy between the quartz data sets and the olivine data set increases as a function of increasing critical misorientation angle. Therefore, we used a critical misorientation angle of 1°in our calibrations.
We fit a power law equation using least squares linear regression on the logarithmic transformation of mean line-intercept length against equivalent stress, yielding the following equations: Equations 1 and 2 represent piezometers with and without the calibration of Holyoke and Kronenberg (2010) discussed above in section 2.1.1, respectively. Uncertainties are reported as 95% confidence intervals. Here we use b = 0.51 nm, 0.50 nm, or 0.50 nm and μ = 42.0 GPa, 77.8 GPa, or 62.6 GPa for quartz, Fo 90 , and Fo 50 , respectively (see Text S2 for discussion).

Discussion
The organization of dislocations into subgrain boundaries is driven by the associated reduction in the strain energy of the system (e.g., Raj & Pharr, 1986;Read, 1953, p. 226). During deformation, the mean subgrain size varies systematically with the applied stress. Normalizing the stress by the shear modulus and the subgrain size by the Burgers vector collapses mineral-specific piezometers into a single relationship (e.g., Twiss, 1986). Our results thus define a subgrain-size piezometer that can be used on multiple minerals. As our piezometer is calibrated by EBSD, it differs from calibrations obtained by different measurement techniques (Durham & Goetze, 1977;Goetze, 1975;Karato et al., 1980;Mercier et al., 1977;Toriumi, 1979). We also provide a set of tests that assess the suitability of a data set for analysis by this method based on factors including the number of linear intercepts, the step size, and the map area. The ability to perform these tests routinely in an automated and objective manner provides clear benefits over existing quality control measures (e.g., Humphreys, 2001Humphreys, , 2004Valcke et al., 2006).
Subgrain-size piezometry has several benefits compared to grain-size piezometry. First, subgrain-size piezometry is insensitive to the presence of secondary minerals and the pinning of grain boundaries (Hansen & Warren, 2015;White, 1979). As most rocks are polymineralic and the presence of secondary phases leads to smaller-than-expected recrystallized grains, the majority of existing stress estimates were derived from spatially restricted monomineralic rocks or domains. In contrast, in polymineralic rocks, subgrain-size piezometry should record the average stress supported by the analyzed phase, regardless of the fraction of secondary phases. Thus, the subgrain-size piezometer can be used to analyze the stress supported by each phase deforming by dislocation creep in polymineralic rocks. Consequently, estimates of stresses supported by the lithosphere based on grain-size piezometery (e.g., Behr & Platt, 2014; Weathers et al., 1979; 10.1029/2020GL090056 White, 1979) can be compared to new data acquired using subgrain-size piezometry on more representative rock types.

Geophysical Research Letters
Grain-size piezometry also requires the user to distinguish between recrystallized and relict grains, the proportions of which depend on total strain. Previously, the bimodal grain-size distribution of partially recrystallized rocks allowed the characterization of recrystallized and relict grains on the basis of size (e.g., Post & Tullis, 1999;Stipp & Tullis, 2003). This method truncates the grain-size distribution, thereby modifying the mean grain size and the resulting stress calculated from that grain size. More recent work used the degree of intracrystalline lattice distortion within each grain to identify recrystallized grains on the assumption that relict grains have a greater degree of internal distortion than recrystallized grains (Cross et al., 2017). Although this new method provides a working calibration, it adds additional steps and assumptions to the analysis. In deforming materials with active subgrain formation, both recrystallized and relict grains develop subgrain boundaries that fall on a single piezometric relationship (Ross et al., 1980). Because our calibration contains subgrains both from relict and recrystallized grains, as long as both sets of grains contain subgrains in the sample to be analyzed, no distinction between them is necessary in measuring subgrain size. Instead, we only need to define a critical misorientation angle. A 1°critical misorientation angle is consistent with previous observations in NaCl that subgrain boundaries tend to have misorientation angles <2°at strains of 15% and <5°at strains of 50% (Pennock et al., 2005). In addition, Trimby et al. (1998) observed that subgrains in relict and recrystallized grains of quartz were the same size based on a 1°critical misorientation.
Finally, subgrain-size piezometry is independent of flow geometry. In contrast, samples deformed in general shear and axial compression may exhibit different grain sizes at the same equivalent stress. Experiments by Heilbronner and Kilian (2017) revealed that quartzites deformed in general shear developed a larger grain size at a given stress than would be expected from piezometers calibrated with experiments conducted in axial compression. In addition, the compilation of data for quartz presented by Tokle et al. (2019) demonstrates experiments conducted in general shear yield a different grain-size piezometer than experiments conducted in axial compression. To explore whether the flow geometry also affects subgrain sizes, we regrouped the data according to whether the samples were deformed in a pure shear (i.e., axial compression) or general or simple shear, geometry. In Figures 3b and 3d, there is no obvious dependence of the mean line-intercept length on the deformation geometry. Therefore, we suggest that our single subgrain-size piezometer can be applied to a wide range of rocks, regardless of flow geometry.
It is necessary to highlight two potential limitations of subgrain-size piezometry. First, it appears that the observed relationship between subgrain size and stress is not applicable to deformation at high stresses ( Figure 3). The relationship between subgrain size and stress may change at higher stresses as the recrystallization mechanism changes, similar to the relationships observed for the size of recrystallized grains of quartz (Stipp & Tullis, 2003), which is why the three samples deformed in this high-stress regime were not included in our calibrations. Similarly, differences in the processes of subgrain formation or in the mechanisms of dynamic recrystallization might require different calibrations at significantly lower stresses (e.g., Stipp et al., 2010;Valcke et al., 2007;Valcke & Bresser, 2015). Therefore, we recommend that our piezometer should be applied within the normalized subgrain size and stress range over which it was calibrated. This range corresponds to mean line-intercept lengths of 1-9 μm for olivine and 1-18 μm for quartz and a similar range for other silicate minerals. These limits on subgrain size correspond to stress ranges of 60-420 and 20-240 MPa for olivine and quartz, respectively.
Second, the potential for modifying subgrain size through static annealing or stress relaxation after deformation remains poorly constrained. Ross et al. (1980) suggested that subgrains record the highest stress in rocks subjected to complex deformation histories. However, we demonstrate that a strong correlation between subgrain size and final stress is apparent, even in cases in which the larger stresses were experienced earlier in the deformation history (Hansen et al., 2012;Holyoke & Tullis, 2006). Furthermore, experiments on steel by Qin et al. (2003) revealed that subgrains were only able to coarsen with additional strain while under stress of a reduced magnitude, whereas they did not coarsen during static annealing. These observations suggest that subgrain sizes best reflect the last increments of deformation, although further work exploring the effect of stress changes is clearly required to fully investigate the mobility of subgrain boundaries in minerals. In comparison, the response of the grain size to non-steady-state creep in geological materials has been the focus of recent field studies (e.g., Campbell & Menegon, 2019) and experimental work (e.g., Kidder et al., 2016) and, as such, is better constrained than that of the subgrain size. If the subsequent modification of subgrain size occurs on a different timescale than the modification of other microstructural features, such as dislocation density or recrystallized grain size, then combined analysis using multiple piezometers may yield information about complex stress histories (Kohlstedt & Weathers, 1980;White, 1979).