Assessment of quartz grain growth and the application of the wattmeter to predict recrystallized grain sizes

We investigated relationships between the recrystallized grain size and stress in experimentally deformed water-added quartz aggregates. For stresses 100 MPa there is a variation in the measured recrystallized grain size for a given stress. We found this variation correlates with a change in the c-axis fabric in general shear experiments, where samples with larger recrystallized grain sizes for a given stress have dominantly prism <a> c-axis fabrics and samples with smaller recrystallized grain sizes for a given stress have dominantly basal <a> c-axis fabrics. The dislocation creep flow law also changes at conditions where these two c-axis fabrics form (Tokle et al., 2019). Using the wattmeter model (Austin and Evans, 2007), we quantify piezometric relationships for samples that develop prism <a> and basal <a> caxis fabrics, respectively. The wattmeter model is sensitive to grain growth kinetics; we formulated a new grain growth law for quartz based on reanalysis of microstructures in samples from previous work (Tullis and Yund, 1982). The activation enthalpies and water fugacity exponents for our grain growth law and dislocation creep flow laws are the same within error, suggesting the recrystallized grain size versus stress relationships are nearly independent of temperature and water fugacity, consistent with laboratory observations. The wattmeters successfully predict the recrystallized grain size versus stress relationships of all water-added experimental quartz samples. Our results provide support for the use and extrapolation of the wattmeter model to both experimental and geologic conditions to investigate the stress state and grain size evolution of quartz rich rocks. Plain Language Summary Stress is a fundamental rheological parameter in understanding how rocks deform; however, stress is hard to quantify in rocks that flow rather than fracture at geologic conditions. The relationship between the recrystallized grain size and stress is established through rock deformation experiments where both the stress and grain size can be measured. Therefore, the stress can be estimated in naturally deformed rocks by measuring the recrystallized grain size and applying the experimentally calibrated relationship. Here we show new evidence for different recrystallized grain size versus stress relationships in quartz, and that these different relationships correlate with the crystallographic orientations of grains in samples deformed in the shear geometry. To quantify these new observations, we use the wattmeter model, which models the recrystallized grain size of a deforming aggregate based on the balance between the rates of grain growth and grain size reduction, to predict the quartz recrystallized grain size. This model is sensitive to grain growth kinetics; therefore, we formulated a new grain Preprint submitted to Journal of Geophysical Research – Solid Earth and EarthArXiv growth law for quartz. Our calibrated wattmeters predict the recrystallized grain size versus stress relationships in experimentally deformed wet quartzite samples. Our wattmeter model can therefore be used to estimate stress magnitudes in naturally deformed quartz-rich rocks.


Introduction
Due to the abundance of quartz in the continental crust, quartz rheology is fundamental to our understanding of many geodynamic processes. Grain size is an important parameter in the rheology of many materials, including quartz. In addition, a practical technique to infer stress magnitudes in the continental crust is the piezometric relationship between recrystallized grain size and stress, expressed as: where d is recrystallized grain size,  is differential stress, and Ao and k are constants (Twiss, 1977; Stipp and Tullis, 2003). The most widely applied recrystallized grain size piezometer for quartz was empirically developed based on recrystallized grain size measurements from 12 deformation experiments conducted on "as-is" quartzite (i.e., without pre-drying or adding water) in axial compression at temperatures between 800-1100C, where Ao and k are 3631 and 1.26, respectively (Stipp and Tullis, 2003); we refer to this relationship as the S&T piezometer. Stipp and Tullis (2003) quantified a second piezometric relationship at higher stresses (> 350 MPa) and smaller grain sizes (≲ 2 m) based on axial compression experiments of as-is novaculite conducted by Bishop (1996). Furthermore, Stipp et al. (2006) conducted axial compression experiments on water-added and pre-dried quartz samples, showing that over the range of measured stresses and grain sizes there is no difference in the piezometric relationship for water-added, as-is, or pre-dried quartz samples. Kidder et al. (2016) conducted deformation experiments on water-added quartzite aggregates in axial compression and showed similar results to those produced by Stipp and Tullis (2003) and Bishop (1996).
Analyses of samples deformed in general shear suggest that the recrystallized grain size piezometer for quartz is not as simple as presented in Stipp and Tullis (2003) and Twiss (1977). Heilbronner and Kilian (2017) reanalyzed as-is and water-added samples from Heilbronner and Tullis (2002;  The recrystallized grain size versus stress relationship has important implications for our understanding of stress in the continental crust. For example, one could fit the compiled data set in Figure 1, suggesting that the recrystallized grain size is not as strongly dependent on stress as indicated by the S&T piezometer. In contrast, we illustrate that the variability in recrystallized grain size versus stress relationship can be reconciled by applying the wattmeter (Austin and Evans, 2007) and accounting for differences in quartzite flow laws at the experimental conditions. We analyze recrystallized grain sizes and mechanical data from five studies on experimentally deformed wet quartzites (Figure 1c,d). We develop two wattmeters for quartz, utilizing the two flow laws determined by Tokle et al. (2019). As part of our analysis, we reformulate a grain growth law for quartz aggregates, which is a fundamental component of the wattmeter model. Using our newly formulated wattmeters, we extrapolate to crustal conditions to evaluate stress estimates based on published grain size and c-axis fabric data.

Figure 1.
Plots of log equivalent stress versus log grain size color-coded by a) study and b) whether the starting material was pre-dried, as-is, or water-added. c) plot of log stress versus log strain rate showing the two flow laws derived by Tokle et al. (2019) and the transitional region color-coded, red, blue, and black respectively. d) plot of log equivalent stress versus log grain size for the water-added samples color-coded by their c-axis fabric and/or where the sample plots in stress versus strain rate space. In c) and d) the pole figures are oriented with the shear plane in the E-W orientation with a dextral sense of shear.

Experimental data
We analyze recrystallized grain sizes and mechanical data from five different experimental studies; two employed an axial compression deformation geometry (Stipp et  showed that the RMS grain sizes determined from EBSD and CIP were the same within error over the stress range of 48 to 177 MPawith the exception of sample W1143 (see Figure 4a in Cross et al. 2017 supplementary Table S1.

Deviation in the recrystallized grain size versus stress relationship
The recrystallized grain size versus stress data show an increasing range of deviation from the S&T piezometer at stresses greater than ~100 MPa, with select samples showing a larger grain size for a given stress ( Figure 1). These deviations are apparent for both water-added and as-is experiments, indicating that the scatter does not reflect a systematic effect of water Preprint submitted to Journal of Geophysical Research -Solid Earth and EarthArXiv content ( Figure 1b). Figure 1 also shows that deviation from the piezometer cannot be systematically explained by differences in deformation geometry.
The deviation of recrystallized grain size versus stress data from the S&T piezometer correlates with differences in the crystallographic preferred orientation (CPO). As shown in

The Wattmeter
To investigate the origin for different recrystallized grain size versus stress relationships presented in Figure 1, we apply the wattmeter model Evans, 2007, 2009). The wattmeter is a relatively simple grain size evolution model that is practical to apply. The wattmeter is defined as a scaling relationship representing a dynamic balance between the rates of grain growth and grain size reduction, where the grain size evolution rate is expressed as Evans, 2007, 2009): The scaling factor  is defined as the fraction of total energy input during dislocation creep () that is not dissipated as heat, which is available to change the internal energy through the creation of crystal defects (Austin and Evans, 2007;. See Table 1 for definition of all symbols and their values. Setting the grain size evolution rate to zero (̇ = 0), the steady state grain size is defined as: and by inserting the dislocation creep flow law into the strain rate term and expanding the kg term from the grain growth law into equation 3, the steady state grain size becomes:  (Stipp and Tullis, 2003) or theoretically estimated (Twiss, 1977). Equation 4 illustrates that the slope, k', of the recrystallized grain size versus stress relationship is related to the stress exponent (n) for dislocation creep and the grain growth exponent (p) from the grain growth law, which indicates that changing n or p will change the slope of the recrystallized grain size versus stress relationship. This effect is consistent with our observation illustrated in Figure 1d; assuming the grain growth exponent is constant, the slope of the recrystallized grain size versus stress relationship will be steeper for samples that define the n = 4 flow law than the n = 3 flow law. Equation

Reanalysis of a wet quartz grain growth law
Improving our understanding of grain growth kinetics for quartz will improve our ability to model quartz grain size evolution. The most widely applied grain growth law for wet quartz   Following these observations, we formulated a new grain growth law for wet quartz.
We reanalyzed the microstructures in the samples from the grain growth experiments conducted by Tullis and Yund (1982). We found that approximately 20% of the samples used  2019) that were conducted at the same starting water content (1-2 wt%) as Tullis and Yund (1982). All of the experimental grain growth data used in our analysis are listed in supplementary Table S2. We define our grain growth law for wet quartz by equations 5 and 6 (see Table 1 for symbols and descriptions):  Figures 3a and S1). Supplementary Figure S3 shows the variation in the slope for values of p from 2 to 4. To determine Ag, rg, and Qg, we perform a least squares linear regression assuming p = 3. The resulting parameter values are listed in Table 2. In Figure 3, we plot the derived grain growth law together with the data, showing that the grain growth law provides a good fit to the experimental data over a range of pressure/water fugacity and temperature. We use equation 7 to estimate the water fugacity at crustal conditions, where 2 is the water activity and A1, A2, and A3 are empirically fit constants. Equation 7 was formulated by Shinevar et al. (2015), who empirically fit the thermodynamic database of Pitzer and Sterner (1994) to determine a single equation that provides a good fit to the fugacity data for a wide range of crustal geotherms and determined A1 = 5521 MPa, A2 = 31.28 kJ/mol, and A3 = -2.009  10 -5 m 3 . Equation 7 is also used to determine the water fugacity in the flow laws.

Discussion
Motivated by our observations in Figure 1, we assess the utility of the wattmeter model in predicting quartz recrystallized grain sizes and its implications for estimating stress at crustal conditions. First, we discuss the comparison between our reformulated grain growth law and  grain growth law predicts grain growth from 1 m to 80 m in ~30 kyr at 450C and ~10 kyr at 500C while our grain growth law predicts a ~650 kyr at 450C and ~100 kyr at 500C ( Figure 4).

Predicting the recrystallized grain size using the wattmeter
With our newly formulated grain growth law, the wattmeter accurately predicts the range of recrystallized grain sizes for all of the water-added experimental samples. To determine the efficacy of applying the wattmeter, we first use equation 3 with the known experimental data consisting of stress, strain rate, pressure, and temperature, to predict the steady state grain size. We find that the wattmeter, with a constant value of the combined term / in equation     Supplementary Figures S7 and S8 show the comparison between the recrystallized grain size versus stress measurements and the wattmeters at all temperature and pressure conditions. The recrystallized grain size versus stress measurements are color-coded based on Figure 1d.

The influence of temperature and water fugacity
The wattmeter model provides a prediction for how temperature and water fugacity impact the steady state grain size and grain size evolution of quartz. From equation 4, if Qg = Q and rg = r then no effect of temperature or water fugacity on the stable grain size is predicted. In contrast, if Qg  Q or rg  r then variations in temperature or water fugacity are predicted to modify the stress versus grain size relationship.   Table 2. In plots b,c,d Qg-Q represents the activation enthalpy of grain growth minus the activation enthalpy of creep in kJ/mol while rg-r is the water fugacity exponent for grain growth minus the water fugacity exponent for creep. The Stipp and Tullis (2003) piezometer is plotted for reference in all plots. The  values for the corresponding wattmeters are listed in Figure 6.
In Figure 8, we illustrate the effects of temperature (using Qg -Q) and water fugacity (using rg -r) on the stress versus grain size relationships at both experimental and crustal conditions. We use the Tokle et al. (2019) flow laws and compare results using the three grain growth laws listed in Table 2. At experimental conditions, T = 900C and P = 1.5 GPa, the wattmeters calculated using all three grain growth laws are nearly identical, which is expected because all three grain growth laws and  were constrained at or near this experimental condition (Figure 8a). The wattmeters calibrated with our grain growth law have a Qg -Q less than 20 kJ/mol and a rgr less than 0.40 for both flow laws (Figure 8b).
This results in a slight shift in the wattmeters to smaller recrystallized grain sizes for a given shift in the wattmeters to smaller grain sizes for a given stress at crustal conditions ( Figure   8d). The activation volume for quartz grain growth has not previously been calculated.

A few remarks on the activation volume for quartz rheology
Using our compiled dataset for grain growth, we performed linear regression fits of equations 5 and 6 assuming a grain growth exponent of p = 3.0. We determine a low activation volume with a large uncertainty (Vg = 0.5  28 cm 3 /mol), and a water fugacity exponent (rg = 1.4  (Table S3). While this analysis suggests similar activation volumes and water fugacity exponents for the dislocation creep flow laws, to more accurately determine the activation volume and water fugacity exponents will require additional grain growth and deformation experiments focusing on independently determining rg and Vg.

RMS vs. Arithmetic mean calibrated wattmeters
There are a number of ways to define the 2D grain size (Heilbronner and Barrett, 2013). The      provide peak stress estimates for the WMCC that are also consistent with the peak stress estimates from the KTB borehole based on measurements of dislocation density (Dresen et al., 1997) and borehole stress (e.g., Zoback and Townend, 2001). Our wattmeters are also consistent with the range of experimental observations (such as the lack of a discernable influence of temperature on recrystallized grain size) and explains the deviation in stress versus grain size relationships for the experimental data ( Figure 1).

Conclusions
The observations of a concomitant switch in the c-axis fabric and stress versus grain size relationship as well as a switch in the c-axis fabric and flow law relationship provides support linking different deformation mechanisms, piezometric relationships, and c-axis fabrics in quartz. By reformulating the quartz grain growth law, we show there is a modest temperature and water fugacity dependence on the stable grain size. Our wattmeter model is able to explain the different stress versus grain size relationships observed at laboratory conditions while also providing stress estimates consistent with other piezometric models. The results of this analysis provide support for the use and extrapolation of the wattmeter model to both experimental and geologic conditions to investigate the stress state and grain size evolution of quartz rich rocks.