The Effect of Secondary‐Phase Fraction on the Deformation of Olivine + Ferropericlase Aggregates: 1. Microstructural Evolution

To study the microstructural evolution of polymineralic rocks, we performed deformation experiments on two‐phase aggregates of olivine (Ol) + ferropericlase (Per) with periclase fractions (fPer) between 0.1 and 0.8. Additionally, single‐phase samples of both Ol and Per were deformed under the same experimental conditions to facilitate comparison of the microstructures in two‐phase and single‐phase materials. Each sample was deformed in torsion at T = 1523 K, P = 300 MPa at a constant strain rate up to a final shear strain of γ = 6 to 7. Microstructural developments, analyzed via electron backscatter diffraction (EBSD), indicate differences in both grain size and crystalline texture between single‐ and two‐phase samples. During deformation, grain size approximately doubled in our single‐phase samples of Ol and Per but remained unchanged or decreased in two‐phase samples. Zener‐pinning relationships fit to the mean grain sizes in each phase for samples with 0.1 ≤ fPer ≤ 0.5 and for those with 0.8 ≥ fPer ≥ 0.5 demonstrate that the grain size of the primary phase is controlled by phase‐boundary pinning. Crystallographic preferred orientations, determined for both phases from EBSD data, are significantly weaker in the two‐phase materials than in the single‐phase materials.

Similar effects have been demonstrated theoretically (Bercovici et al., 2023;Bercovici & Mulyukova, 2021) and experimentally in samples composed of Ol + Opx (Tasaka et al., 2017(Tasaka et al., , 2020, calcite + anhydrite (Cross & Skemer, 2017), and quartz + albite . Additionally, microstructural developments from experimentally deformed samples have been used to investigate possible mechanisms responsible for efficiently mixing two or more mineral phases in samples in which pinning is effective (Cross et al., 2020;Cross & Skemer, 2017;Tasaka et al., 2017). However, many of these experimental studies focused on two-phase samples with only one to three different proportions of the two phases. The lack of sets of data from samples experimentally deformed to large strains over the entire range of phase proportions in two-phase samples limits the possibilities of testing models of two-phase flow.
In this study, we deformed aggregates of Ol + periclase (Per) with eight different proportions of the two phases to investigate how properties of two-phase rocks vary with the relative amounts of the two-phases present. This paper focuses on the microstructural developments of our two-phase system, while a companion paper examines their mechanical behavior. The proportions of the two phases in our samples covered the range f Per = 0.1 to 0.8. Although Ol and Per do not occur at the same depth in the Earth, both are well-characterized geological materials making them good candidates to serve as an analog system for the investigation of the processes that occur in two-phase systems more generally and to elucidate the physics of deformation of polymineralic materials. In addition, Ol and Per share a structural building block, namely (Mg,Fe)O, similar to the Ol + Opx system, enabling us to test the physical processes responsible for phase mixing as described by Tasaka et al. (2017), Bercovici and Skemer (2017), and Cross and Skemer (2017). However, mechanisms for phase mixing were a focus of our previous study (Wiesman et al., 2018) and will not be a focus of the present study.

Sample Preparation
Two-phase, polycrystalline rocks were synthesized from powders of olivine (Ol, Fo 90 ) mixed with powders of periclase (MgO, Per 100 ) or ferropericlase (Per 90 or Per 70 ). Details of sample preparation given in Wiesman et al. (2018) are summarized here. Note that in Wiesman et al. (2018) we used Fp 90 as an abbreviation for ferropericlase, which is equivalent to Per 90 in the present paper.
Olivine powders were prepared from gem-quality crystals of San Carlos olivine (SC-Ol) with minimal inclusions, crushed to an average particle size of 5.5 ± 2.2 μm. Periclase (Per 100 ) powders were obtained from American Elements with a particle size of 1 to 5 μm. Synthesis of Per 90 powder is described in Wiesman et al. (2018). Following the procedure outlined by Bystricky et al. (2006), Per 70 powders were alloyed from 1 to 5 μm powders of MgO and Fe 2 O 3 mixed in the appropriate molar ratios in a mortar and pestle, then annealed for 12 hr in a one-atmosphere, gas-mixing, alumina-tube furnace at a temperature, T, of 1673 K and an oxygen partial pressure, O 2 , of 10 −2 Pa set by a mixture of CO-CO 2 gases. To ensure complete reaction between the powders, this procedure was repeated three additional times, grinding the powders in an agate mortar and pestle between anneals. To remove adsorbed water, Per 100 powders were dried in a box furnace at T = 1173 K in air for 12 hr, while SC-Ol powders were dried for 12 hr in a gas-mixing furnace at T = 1273 K and O 2 = 10 −5.2 Pa, corresponding to the Ni/NiO buffer.
In total, eight two-phase mixtures were prepared with Ol content ranging from 90 to 20 vol%, including five mixtures of SC-Ol and Per 100 , one of SC-Ol and Per 90 , and two of SC-Ol and Per 70 . The five mixtures of SC-Ol plus Per 100 powders were prepared in vol% ratios from 80 vol% SC-Ol + 20 vol% Per 100 to 20 vol% SC-Ol + 80 vol% Per 100 . Powders of both phases were mixed in an agate mortar and pestle, then tumbled overnight in a neoprene bottle with teflon-coated steel ball bearings to produce two-phase mixtures with the minor phase well dispersed within the primary phase. Each mixture was then annealed at T = 973 K in a one-atm furnace at  PT-1214, PT-1316, PT-1317, PT-1320, PT-1321, and PT-1322 were cold pressed with an 8-mm diameter central Ni post, resulting in thin-walled specimens, while powders for samples PT-1407, PT-1409, and PT-1410 were cold pressed without a central Ni post, resulting in full-cylinder specimens. Examples of both sample types are pictured in Figure 1a. The use of a thin-walled sample diminishes the stress and grain size differences along the sample diameter relative to those in full-cylinder samples, thus simplifying the analysis of the mechanical and microstructural data (Hansen et al., 2012). A thin layer (0.5 g) of NiO powder was cold pressed into the base of each Ni capsule before adding the sample powders to buffer the O 2 at the Ni/NiO buffer. To further densify the powders, an evacuated hot press (Meyers et al., 2017) was performed in a gas-medium apparatus (Paterson, 1990). The hot press was conducted at a confining pressure of P = 300 MPa and T = 1523 K for 2 to 3 hr while a vacuum of 10 to 20 Pa was drawn on one end of the sample to promote densification. Microstructural parameters and hot-pressing conditions for each of the starting materials are summarized in Table 1. After hot pressing, samples were cut into ∼3 mm tall cylindrical disks with an outer sample diameter between 11.0 and 12.6 mm for deformation experiments.

Deformation Experiments
Samples were deformed in a high-resolution, gas-medium deformation apparatus equipped with a torsion actuator (Paterson & Olgaard, 2000). Experiments were performed at P = 300 MPa, T = 1523 K at a constant twist rate while torque and angular displacement were measured with an internal load cell and an external rotary variable differential transformer, respectively. During each experiment the pressure was controlled within ±5 MPa of the target pressure and temperature was controlled within ±2 K of the target temperature. In this study, samples were deformed at a maximum shear strain rate, γ , between 6.5 × 10 −5 and 1.5 × 10 −4 s −1 at the outermost radius of the sample to maximum shear strains, γ, between 1 and 7, typically reaching shear stresses, τ, between 80 and 150 MPa at the end of each experiment. Shear strains were derived from the recorded angular displacement data and confirmed by the displacement of a strain marker on the jacket surface, while shear stresses were calculated from the measured torque data. Further details of the deformation experiments, including the deformation assemblies and the analysis of the mechanical data, are provided in our companion paper and summarized in Section 4.1.1.

Analysis and Reanalysis of Sample Microstructures
After each deformation experiment, the sample was extracted from the deformation assembly, and the metal around the sample was removed on a SiC grinding wheel. Tangential sections (Figure 1b) of each sample were polished on diamond lapping film down to a grit size of 0.5 μm, followed by a chemo-mechanical polish with 40-nm colloidal silica. Prior to microstructural analysis, samples were coated with 4 to 5 nm of carbon to avoid charging in an electron microscope. Crystallographic orientation data were collected with electron backscatter diffraction (EBSD) analyses using an Oxford Symmetry detector on a JEOL 6500 scanning electron microscope (SEM). EBSD patterns were collected at an accelerating voltage of 20 kV, a beam current of 65 μA, and a step size of 0.2 -0.3 μm. Patterns were processed in the AZtec HKL software to remove wild spikes, individual pixels with different orientations from their neighbors, and further analyzed using the MTEX toolbox for MATLAB (Bachmann et al., 2010). Grain sizes were calculated from the EBSD data using a threshold of 10° misorientation between neighboring pixels to determine grain boundaries within each phase. The grain size of each phase was calculated from the equivalent circle diameter of the grains identified in the EBSD maps with a correction factor of 4/π applied to convert grain sizes from 2-D maps to 3-D values, assuming spherically shaped particles (Underwood, 1970, p. 91). Subgrain size was calculated from EBSD data using the line-intercept 2) and a thin-walled sample, that is, a sample with an internal Ni post (right, PT-1317 f Per = 1.0). (b) A schematic drawing of a torsion sample with the direction of shear indicated above the sample. After deformation, a tangential section of each sample is exposed and polished for microstructural analysis.
10.1029/2022JB025723 5 of 25 method, following the procedure outlined by Goddard et al. (2020) and using the MATLAB code provided in their supporting information. To allow comparison with the subgrain-size piezometer determined by Goddard et al. (2020), no stereological correction factor was applied to the subgrain sizes. Additionally, the lengths of Ol-Ol, Per-Per, and Ol-Per boundaries were calculated from EBSD maps using the MTEX toolbox.
In the Results Section, the microstructural data from samples originally reported in Wiesman et al. (2018) (i.e., PT-1214, PT-1219, PT-1239, PT-1250, PT-1283, and PT-1324 were reanalyzed. New grain sizes are reported for each of these samples based on EBSD analysis rather than the grain tracing used in Wiesman et al. (2018). This approach allowed a better comparison of microstructural results from these samples with those from the samples deformed in this study. None of the updated results affect our interpretation of the data presented previously.

Microstructural Data
Representative microstructures of samples deformed at γ = 1.2 × 10 −4 to 1.5 × 10 −4 s −1 and of their respective undeformed starting materials are presented as EBSD phase maps in Figure 2. Phase maps of undeformed samples PT-1320, PT-1321, PT-1322, PT-1409a, and PT-1410a in Figures 2d.i and 2f.i-2j.i reveal initially well-mixed microstructures with grains of the secondary phase randomly distributed along grain boundaries, at triple junctions, and at four-grain junctions of the primary phase. Similar well-mixed microstructures are observed after deformation to strains between γ = 5 and 7 in these two-phase samples in Figures 2d.ii and 2f.ii-2j.ii; secondary-phase particles are distributed throughout the primary-phase matrix in each case, and clusters of grains are stretched out at angles between 0° and 20° to the shear direction. In samples with f Per ≥ 0.5, individual grains of both Ol and Per have higher aspect ratios than in the undeformed material and are elongated parallel or sub-parallel to the shear direction. EBSD phase maps for samples deformed at γ = 6.5 × 10 −5 s −1 are displayed in Figure 3. Microstructures for these samples deformed at a slower strain rate exhibit a similar arrangement of secondary-phase grains mixed with the primary phase to the samples deformed at faster strain rates of γ = 1.2 × 10 −4 to 1.5 × 10 −4 s −1 in Figures 2d.i and 2f.i-2j.i discussed above. Prior to deformation, grains of the secondary phase are randomly distributed along grain boundaries, at triple junctions, and at four-grain junctions of the primary phase; the same is true after deformation.

Grain Size
Grain size distributions calculated from equivalent circle diameters from the EBSD maps for Ol and Per are plotted in Figures 4 and 5, respectively. A log-normal distribution was fit to each grain size distribution; reported grain sizes are the arithmetic mean of each distribution. Estimates for the error of each grain size are reported as the standard deviation and standard error, as summarized in Table 2. In the undeformed two-phase samples, the mean Ol grain size, d Ol, ranges from 2.9 to 3.8 μm, and the mean Per grain size, d Per , ranges from 1.6 to 5.2 μm. Samples PT-1214, PT-1407, and PT-1409, which were prepared with Per 90 or Per 70 powder rather than Per 100 powder, have large clusters of Per distributed throughout the Ol matrix (Figures 2b.i, 2c.i, and 2e.i). The average grain sizes in these samples with clusters are d Ol = 7.9, 3.6, and 4.4 μm and d Per = 12.2, 3.0, and 9.0 μm, respectively, although Per grains with sizes as large as 25 to 50 μm are commonly observed. After deformation, Ol grain sizes are between d Ol = 1.7 and 4.4 μm and Per grain size between d Per = 2.2 and 4.4 μm for all two-phase samples. In general, as the Per content in the two-phase sample increases, the Ol grain size progressively decreases from 4.4 to 1.7 μm, while Per grain size increases from 2.5 to 4.4 μm.
In the single-phase Ol sample, the initial grain size of d Ol = 4.6 μm increased to d Ol = 8.4 μm during deformation to γ .

Crystallographic Preferred Orientation (CPO)
Crystallographic orientation data collected from EBSD patterns are plotted in pole figures using one point per grain on lower hemisphere spherical projections for Ol in Figure 6 and for Per in Figure 7. For Ol, the strength of the CPOs is characterized using the J-index, which ranges from 1 for a random distribution to ∞ for a single-crystal, and the M-index, which ranges from 0 for a random distribution to 1 for a single-crystal (Mainprice et al., 2000;Skemer et al., 2005). Neither the J-nor the M-index was used to quantify the strength of the Per CPOs because the multi-maxima patterns that are characteristic of polycrystalline, sheared Per and other face-centered cubic (FCC) materials (Bronkhorst et al., 1992;Heidelbach et al., 2003;Long et al., 2006;Wenk et al., 2009) are considered random based on these analyses due to the spread of orientations between neighboring grains. For comparison among samples containing Ol, all of the Ol CPOs are scaled to the maximum multiples of uniform distribution (MUD) determined for the single-phase Ol sample (Figure 6a.ii). Likewise, the Per CPOs are scaled to the maximum MUD obtained for the single-phase Per 100 sample (Figure 7a.ii). Additional CPOs for both Ol and Per in samples deformed at a slower strain rate of γ = 6.5 × 10 −5 s −1 are plotted in Figure S1 in Supporting Information S1.   . Grain size distributions for Ol in all samples normalized to the total number of Ol grains (N tot ) measured for undeformed samples, column (i), and deformed samples, column (ii). Distributions of grain size were fit with a log normal distribution, the mean of which is the reported grain size. Grain size histograms in each row are for Ol volume fractions,  (010)[100] slip system with slip also occurring on (001)[100] (Hansen et al., 2014). The CPO in PT-1324 has a J-index of 10.9 and an M-index of 0.3, slightly less than previously reported M-index values that typically lie between 0.4 and 0.5 for samples of SC-Ol and Fo 50 deformed to similar strains (Hansen et al., 2012;Skemer et al., 2005). The smaller value for the M-index reported here for our single-phase Ol sample is likely due to the pencil glide fabric with activation of two slip systems, rather than a pure A-type fabric for which the orientations of the [010] and the [001] axes are more clustered than   <100> maxima are normal to the shear plane, while the other two maxima are subparallel to the shear direction. Four of the <110> maxima are in the shear plane, two of which are in the shear direction with the other four at high angles relative to the shear direction. Additionally, two <111> maxima are perpendicular to the shear  plane. The remaining six <111> maxima form weak girdles surrounding the shear plane. Wenk et al. (2009) observed similar textures in their experiments on halite deformed to comparable shear strains, and, based on modeling efforts, they argued that {111}<1 10> is still the dominant slip system, despite the patterns observed in pole figures. The CPOs of deformed samples with 0.3 ≤ f Per ≤ 0.8 in Figures 7b.ii-7f.ii have patterns similar to those of the single-phase Per sample. The major differences between the CPOs of Per in two-phase samples and the single-phase sample are the presence of only six maxima in the <110> and <111> pole figures as well as weaker maxima in the <100> directions normal to the shear direction, which indicates that slip on {100}<011> was likely less active in our two-phase samples. In addition, the strength of the Per CPO patterns decreases with decreasing Per content, such that the patterns become random at f Per < 0.3 with no discernible maxima. These observations point toward {100}<011> as the dominant slip systems, with slip also active on {111}<1 10> systems (Heidelbach et al., 2003).
The evolution of the CPOs for both phases with increasing strain is displayed in Figure 8 Figures 8d, 8f, 8h, and 8j, the CPOs for Per 90 are nearly random with only weak maxima in each case, likewise similar to that described above. The most relevant components of the observed fabrics are the weak maxima in <100> and <111> normal to the shear plane and weak maxima in <110> in the shear direction. In the supplementary material, Figure S2 in Supporting Information S1, the evolution of Per CPOs is plotted with strain for single-phase Per samples. We note a change in CPO similar to that observed with increasing strain for Per 80 (Heidelbach et al., 2003) and halite (Wenk et al., 2009); at small strain the most important features of the Per CPO are the <111> maxima normal to the shear plane and <110> maxima parallel to the shear direction, which, at large strains, evolves to strong <100> maxima normal to the shear plane, weaker <111> maxima normal to the shear plane, and <110> maxima parallel to the shear direction.

Summary of Mechanical Behavior
Although the mechanical behavior of our samples is discussed in detail in our companion paper, the interpretation of the microstructural data in the present study requires some knowledge of the mechanical behavior and deformation mechanisms in our samples. Therefore, the results from our companion paper are briefly summarized here. Based on the trends in plots of shear stress versus shear strain (companion paper, Figure 1) and calculated values of the stress exponent, n (companion paper, Figure 3) our samples can be divided into two groups with distinct behaviors: samples with f Per < 0.5, including the single-phase Ol sample, and samples with f Per > 0.5, including the single-phase Per samples. For samples with f Per < 0.5, the mechanical behavior is characterized by a peak stress that was reached at relatively small shear strains, between γ = 0.1 and 0.7, after which stress decreases until a steady-state stress was reached at γ ≥ 5.0. We calculated values for the stress exponents between n = 2.2 and 3.1 for these samples and, along with the mechanical behavior, interpreted this result as evidence of deformation dominated by dislocation-accommodated sliding along grain interfaces, disGBS, in our samples with f Per < 0.5. In samples with f Per > 0.5, the mechanical behavior is characterized by a rapid increase in stress up to γ = 0.1, after which stress remained constant for the rest of the experiment in single-phase Per or continued to increase at a slower rate for the remainder of the experiment in two-phase samples. We calculated values for the stress exponent of between n = 3.5 and 5.1 for these samples and thus concluded that deformation was dominated by dislocation creep of Per in our samples with f Per > 0.5.

Complications Due To Fe-Mg Exchange
One caveat in interpreting the data from this study and the companion study is associated with the differences in Fe content among samples. Because the Fe content of SC-Ol (Fo 90 ) is in equilibrium with that of Per 70 , in samples PT-1214, PT-1320, PT-1321, PT-1322, PT-1409a, and PT-1410a prepared using Per 90 and Per 100 , Fe-Mg exchange took place between Ol and Per. As mentioned at the end of the Introduction, these experiments were originally designed to examine the physical processes responsible for phase mixing (Wiesman et al., 2018), in which case the Fe content should not be a concern. To confirm this expectation, we prepared two samples with Per 70 .
Electron microprobe analyses provided two important results. First, Fe-Mg exchange between Ol and Per was completed during hot-pressing, that is, before a deformation experiment was initiated. Second, the compositions of both phases were analyzed to determine the range of Fe content among our samples; for samples prepared from SC-Ol and Per 90 or Per 100 , the Ol compositions ranged between Fo 95 and Fo 97 and the Per compositions ranged between Per 80 and Per 99 . Since the effects of Fe-Mg composition on the mechanical behavior and microstructural evolution have been published for Ol (Qi et al., 2021;Zhao et al., 2009) and can be estimated for Per (Bystricky et al., 2006;Heidelbach et al., 2009;Long et al., 2006;Stretton et al., 2001), we have a point of comparison for the effect of differences in Fe-content on the results obtained in the present study. Most relevant to the present study are the results of Qi et al. (2021), who investigated grain size evolution and CPO development during deformation of Fe-rich Ol aggregates, and those of Long et al. (2006), who examined the effect of Fe-content on CPO development in Per. These results are discussed further in Sections 4.3 and 4.4 on grain size and CPO development, respectively, in our samples.

Evolution of Phase-Boundary Density
The degree of phase mixing in our samples can be quantified by examining the densities of grain and phase boundaries before and after deformation (Heilbronner & Barrett, 2014, pp. 351-359). Heilbronner and Barrett (2014) separate microstructures into three categories based on the fraction of phase boundaries, that is, the length of phase boundaries out of the total length of grain plus phase boundaries, in each sample. These categories include: (a) clustered, in which grains of the secondary phase are more often neighbored by grains of the same phase than of different phases, (b) anti-clustered or ordered, in which grains of the secondary phase are typically neighbored by different phases, and (c) random, in which grains of the secondary phase follow a mixture of both clustered and anti-clustered tendencies. Mathematically, a random distribution may be described by a binomial distribution that details the probability that any given boundary between neighboring grains is a phase boundary. This distribution is plotted in Figure 9 as a function of f Per along with the fraction of phase boundaries measured in each of our two-phase samples. Samples with clustered microstructures have phase-boundary fractions less than that predicted by the random distribution, while samples with anti-clustered microstructures have phase-boundary fractions greater than the random distribution.
During deformation, the fraction of phase boundaries evolves toward that of a random distribution in each of our samples, regardless of whether the initial distribution of phase boundaries was clustered or anti-clustered. For samples with f Per < 0.5, this evolution is consistent with the microstructural evolution demonstrated in Figures 2b, 2c, and 2e in which large Per grains or clusters of Per grains have undergone significant grain size reduction resulting in an increase in the number of phase boundaries. For the remaining samples, microstructural changes corresponding to changes in phase-boundary density can still be observed but are less obvious in some samples due to initially small grain sizes and fewer clusters of grains or large grains that existed in the starting materials. For example, in the sample with f Per = 0.7 in Figure 2h, domains of fine-grained Per that are relatively free of Ol grains in the starting material evolve such that small, recrystallized grains of Ol occur throughout the Per matrix after deformation. Unfortunately, due to the initial dispersion of the phases amongst one another in our undeformed samples, such as that pictured in Figures 2d.i and 2f.i-2j.i, it is difficult to determine precisely where new microstructural features-such as new, small grains-originated in the deformed samples. We therefore leave discussion of the physics of grain mixing for a future study.
Based on the results from the present study, we predict that phase-boundary density evolves toward a random distribution due to deformation-induced phase mixing and represents the development of a well-mixed microstructure. Yet, observations from previous experiments on two-phase samples carried out to increasing strains by Tasaka et al. (2017) and Wiesman et al. (2018) proposed that the proportion of phase boundaries increases with increasing strain, resulting in an anti-clustered distribution of the two-phases at γ > 10. However, there is relatively little change in the absolute phase-boundary density measured after γ ≥ 4 in the samples from both previous studies (Tasaka et al Figure 8). Additionally, the measured Figure 9. Fraction of phase boundaries in each sample versus f Per . The black curve represents a random distribution of phase boundaries for a two-phase sample (Heilbronner & Barrett, 2014, pp. 355-359) The region below the curve denotes a clustered distribution of the two phases, while the region above the curve represents an anti-clustered distribution of the two-phases. Data for undeformed samples are plotted as triangles, and data for samples deformed to γ > 4 are plotted as circles. f Per reported for each sample is derived from the powder weight added for both phases during cold pressing and converted into a volume using the density of each phase.
phase-boundary densities in the deformed samples with γ ≥ 4 from Wiesman et al. (2018) are likely within error of the value predicted for a random distribution of phase boundaries for their two-phase composition. It is not clear why the high-strain samples from Tasaka et al. (2017) lie in the anti-clustered field with phase-boundary densities greater than predicted by the random distribution. Their observation may be due to a difference between materials (Ol + Opx vs. Ol + Per) or a difference in measurement technique (grain tracing vs. EBSD). Although grain size reduction continues with increasing strains in samples from both previous studies, the observation that phase-boundary density does not change significantly suggests a well-mixed microstructure is reached by γ = 4 for which phase-boundary pinning is effective at hindering grain growth processes. A recent theoretical model suggested that phase-boundary density approaches a steady-state piezometer once pinning effects control the grain-size evolution in two-phase materials (Bercovici et al., 2023), which may explain the lack of change in phase-boundary density observed experimentally above certain strains.

Grain Size
For single-phase materials, a grain size piezometer is often used to relate the observed grain size to the steady-state stress reached after deformation to strains of γ > 3. This piezometer represents a balance between grain growth and dynamic recrystallization occurring in the sample. A piezometer for Ol was determined by Van der Wal et al. (1993) using data on the size of newly recrystallized grains in coarse-grained Aheim and Anita Bay dunites deformed in triaxial compression by Chopra and Paterson (1981) and in SC-Ol single crystals deformed to large strain in compression by Karato et al. (1980). A piezometer for Per was determined by Takeuchi and Argon (1976) using data acquired by Hüther and Reppich (1973) from compression experiments on Per 100 single crystals oriented with a {100} face normal to the compression direction. The grain size data from both single-phase Ol and Per samples in this study are smaller than predicted by their respective piezometers; the grain size of Ol is approximately 2 times smaller than predicted by the piezometer for Ol, and the grain size of Per is approximately 1.3 times smaller than predicted by the piezometer for Per. This discrepancy is likely due in part to differences in the techniques used to acquire microstructural images from which grain sizes were measured (i.e., EBSD vs. optical microscopy) (Hansen et al., 2011). For Ol, instead of comparing our data to the piezometer determined by Van der Wal et al. (1993), we can compare our results to the piezometer determined by Meyers (2023) using longitudinal axial (radial) sections from polycrystalline samples of SC-Ol deformed in torsion. Unlike Van der Wal et al. (1993), Meyers (2023) calibrated their piezometer using Ol grain sizes measured from EBSD maps instead of optical microscopy. Their piezometer is plotted as a shaded region of data in Figure 10a, and the grain size in our single-phase Ol sample is within error of that predicted by their piezometer for the same stress.
In two-phase samples, however, grain growth via grain-boundary migration is inhibited by the presence of the secondary phase, changing the balance between grain growth and recrystallization that is built into the single-phase piezometer. This effect has been predicted to occur during deformation (Bercovici & Mulyukova, 2018Bercovici & Ricard, 2016;Herwegh et al., 2011) and has been observed in experimentally deformed two-phase systems (Cross & Skemer, 2017;Tasaka et al., 2017Tasaka et al., , 2020. As demonstrated in Figure 10, both d Ol and d Per in two-phase samples from this study are smaller than predicted by the respective single-phase piezometers at a given stress. For Ol, grain sizes measured in our samples are 2.5 to 10 times smaller than predicted by the piezometer determined by Van der Wal et al. (1993) and 1.5 to 5 times smaller than predicted by the piezometer determined by Meyers (2023), while for Per, grain sizes measured in our samples are 2 to 4 times smaller than predicted by the piezometer calculated by Takeuchi and Argon (1976). Additionally, grain sizes of both phases typically remained the same or decreased during deformation of our two-phase samples, whereas grain size approximately doubled during deformation in both of our single-phase end-member samples, despite being deformed at the same experimental conditions.
As mentioned at the end of Section 4.1.2, the Fe-Mg compositions of Ol and Per may have influenced the evolution of grain size in our samples. Qi et al. (2021) observed that deformed Ol aggregates with large Fe-contents in the range of Fo 0 to Fo 70 had larger grain sizes at a given stress than values predicted by the piezometric relationship for Fo 90 (Qi et al., 2021, Figure 12). Therefore, it is possible that some of the differences in Ol grain size in our two-phase samples are due to differences in Fe-Mg content of Ol, such that smaller grain sizes would be expected for the Fe-poor Ol in our samples than would be predicted by the Fo 90 piezometer (Meyers, 2023). It is difficult to be quantitative about the effect of composition on grain size in our samples without a piezometric relationship for Ol with compositions in the range of Fo 95 to Fo 97 . However, we note that grain sizes for the range of compositions examined by Qi et al. (2021) were within a factor of two of one another at the stresses relevant to the present study of 100 to 300 MPa. We noted earlier that grain sizes in our samples are up to five times smaller than predicted by the single-phase piezometer for Fo 90 from Meyers (2023), so it is unlikely that all of the differences in the grain size of Ol in our samples are due to differences in Fe-Mg content. For Per, if a similar trend in grain size with increasing Fe content is assumed, an increase in the Fe content of Per would result in larger grain sizes for a given stress in the piezometric relationship. Therefore, the piezometer plotted in Figure 10b for Per 100 represents a lower-bound on the grain size of Per in our samples. The observation that Per grain sizes in each of our two-phase samples are smaller than predicted by this piezometric relationship indicates that an increase in the Fe content of Per in our two-phase samples was not responsible for the discrepancy in grain size.

Subgrain Size
Instead of relating grain size and stress for two-phase materials, subgrain size may be a more appropriate variable for predicting stresses in piezometric relationships as subgrain size may be unaffected by grain boundary pinning that occurs in two-phase materials (Goddard et al., 2020;Hansen & Warren, 2015). Goddard et al. (2020) calibrated a subgrain size piezometer using EBSD data from experiments on SC-Ol, iron-rich Ol (Fo 50 ), and quartz. Their subgrain piezometer is plotted in Figure 11 along with subgrain sizes calculated for Ol and Per from samples in the present study. Subgrain size and stress measurements were normalized by the Burgers vector (b) and shear modulus (G) of each material, respectively. For SC-Ol at T = 1523 K, we used values of b = 5.0 × 10 −4 μm (Goddard et al., 2020) and G = 61.8 GPa (Zhao et al., 2018), while for Per at T = 1523 K, we used values of b = 2.98 × 10 −4 μm (Frost & Ashby, 1982) and G = 105.8 GPa (Fan et al., 2019).  (Karato et al., 1980) and triaxial compression experiments on natural dunites ( Van der Wal et al., 1993). The black dashed line is the piezometric relationship fit through their data. The dark gray shaded area represents data from longitudinal axial sections of polycrystalline samples of SC-Ol deformed in torsion (Meyers, 2023). In (b), the black dashed line is the piezometric relationship for Per determined by Takeuchi and Argon (1976). Colors of each data point corresponds to the fraction of periclase in each sample.
For both single-phase Ol and single-phase Per, the calculated subgrain sizes match the prediction from the piezometer within error. Importantly, this result demonstrates that the subgrain size piezometer from Goddard et al. (2020) correctly predicts the subgrain size for single-phase Per, despite not using Per as one of the materials in the calibration of their piezometer.
The subgrain sizes of Ol in our two-phase samples in Figure 11 can be separated into two groups. The first group consists of samples PT-1420 and PT-1425 in which the subgrain size of Ol also matches the prediction from the subgrain size piezometer within error. These two samples were prepared with Per 70 as the second phase, which was initially organized into large (>25 μm) Per grains or large clusters of Per grains (Figures 2b, 2c, and 9). The second group consists of the remainder of the samples in which the subgrain size of Ol is smaller than predicted by the subgrain size piezometer. Unlike the two samples in the first group, the samples in the later group were prepared with Per 100 or Per 90 ; additionally, grains of the two phases are more intimately mixed with one another in the initial microstructures forming an anti-clustered microstructure (e.g., Figures 2d, 2g, and 9). The different distributions of phases in these two groups of samples produced small Ol grain sizes before deformation in the second group of samples, which were kept small during deformation due to pinning at phase boundaries and resulted in grain sizes smaller than the predicted subgrain sizes, as is observed in Figure 11. The observation that the subgrain size in Ol grains is smaller than predicted supports the argument that Ol deformed by dislocation-accommodated siding along interfaces in the regime for which grain size is smaller than the subgrain size rather than dislocation creep, as noted above and in our companion paper. This result also suggests that subgrain size piezometry may not be appropriate for the most well-mixed and fine-grained of materials, such as ultramylonites, in which grain size is pinned at a smaller size than the subgrain size.
A similar distinction can be made for Per subgrain sizes in our two-phase samples; samples PT-1420 and PT-1425 with clustered microstructures match the subgrain size predicted by the piezometer, while the remaining samples with anti-clustered microstructures have subgrain sizes slightly smaller than, but roughly parallel, to the piezometer. Unlike the results for Ol discussed above, subgrain sizes for Per all fall within error of the subgrain size piezometer from Goddard et al. (2020), suggesting that subgrain sizes of Per in our two-phase samples are well-described by the single-phase, subgrain size piezometer and that grains of Per in our samples are large enough to accommodate subgrains. The observation that the subgrain size of Per in our samples matches that predicted by the piezometer is further confirmation that Per deformed by dislocation creep (see companion paper). A kernel average misorientation map for Per in sample PT-1416 with f Per = 0.6 is plotted in Figure S3 in Supporting Information S1 as an example of the distribution of subgrain boundaries.
Finally, we note that variations in Fe content of both the Ol and the Per among our samples will affect the value of the shear modulus used to normalize the data, which was not accounted for in Figure 11. The effect of removing Fe from Ol due to the use of Per 100 and Per 90 in many of our samples results in a slightly larger shear modulus (Mao et al., 2015;Zhao et al., 2018). For example, the shear modulus for Fo 97 , the smallest Fe content in Ol among our samples, is G = 63.9 GPa, an approximately 4% increase from the value for Fo 90 . This increase does Figure 11. Subgrain size piezometer for (a) Ol and (b) Per plotted as subgrain size normalized by the Burgers vector versus stress normalized by the shear modulus for each phase. Light gray data points in (a) are from samples of SC-Ol deformed in compression and whose subgrain size was analyzed by Goddard et al. (2020). The solid black line in (a) and (b) is the piezometric for subgrain sizes in multiple mineral phases determined by Goddard et al. (2022). Colors are the same as in Figure 10. not significantly change the difference between the normalized data for Ol from our two-phase samples and the piezometer. The effect of adding Fe to Per, however, results in a smaller shear modulus (Fan et al., 2019). For example, the shear modulus for Per 70 , the largest Fe content in Per among our samples, is G = 68.1 GPa, a 35% decrease from the value for Per 100 . The effect of this decrease in the shear modulus is to bring our normalized data for Per in our two-phase samples into closer agreement with the sub-grain size piezometer.

Zener Relationship
For two-phase materials, the Zener relationship relates the ratio of grain sizes of the two phases to the fraction of the secondary phase, f II , as where d I and d II are the grain sizes of the primary and secondary phases, respectively, m is an exponent that is determined by the geometrical distribution of secondary phase particles in the primary phase matrix as described below, and c is a constant (Evans et al., 2001;Manohar et al., 1998). Although the Zener relation in Equation 1 was originally conceived to describe the results of static annealing experiments (Manohar et al., 1998;Smith, 1948), studies on deformed natural and experimental samples have used the Zener relation to describe the grain sizes in their rocks (e.g., Herwegh et al., 2011;Linckens et al., 2011;Tasaka et al., 2020). As mentioned in Tasaka et al. (2020), the Zener relation does not change significantly between deformation and annealing experiments and is, therefore, applied to the results from our study.
To investigate the application of the Zener relationship to our samples, the ratio of grain sizes between Ol and Per is plotted versus secondary phase fraction in Figures 12a and 12b for samples with f Per ≤ 0.5 and for those with f Per ≥ 0.5, respectively. The Zener relationship from static annealing experiments for the forsterite (Fo) + enstatite (En) system from Hiraga et al. (2010) is included in Figure 12 for comparison. For samples with f Per ≤ 0.5, a linear least-squares fit to the data yields m = 0.27 ± 0.06 and c = 0.78 ± 0.06, while for samples with f Per ≥ 0.5, m = 0.67 ± 0.25 and c = 0.67 ± 0.20. The fact that the samples in this study can be described by a Zener relation indicates that the grain size of the primary phase is controlled by the presence of the secondary phase.  Figure 2; for example, in the sample in Figure 2d.ii with f Per = 0.2, small, equant Per grains are found primarily at Ol triple and quadruple junctions, while in the sample in Figure 4h.ii with f Per = 0.7, elongate Ol grains are primarily located along Per grain boundaries with some small Ol grains at Per triple and quadruple junctions. As a comparison to other two-phase systems, previous experimental studies on aggregates of Fo + En by Hiraga et al. (2010) and Tasaka and Hiraga (2013) determined values of m = 0.59 and m = 0.52, respectively. Although the value of m = 0.67 determined for samples in this study with f Per ≥ 0.5 is within error of the values for m in the Fo + En system, the value of m = 0.27 calculated for samples in this study with f Per ≤ 0.5 differs significantly from their values of m. Based on the work of Evans et al. (2001), the differences in values of m for samples with f Per ≤ 0.5 in this study likely reflect microstructural differences between samples in our study and the samples in the study of Hiraga et al. (2010). These differences can be observed by comparing the microstructures in Figure 2d to those in Figure 2 in Hiraga et al. (2010) and Figure 1 in Tasaka and Hiraga (2013), in which more secondary phase grains are present along Ol grain boundaries in their samples due to the similar grain sizes of the Fo and En phases. Additionally, the significantly larger size of Per grains relative to the Ol grains in our samples with microstructures displayed in Figures 2b, 2c, and 2e contributes to the smaller value of m determined for our samples.

Evolution of CPO
As mentioned in Section 3.1.2, notable differences exist between the CPOs of deformed single-phase samples and those of deformed two-phase samples. In the single-phase Ol sample, the CPO displays a strong pencil glide type fabric, while in two-phase samples the Ol CPOs exhibit a relatively weak A-type fabric with a weak, secondary C-type fabric. For Per CPOs, there are fewer maxima in each pole figure for the two-phase samples than for the single-phase sample, and the intensity of the <100> maxima normal to the shear plane is weaker in two-phase samples. Additionally, as f Per decreased, the Per CPOs in deformed two-phase samples became more random.
The presence of weak CPO patterns for Ol in two-phase samples may have been due to differences in Fe content between two-phase samples and our single-phase samples. However, we note that the CPOs for Ol in our samples prepared with Per 70 (Figures 6b.ii and 6c.ii), for which Fe-Mg exchange did not occur with SC-Ol, are the same as the CPOs in samples prepared with Per 100 or Per 90 (Figures 6d.ii-6j.ii), which underwent Fe-Mg exchange. Therefore, Fe content did not affect the strength of Ol CPOs in our samples. Instead, the weak CPO patterns for Ol in two-phase samples suggest that mechanisms in addition to glide of dislocations on the (010)[100] slip system contributed to strain accommodation in our two-phase samples. Langdon (2006) suggested that interface sliding involving grain rotation could contribute more to the overall deformation than dislocation glide in the dislocation-accommodated interface sliding regime in which grain size is smaller than the subgrain size. Tasaka et al. (2017) also reported weaker Ol CPOs at larger strains (γ > 17) than at lower strains (γ < 4) in their two-phase samples, a result that they attributed to the transition from dislocation-accommodated interface sliding with grain size larger than the subgrain size to dislocation-accommodated interface sliding with grain size smaller than the subgrain size. Based on the analysis of Ol deformation in our samples presented in our companion paper and the analysis of Ol grain sizes and subgrain sizes described above in Section 4.3.2, the interpretation that deformation was dominated by dislocation-accommodated interface sliding with grain size smaller than the subgrain size is consistent with our results. Finally, we note that as f Per increases and the grain size of Ol decreases in our samples, it is also possible that diffusion creep may be accounting for a larger proportion of the deformation of Ol alongside dislocation-accommodated interface sliding, which also contributes to the formation of weak CPOs.
In Section 3.1.2, we also noted the presence of weak secondary maxima in many of the Ol CPOs for two-phase samples. The secondary maxima constitute a C-type fabric in addition to the primary A-type fabric. Secondary maxima have been observed in Ol in the presence of melt (Qi et al., 2018) and in hydrated polycrystalline samples (Wallis et al., 2019), however, the cause of the secondary maxima in our samples is not clear. One possibility is that the relative strengths of the (010) [100] and (100)[001] slip systems changed. The relative strengths of slip systems in Ol depend on the activity of Opx (Bai et al., 1991). However, similar sets of secondary maxima were observed in Ol CPOs from experiments on Ol + Opx (Tasaka et al., 2017(Tasaka et al., , 2020 as were documented in the present study, so this feature of the Ol CPOs does not seem to be dependent on silica activity buffering conditions. Another possibility is that, in addition to inhibiting grain growth, the presence of secondary phases also inhibits grain rotation, which would tend to preserve the grain orientations of the starting material. Because of the weak initial CPO present in the hot-pressed material, pinning grain orientations could result in such secondary maxima in the deformed samples. In the Per CPOs, the number and arrangement of maxima for two-phase samples is similar to that observed at smaller strains (γ ≈ 1) in Per 80 (Heidelbach et al., 2003). These authors interpreted their CPOs in terms of dislocation activity dominated by the {111}<1 10> slip systems, which evolved to dislocation activity dominated by the {100}<011> slip systems by γ ≈ 3. Interpreting our results as a change in the dominant slip system in two-phase sample would also be consistent with the change between one-and two-phase samples in the relative strength of the maxima normal to the shear plane. As for Ol, the observation of weaker Per CPOs and nearly random Per CPOs in the case of samples with f Per < 0.5 may indicate that a larger proportion of the strain was accommodated by diffusion creep or interface sliding in addition to dislocation motion in these sample, thereby resulting in a randomized CPO. As another possible explanation for the weak Per CPOs, we note that Fe concentration can affect CPO development in Per, as demonstrated by Long et al. (2006). Rather than a weaker CPO, however, these authors observed a change in CPO pattern with increasing Fe content, namely a decrease from five maxima to four maxima in the <001> axes and an increase from four maxima to six maxima in the <111> directions. Long et al. (2006) also noted that the CPOs of their Fe-rich samples (Per 0 and Per 25 ) were similar to the high-strain fabrics reported by Heidelbach et al. (2003) for Per 80 , making the authors uncertain if the changes in CPO that they observed were due to increased Fe content or to degree of fabric development with strain in their samples. We noted a similar evolution of Per CPO with strain in our single-phase samples ( Figure S2 in Supporting Information S1) and, most importantly, that Per had similar CPO patterns in each of our two-phase samples with only the strength, not the number, of maxima changing. Therefore, it is unlikely that differences in composition from sample to sample contributed to weaker CPOs.

Conclusions
1. In our two-phase samples, deformation results in a change in the proportion of phase boundaries. With increasing strain, the density of phase boundaries evolves toward the binomial distribution that defines a random distribution of the two phases, rather than continuing to increase to an anti-clustered distribution. 2. Phase-boundary pinning is effective in our two-phase system of Ol + Per. Pinning results in smaller grain sizes in two-phase samples than in single-phase samples deformed under the same thermomechanical conditions and results in Ol grain sizes smaller than the predicted subgrain sizes. The grain-size data for samples with f Per ≤ 0.5 and for samples with f Per ≥ 0.5 are well-described by Zener-pinning relations. 3. The presence of multiple phases influences the CPO of each phase, which manifests as weaker fabrics and a change in the relative strength of maxima in the pole-figures. These results suggest an increasing contribution to deformation from diffusion creep and/or the activation of other, "harder", slip systems to accommodate the deformation.