Garnet yield strength at high pressures and implications for upper mantle and transition zone rheology

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2007jb004931, 2007 Garnet yield strength at high pressures and implications for upper mantle and transition zone rheology Abby Kavner 1,2 Received 8 January 2007; revised 10 June 2007; accepted 16 August 2007; published 29 December [1] Garnet helps control the mechanical behavior of the Earth s crust, mantle, and transition zone. Here, measurements are presented suggesting that garnet, long considered to be a high-viscosity phase, is actually weaker than the other dominant components in the transition zone. The mechanical behavior of garnet at high pressures was examined using radial diffraction techniques in the diamond anvil cell. The yield strength of grossular garnet was inferred from synchrotron X-ray measurements of differential lattice strains. The differential stress was found to increase from 1.3 (±0.6) GPa at a hydrostatic pressure 5.8 (±1.1) GPa to 4.1 (±0.4) GPa at 15.7 (±1.0) GPa, where it was level to 19 GPa. The strength results are consistent with inferred strength values for majorite garnet from measurements in the diamond cell normal geometry, bolstering the idea that garnetstructured materials may all have similar strengths. In this low-temperature, high differential stress regime, garnet is shown to be significantly weaker than anhydrous ringwoodite and to have a strength similar to hydrous ringwoodite. This result suggests that the presence of water in the transition zone may not be required to explain a weak rheology, and therefore models of transition zone behavior built assuming that garnet is the high-strength phase may need to be revised. Citation: Kavner, A. (2007), Garnet yield strength at high pressures and implications for upper mantle and transition zone rheology, J. Geophys. Res., 112,, doi: /2007jb Introduction 1 Earth and Space Science Department, University of California, Los Angeles, California, USA. 2 Institute for Geophysics and Planetary Physics, University of California, Los Angeles, California, USA. Copyright 2007 by the American Geophysical Union /07/2007JB004931$09.00 [2] The garnet component of the Earth s crust, mantle, and transition zone has been long thought to control aspects of the mechanical behavior of the mantle, because of garnet s inferred high strength in comparison with the other mineral components. Although the garnet structure is not the most predominant mantle mineralogy, garnet-rich compositions are inferred for continental crust material and the crust of subducting slabs. In addition, garnet becomes increasingly important with depth throughout the upper mantle and transition zone, since the pyroxene component dominant within subducting slabs transforms into majoritic garnet [Ita and Stixrude, 1992]. Garnet s role in crustal strengthening has long been acknowledged by field observations. These observations have been bolstered by a variety of laboratory experiments confirming the high strength of garnet and garnet bearing rocks relative to other crustal minerals and assemblages [e.g., Ingrin and Madon, 1995; Jin et al., 2001; Ji et al., 2003; Li et al., 2006a; Whitney et al., 2007; Zhang and Green, 2007]. Geophysical models for crust and mantle behavior have been developed on the basis of these strength measurements such as arguments for delamination of the continental crust, and separation of the crustal component of subducting slab within the transition zone [e.g., van Keken et al., 1996; Karato, 1997; Jin et al., 2001]. [3] Macroscopic rheology of the Earth is governed by the microscopic mechanical behavior of the component minerals. Therefore laboratory measurements of the mechanical properties of the relevant mineral assemblages are required to constrain the mechanical behavior of the mantle. Currently, most laboratory methods to assess mechanical behavior are limited because they cannot access all of the relevant conditions: the pressure, temperature, and strain rate corresponding to mantle processes. Therefore a variety of techniques have been used to assess the mechanical behavior of minerals under conditions accessible in the laboratory, and scaling laws have been developed to extrapolate the experimental results to deep Earth conditions, including rock mechanics measurements and the newly developed D-DIA press. While each of these techniques has a unique set of strengths and weaknesses, none can access the pressures corresponding to the lower part of the upper mantle and the transition zone. [4] Within the last several years, advancements in synchrotron-based radial diffraction techniques have been exploited in studies to measure the differential stress of materials at high pressures in the diamond anvil cell, of a variety of metals [Duffy et al., 1999a, 1999b; Kavner and Duffy, 2003; He and Duffy, 2006], oxides [Merkel et al., 2002; He et al., 2004; Speziale et al., 2006], silicates [Kavner and Duffy, 2001; Shieh et al., 2002, 2004; Merkel 1 of 9

2 other types of measurements. Finally, potential implications for the rheology of the Earth s mantle are discussed. Figure 1. Representative energy-dispersive X-ray diffraction patterns for grossular garnet, at highest compression. Data collected from diamond anvil cell (DAC) normal direction (gray line at top), minimum strain direction (gray line in middle), and maximum strain direction (black line at bottom) are shown. Indexed diffraction peaks are shown with arrows to indicate peak shifts in different patterns. The large peak at 32.0 kev is due to the Be gasket. Patterns are all normalized to the intensity of the (420) peak; 2q = for all patterns. et al., 2003; Kavner, 2003], and ultrahard materials [Amulele et al., 2006; Mao et al., 2006; Chung et al., 2007]. Many of these radial diffraction studies have been motivated by the necessity to understand the relationship between mineral strength and equation of state measurements at high pressures, in order to account for inconsistencies among different experiments that are likely introduced by nonhydrostatic strength [Kinsland, 1978; Kavner et al., 2000; Conil and Kavner, 2006]. [5] Besides their technical importance, these studies have also provided a unique window into additional mechanical behavior, especially yield strength, of Earth-relevant oxides and silicates. What can these measurements tell us about the Earth s mantle? Can we learn about material systematics relevant to the mantle using measurements exploiting different stress and strain rate regimes? Do other methods of rheology measurements, such as longstanding rock mechanics measurements [Evans and Goetze, 1979; Goetze and Evans, 1979; Kohlstedt et al., 1995; Jin et al., 2001; Raterron et al., 2004], and/or the newly developed deformation DIA press [Wang et al., 2003; Li et al., 2006a, 2006b] yield similar values for absolute and relative mineral strength despite significant differences in temperature, strain rate, and differential stress regimes? [6] To address these questions, the elastic limit of a grossular garnet subjected to large hydrostatic and differential stress was measured using radial diffraction techniques in the diamond anvil cell. Here the new results on grossular strength at high pressures are presented in comparison to other radial diffraction results on mantle relevant materials. These radial diffraction results are then shown in the context of other laboratory rock mechanics techniques, to evaluate the specific contribution of radial diffraction with respect to 2. Interpretation of Radial Diffraction Data [7] In a radial diffraction experiment, a polycrystalline sample is compressed without a pressure transmitting medium in a diamond anvil cell, and the lattice strains are measured through the side of the gasket via powder X-ray diffraction techniques, and analyzed as a function of orientation between minimum and maximum stress direction. The goal is to examine the lattice strain behavior of materials subjected to large differential stresses. Analysis of radial diffraction data for cubic materials is relatively straightforward via linear strain theory and the theory has been developed in a series of papers [Singh, 1993; Singh et al., 1998] with several examples of applications (see earlier citations). The assumptions, which will be explored in more detail in the discussion section, are that the sample consists of hundreds to thousands of randomly oriented crystallites, which are collectively subject to a homogeneous, cylindrically symmetric stress state (Figure 1). The diffraction geometry is oriented radially through an X-ray transparent gasket, rather than through the diamond normal direction. In the energy dispersive geometry, as in these experiments, X- ray diffraction patterns are then collected as a function of angle y about the axis of the incoming X-ray beam. [8] For a mineral with isometric crystal symmetry, the linear elastic strain theory predicts that lattice strain measured as a function of rotation angle y should lie along a curve determined by eðhklþ ¼ e hydro þ t 3 1 G agg ðhkl Þ 1 3 cos2 y ; ð1þ where e(hkl) is the measured strain of each lattice plane (each hkl), e hydro is the hydrostatic component of the lattice strain, and t is the supported differential stress. The strain here is defined so that it is negative under compression. The appropriate aggregate shear modulus G agg (hkl) is in general a function of the lattice plane (hkl) and is bounded by the moduli for constant strain and constant stress conditions, given by n o G agg ðhklþ ¼ a½2g R ðhklþš 1 þð1 aþ½2g V Š 1 ; ð2þ where G R (hkl) and G V are the lattice-dependent Reuss (constant stress) and Voigt (constant strain) aggregate shear moduli, and a has a value between 0, corresponding to the constant strain condition, and 1, corresponding to constant stress. Note that for an elastically isotropic material G R and G V are identical, and even for an elastically anisotropic material, the average of all of the (hkl)-dependent lattice strains analyzed using the Reuss bound on the shear modulus yields a result similar to that obtained with the Voigt bound [e.g., Kavner and Duffy, 2003]. For simplicity, in this study garnet is treated as if it were elastically isotropic; this assumption is examined in section Experimental Procedure [9] A natural polycrystalline grossular-rich garnet with a composition of Grs 87 And 9 Pyp 2 Alm 2 [Jiang et al., 2004] 2 of 9

3 [11] Equations (1) and (2) are appropriate only if the cylindrically symmetric sample stress tensor is aligned with the diamond cell axis. To test this assumption, and to quantify any rotation of the sample tensor with respect to the diamond cell tensor, the fit of the lattice parameter versus y data was performed with a function that includes the possibility of a phase shift, y phase, aðhklþ ¼ a hydro þ 1 h i 3 a ampðhklþ 1 3 cos 2 y y phase : ð3þ Figure 2. Lattice parameters for the (400) diffraction line as a function of y at each of the pressure steps. The 1 3cos 2 y fits are shown. was loaded in a diamond cell sample chamber consisting of a preindented beryllium gasket with a drilled 100 mm hole. To maximize nonhydrostatic stresses, no separate pressure medium was used within the sample chamber. White radiation diffraction experiments in the radial geometry were performed at beam line X17C of the National Synchrotron Light Source, following earlier procedures [Kavner, 2003; Kavner and Duffy, 2001, 2003]. The incident X-ray beam was focused to a spot size of approximately 10 mm 15 mm, and the diffraction angle was maintained at a constant 2q = 12.44(1) which was calibrated using a gold foil. The detector energy was calibrated using a series of X-ray fluorescent standards. Energy dispersive diffraction patterns were obtained in the radial geometry, with the X-ray beam passing through the side of the gasket, and between the two diamond faces. At each of the four pressure steps, several diffraction patterns were obtained as a function of rotation angle, y, which is the rotation of the diamond cell about the bisector line between the incoming and diffracted X-ray beam. At y = 0 the crystallites closely oriented to sense the maximum lattice strain are sampled (the lattice response to the maximum principle stress along the diamond axis); when y = 90, the measured strain closely corresponds to the minimum stress direction in the sample (the lattice response to the minimum principle stress, in the radial (gasketting) direction). Before each diffraction pattern was collected, the sample was laterally scanned in two dimensions orthogonal to the incoming beam to maintain the diffracting volume at the sample center. [10] At each pressure step, garnet diffraction patterns were collected at five to ten different y angles, and each peak was indexed with the appropriate Miller indices (Figure 1). Lattice parameters were determined separately for each peak via a Gaussian fit for the center energy, and by calculating the corresponding d-spacing using Bragg s law adapted for energy dispersive diffraction. Separately for each (hkl), the lattice parameter was plotted as a function of rotation angle, y (e.g., (400) peak in Figure 2). Since garnets have a cubic crystal structure, each d-spacing provides an independent measurement of the garnet lattice parameter. Therefore, for each lattice plane, and at each pressure step, three independent parameters are obtained from the data: a hydro, the hydrostatic value of the lattice parameter; a amp (hkl), the amplitude corresponding to the difference between the lattice parameters at the maximum and minimum stress directions; and a phase shift, y phase. These fit parameters for each data set are shown in Table Results and Discussion 4.1. Strain Data [12] At each compression step, the pressure was calculated by fitting the average hydrostatic value of the lattice strains to a Birch-Murnaghan equation of state using the bulk modulus K 0T = 163.8(5) GPa and dk/dp = 3.9(2) [Jiang et al., 2004]. Strains are calculated from an average of the hkl-dependent lattice parameter data (with the exception of the (444) lattice plane, as described below), using a finite strain definition where compressional strains are negative, and a room pressure lattice parameter of a 0 = Å [Jiang et al., 2004]. The lattice parameters at maximum, minimum, and hydrostatic stress conditions are shown in Table 1, and values for the (400) line are plotted in Figure 2. [13] Deviations from cylindrical symmetry can be assessed by examining the phase shift, y phase, which is indeed present in the grossular data, and it increases to an average of 17 degrees at the highest pressure (Table 1 and Figure 3). This indicates a rotation of the stress tensor of grossular with respect to the diamond cell axis. This effect also was observed in radial diffraction experiments on Pt [Kavner and Duffy, 2003] but not with ringwoodite [Kavner and Duffy, 2001] or hydrous ringwoodite [Kavner, 2003]. Although the full three-dimensional stress tensor information could not be established in these experiments, it is likely that the stress environment can no longer be considered uniaxial. Ultimately, this will render the strength measurements to be a lower bound on the maximum differential stress experienced by the grossular garnet. However, the increasing phase shift suggests mechanical deformations induced by plastic deformations within the sample, further bolstering the argument that the sample has reached its yield strength. [14] The lattice parameters at maximum stress considerably underestimate the bulk modulus at each pressure step, and the minimum stress lattice parameters overestimate the bulk modulus. As shown by Figure 3, these misestimates can be significant: a bulk modulus of 204 GPa is inferred if only data obtained in the minimum stress direction were analyzed (holding V 0 and dk/dp constant); a bulk modulus as low as 114 GPa is inferred when only the maximum 3 of 9

4 Table 1. Results of 1 3cos 2 y Fits From Radial Diffraction Data a Pressure, GPa t, GPa hkl a hydro, Å a amp = a min a max, Å Phase Shift, degrees 5.8(±1.1) 1.3(±0.6) (±0.025) (±0.019) 0 b (±0.007) 0.013(±0.012) (±0.008) 0.060(±0.011) (±0.02) 0.053(±0.014) (±0.009) 0.097(±0.013) (±0.03) 0.067(±0.035) (±0.017) 0.193(±0.02) (±0.036) 0.133(±0.028) (±0.064) 0.038(±0.028) (±0.018) (±0.009) 15.7(±1.0) 4.1(±0.4) (±0.018) 0.192(±0.027) 2(±9) (±0.016) 0.161(±0.011) (±0.011) 0.193(±0.009) (±0.021) 0.233±(0.012) (±0.007) 0.212(±0.003) (±0.025) 0.161(±0.017) 17.9(±0.5) 4.2(±0.3) (±0.006) 0.189(±0.026) 9(±4) (±0.009) 0.199(±0.006) (±0.0039) 0.191(±0.002) (±0.015) 0.204(±0.010) (±0.009) 0.173(±0.006) (±0.009) 0.107(±0.005) (±0.010) 0.221(±0.008) (±0.019) 0.148(±0.011) 19.0(±1.4) 4.0(±0.6) (±0.023) 0.175(±0.046) 15(±13) (±0.010) 0.163(±0.007) (±0.014) 0.159(±0.008) (±0.011) 0.215(±0.007) (±0.024) 0.120(±0.016) (±0.030) 0.245(±0.021) (±0.030) 0.093(±0.019) (±0.031) 0.198(±0.023) (±0.016) 0.127(±0.010) a At each pressure step, the first line (in bold) contains the average values for all of the diffraction lines except the (444) line. Data for individual lines are also shown. b The phase shift for 5.8 GPa data is constrained to 0. lattice strains are analyzed. Two factors make this anomalous result important to note. First, the diamond cell normal direction, which is the usual geometry for the collection of X-ray diffraction data, yields strain data from close to minimum stress direction. Therefore equation of state measurements obtained in the diamond cell normal geometry err on the side of underestimating the hydrostatic strain, therefore providing an overestimate of the bulk modulus (and/or its pressure derivative). Second, elastically isotropic materials such as garnet do not necessarily show an (hkl) dependence at the minimum strain direction. Therefore lattice parameter dependence, often invoked as a marker for the presence of nonhydrostaticity [e.g. Shim et al., 2000; Kavner and Duffy, 2003], is a second-order effect at best and not necessarily even present, as in the case of isotropic materials. Clearly, this experiment was designed to maximize nonhydrostatic stresses; therefore the results are likely an exaggeration of smaller effects relevant to most quasihydrostatic experiments Lattice Anisotropy [15] Any dependence of the minimum and maximum strains on the lattice planes, as quantified by their angle cosines, (G(hkl)) (the angle cosines corresponding to the lattice planes, equal to (h 2 k 2 + k 2 l 2 + h 2 l 2 )/(h 2 + k 2 + l 2 ) 2 ) provides an indication of lattice anisotropy. However, there is a well-documented ambiguity in the interpretation of this lattice anisotropy, which can arise owing to either elastic or plastic (strength) effects [Weidner et al., 2004]. Our data show that with one exception (described in more detail below), an analysis of the lattice plain behavior in the minimum and/or maximum stress directions yields no systematic strain versus G(hkl) dependence, indicating that these experiments do not resolve the presence of any lattice anisotropy. Therefore we do not observe the increase in elastic anisotropy with pressure measured in recent Brillouin spectroscopy studies [Jiang et al., 2004]. However, this low-anisotropy result is in agreement with an earlier study on majoritic garnet in which no lattice-dependent anisotropy 4 of 9

5 Figure 3. Measured lattice parameter (left axis) and inferred volume (right axis) as a function of pressure for garnet. Hydrostatic orientation (solid squares) and maximum and minimum stress orientations (open squares) are shown. Bold curve shows the P(V) equation of grossular garnet from Jiang et al. [2004]. Other curves show the best fit equations of state through the maximum and minimum strain data, holding V 0 and K 0 0 constant. was observed in the minimum stress direction [Kavner et al., 2000]. [16] Anomalous behavior was observed in the (444) diffraction line, which yielded a consistently higher lattice parameter than the other diffraction lines. In addition, at the higher pressure steps, the difference between the lattice parameters inferred for the maximum and minimum stress directions for the (444) peak is significantly less than that for the other peaks, indicating that this plane shows less elastic strain (Table 1). A simple interpretation would be that the (444) plane is weak: it cannot support significant differential stress perhaps because it involved with a preferred slip system in garnet [e.g., Mainprice et al., 2004]. Because we only see this anomaly in the (444) peak, and there are no systematic lattice-plane dependent behavior within resolution of our data, we eliminated the (444) peak from our analysis of hydrostatic pressure and differential stress Supported Differential Stress [17] At each pressure step, the differential stress supported by grossular as determined by lattice strains is given by t ¼ 2G V je max e min j; ð4þ where e max and e min are the lattice strains in the maximum and minimum stress directions. To calculate the shear modulus at pressure, we use a finite strain extrapolation with G V = 104.2(0.3) GPa with dg/dp = 1.1 (1) [Jiang et al., 2004]. The elastically supported differential stress of grossular garnet starts at about 1.3 (±0.6) GPa at P hydro = 5.8 (±1.1) GPa and then rapidly increases to a maximum of 4.1 (±0.4) GPa at P hydro = 15.7 GPa, where it remains approximately constant upon further compression to P hydro = 19.0 (±1.4) GPa. The gasket failed upon further compression, terminating the experiment. The differences in measurements with and without the (444) peak are minimal. For example, at the highest pressure step, the pressure analyzed with the (444) peak is 26.5 GPa, compared with 27.8 GPa without the (444) line; the differential stress measured with the (444) peak is 4.05 GPa, compared with 3.17 without the (444) line. [18] Although the theory used to interpret radial diffraction strains has been developed for a homogeneous material subjected to a uniform stress state, it is likely that in reality the differential stress is nonuniform within the sample. Indeed, earlier measurements of mineral strength at high pressures exploit the measured stress gradient to estimate yield strengths [Meade and Jeanloz, 1988a, 1988b, 1990a, 1990b]. Most radial diffraction analysis is performed assuming a uniform stress state. Studies which have compared strengths inferred by radial diffraction methods and by stress gradient methods have shown that the two methods are in rough agreement [Kavner and Duffy, 2001]. In these radial diffraction experiments, it is likely that some averaging over a stress gradient within the diffracting volume occurs; however, the experimental protocol was designed to insure that the same diffracting volume is maintained as the sample is rotated about y. The volume of grossular sampled by the X-ray diffraction beam is approximately constant with the 10 mm 10 mm beam size. Therefore it is likely that the X-ray beam is sampling a variety of pressures. Therefore both the hydrostatic and deviatoric components of the strain determined from the radial diffraction data are interpreted as values spatially averaged over the center volume of the sample. [19] We interpret the measured differential stress to be a lower bound on the material s actual yield strength: the differential stress required to induce plastic deformation, based on the von Mises yield criterion [Dieter, 1986]: t ¼ s max s min s yield : [20] Interpreting the t values directly as a yield strength requires that the sample has undergone plastic deformation in additional to the X-ray observed elastic deformation (i.e., sample is beyond its elastic limit). In these experiments, we were not able to directly measure plastic strain. However, several lines of evidence suggest that plastic deformation has indeed occurred by at least the highest pressure point. First, the observed differential stress levels off to a constant value after a steep initial increase. Also, the gasket thickness was observed to decrease from an initial thickness of 40 mm to a final thickness of 20 mm, indicating a 50% strain in the axial dimension. In addition, the experiment was terminated because the gasket hole expanded during compression, contributing to significant (10 s of %) in the radial dimension. Both of these strains are significantly larger than the elastic strains, and provide an indication of the presence of plastic deformation during the experiment. Other potential indicators of plastic deformation include: broadening of the diffraction peaks [Weidner, 1998] and the development of lattice preferred orientation [Merkel, 2006]. In addition, the observed rotation of the principle ð5þ 5 of 9

6 Figure 4. A summary of radial diffraction measurements of differential stress inferred from lattice strain for grossular garnet and other mantle materials. Abbreviations and references are as follows: RW, ringwoodite [Kavner and Duffy, 2001]; h-rw, hydrous ringwoodite [Kavner, 2003]; MgO [Duffy et al., 1995]; SiO 2, stishovite [Shieh et al., 2002]; Maj, majorite [Kavner et al., 2000]; olivine [Uchida et al., 1995]; CaSiO 3 [Shieh et al., 2004]. strain direction with respect to the diamond cell axis may be an indicator of plastic deformation within the sample chamber Comparison With Radial Diffraction Measurements on Other Silicates and Oxides [21] A comparison of differential stress of mantle-relevant silicate and oxides measured by radial diffraction at room temperature is summarized in Figure 4. The garnet measurements described by this data set are in qualitative agreement with yield strengths inferred from earlier measurements of majorite garnet under nonhydrostatic conditions in the diamond cell normal direction [Kavner et al., 2000]. However, another study suggests some compositional dependence on garnet strength, with hardness, fracture toughness, and indentation moduli values consistently lower for grossular than for pyrope garnet [Whitney et al., 2007]. Karato et al. [1995] point out that normalized yield stress values (the ratio of the yield stress to shear modulus) converge for the garnet series. [22] Our measured differential stress for garnet initially increases strongly, after which it levels off at pressures above 15 GPa (Figure 4). This increase and leveling is seen for other materials as well in this pressure range including olivine [Uchida et al., 1995] and MgO [Duffy et al., 1995]. This may occur owing to work hardening or perhaps because the sample has reached a maximum supported stress, or this may occur owing to an incipient phase transformation, such as that observed in SiO 2 as it undergoes the coesite/stishovite transformation [Shieh et al., 2002]. A comparison of supported differential stress determined by radial diffraction techniques shows that garnet is significantly weaker than ringwoodite [Kavner and Duffy, 2001], the other dominant mineral component of the transition zone [e.g., Ita and Stixrude, 1992]. In fact, the garnet yield strengths are close to those values measured for hydrous ringwoodite [Kavner, 2003]. The garnet strengths inferred here are lower than those measured for MgO [Duffy et al., 1995] and olivine below about 15 GPa [Uchida et al., 1995] and are commensurate with similar measurements on stishovite [Shieh et al., 2002], calcium silicate perovskite [Shieh et al., 2004], and olivine strength values extrapolated above 15 GPa Comparison With Other Garnet Measurements [23] The key challenges for mantle rheology studies are cross-comparing measurements obtained using different techniques, and extrapolating experimental results to the stress, temperature, and strain rate conditions of the deep Earth. This requires knowing the dominant deformation mechanism [e.g., Frost and Ashby, 1982] not only relevant for the experimental conditions but also applicable in the Earth s mantle. Deformation mechanisms relevant to the mantle are discussed in the literature [e.g., Karato and Wu, 1993]. Because of the high differential stress and lowtemperature conditions of the radial diffraction measurements, it is likely that dislocation glide is the dominant deformation mechanisms in these samples. However, as discussed below, grain size-controlled grain boundary creep is another possible limiting deformation mechanism. A summary of the state-of-the-art knowledge of garnet rheology measurements is provided by Li et al. [2006a]. Direct cross-comparison of rheology measurements using different techniques can only be performed if the dominant deformation mechanism is the same, and its temperature, stress, and strain rate dependences are well determined. [24] Within a single deformation mechanism, a flow law relating the strain rate, stress, and thermal activation energy can be used to extrapolate behavior to different strain rates, stresses, and temperatures: _e ¼ As n e Q RT ; where _e is the strain rate, A is a constant (which may contain additional information, for example grain size), s is the stress, n is the stress exponent which is dependent on the deformation mechanism, Q is an activation energy, also mechanism dependent, and R and T are the gas constant and temperature. At low temperatures the strain rates predicted by material flow laws are dominated by the activation energy of the defects responsible for deformation, which ranges from 270 (±40) KJ/mol [Li et al., 2006a] to 347 (±46) KJ/mol [Wang and Ji, 2000]. [25] Because the low temperatures of the radial diffraction measurements cause the exponential term in equation (6) to dominate, our strength results are not inconsistent with flow laws determined for pyrope garnet [Li et al., 2006a]. [26] Although the absolute value of our strength measurements are not inconsistent with the high-temperature flow laws, the radial diffraction systematics seem to contradict previous studies indicating that garnet and garnet-bearing rocks have systematically higher strength than other upper mantle and transition zone minerals [e.g., Ingrin and ð6þ 6 of 9

7 Madon, 1995; Karato et al., 1995; Jin et al., 2001; Li et al., 2006a; Whitney et al., 2007; Zhang and Green, 2007]. However, in a recent study creep strength of garnetite was found to be comparable to feldspar and olivine in recent deformation creep experiments at 0.1 MPa pressure and temperatures in the range K [Wang and Ji, 2000]. In contrast to the dislocation-controlled experiments cited above, Wang and Ji [2000] show a grain-size dependence on the measured creep of garnetite, suggesting that if grain boundary diffusion creep is the dominant mechanism for deformation, then garnet strength may not be limiting the rheology. This experimental result is bolstered by some field observations showing superplasticity in garnet in a subduction environment [Terry and Heidelbach, 2004]. Therefore, although previously it was assumed that radial diffraction measurements are examining dislocation glidemoderated rheologies, it is possible that grain boundary creep may be playing a role in limiting the strength in these measurements Strengths and Weaknesses of the Radial Diffraction Approach [27] Radial diffraction in the diamond cell is unique in that it can access the stable mineralogy at mantle relevant high pressures, and it has the potential to access two very different strain rates. The technique s major drawbacks are that the data are obtained at room temperature and under a large differential stress, especially compared with the mantle. [28] Two very different strain rates are accessed by the diamond cell experiments. The differential stress is loaded with a very high strain rate: the diamond cell is compressed on a minute-long timescale, during which time 1% elastic strains are generated, and likely (unmeasured) plastic strain is generated as well. This compression corresponds to strain rates on the order of 10 4 s 1. In addition, a second strain rate can be accessed if the elastic relaxation of the sample is measured via X-ray diffraction peak shifts over the course of many hours. This transient creep, corresponding to strain rates of 10 9 s 1, was observed in Pt metal [Kavner and Duffy, 2003]. This corresponds to strain rates of 10 9 s 1, much lower than the D-DIA, which routinely accesses strain rates of s 1 [Wang et al., 2003]. In this experiment we observed no transient creep response of grossular over a 24-h period, which provides a lower bound limit on a creep dynamic viscosity, which is determined by the ratio of the shear stress to the strain rate. Using values for shear stress 4 GPa, and a bound on the strain rate of 10 9 s 1, a lower bound room temperature dynamic viscosity of Pa s is inferred. An interesting implication to this bound is that it implies that viscosities up to this value are resolvable using this diamond cell technique. [29] Radial diffraction is an important technique for ascertaining the extent of nonhydrostaticity in diamond cell samples. Although the observed differential stresses are significantly larger than in a diamond cell experiment designed to measure hydrostatic pressure-volume equation of state, the resulting errors in determining bulk modulus parameters can be directly scaled down, since they arise mostly owing to linear elastic effects. Here we show that a differential stress ranging from 1 to 4 GPa over 20 GPa overestimates the bulk modulus by almost 25%, measuring a value 200 GPa for a material with a bulk modulus closer to 163 GPa if the X-ray diffraction were performed through the diamond window (Figure 4). This error is consistent with an inference for anomalously high bulk modulus measurements from data collected from majorite garnet [Kavner et al., 2000]. Therefore a smaller and more likely differential stresses of GPa in a quasi-hydrostatic experiment could still overestimate bulk modulus by several percent. In addition, the tradeoffs between the bulk modulus and its first derivative could cause an anomalously high value of the pressure derivative of the bulk modulus if the bulk modulus value were fixed to an independent measurement, as was also observed in the majorite garnet data. 5. Conclusions [30] No single technique developed so far can simultaneously access the high pressures, high temperatures, low differential stresses, and low strain rates relevant to mantle processes. This current work will fit into a larger-scale cross calibration of systematic mineral rheology using different techniques at low temperatures such as micro-indentation and nano-indentation [e.g., Whitney et al., 2007] and at high temperatures using slow strain rate deformation techniques such as the D-DIA. Ultimately, these measurements will provide the information that will be useful to extrapolate rheology measurements to deep Earth conditions. If the radial diffraction strength systematics among materials, their relative strengths, hold true with increasing temperature and lower differential stresses, then this result suggests that the garnet phase within the upper mantle and transition zone may not be significantly rheologically stronger than the other components. [31] Taken at face value, these results imply that garnet may not be as strong a phase as has been long inferred. The relatively high differential stress in radial diffraction experiments and the significantly lower temperatures preclude a direct comparison with the Earth s mantle. However, this conclusion is consistent with earlier conclusions that garnetrich materials may not be anomalously strong, inferred from results obtained using creep experiments at higher temperatures but lower pressures [Wang and Ji, 2000]. This may imply that garnet rich compositions, such as an eclogite layer within a subducting slab, and the slab itself as it transforms with depth, may actually be weaker than the surrounding layer. The transition zone, which is rich in garnet and majoritic garnet phases, may be weaker than the upper and lower mantles, an idea that has been previously ascribed to the presence of water. Given the large dichotomy in yield strength between ringwoodite and garnet, similar in scale to that between hydrous and anhydrous ringwoodite, a mineralogically based viscosity filtering effect between upper and lower mantle is allowed without necessarily invoking the presence of water. [32] Garnet deformation can be seismically observed in the mantle and the transition zone if garnet s single crystal elastic constants are anisotropic and if deformation induces a lattice-preferred orientation. In a polycrystalline material, lattice-preferred orientation can develop only if strains can be accommodated by rotations of single grains. The observed development of some preferred orientation in these 7 of 9

8 experiments, indicated by nonuniform relative intensities in the diffraction patterns (e.g., the difference between the intensity of the (640) and (444) peaks in the minimum and maximum strain directions shown in Figure 1) suggests that that garnet is likely to take on a lattice preferred orientation as a response to the application of an external stress. In addition, Brilluoin scattering results suggest that the elastic anisotropy of garnets increase with pressure [Jiang et al., 2004]. Therefore seismic anisotropy techniques may become increasingly sensitive to garnet deformation behavior in the midmantle [Montagner, 2002; Park and Levin, 2002]. [33] Acknowledgments. A. K. thanks Jingzhu Hu at the National Synchrotron Light Source for aid in the experiments and S. Speziale for providing the grossular garnet sample. Use of the National Synchrotron Light Source, Brookhaven National Laboratory, was supported by the U. S. 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