High-Pressure Deformation of Iron–Nickel–Silicon Alloys and Implications for Earth’s Inner Core

1Department of Earth and Planetary Sciences, Harvard University 2Department of Geology and Geophysics, University of Utah 3Sorbonne Université; Muséum National d'Histoire Naturelle; Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie 4Université Grenoble Alpes; Université Savoie Mont Blanc; CNRS; IRD; Université Gustave Eiffel; ISTerre Corresponding author: Matthew C. Brennan (mcbrennan@g.harvard.edu)


Introduction
Since the discovery of the inner core (Lehmann, 1936) and its identification as a solid iron alloy (Birch, 1952), seismic studies have revealed it to be complexly structured and seismically anisotropic. On average, seismic waves move through the inner core several percent faster on paths parallel to Earth's axis of rotation compared to perpendicular paths (e.g., Deuss, 2014). This anisotropy is well documented by travel time (e.g., Creager, 1992) and normal mode (e.g., Durek & Romanowicz, 1999) observations, and deviations from this large-scale anisotropy have been used to suggest other inner core structural features (e.g., Irving & Deuss, 2011;Ishii & Dziewoński, 2002;Ouzounis & Creager, 2001;Su & Dziewoński, 1995). These secondary structures are frequently invoked as evidence for various proposed mechanisms of the primary anisotropy's development.
As with many instances of seismic anisotropy, the Earth's inner core structure likely arises from preferred orientation of crystallites (i.e., texture). Solid Fe-rich alloys (including those in the Fe-Ni-Si system) are expected to exist in the hexagonal close-packed (hcp) crystal structure (ε-Fe) at inner core pressures (P) and temperatures (T) (e.g., Sakai et al., 2011;Tateno et al., 2010Tateno et al., , 2012Tateno et al., , 2015. Since hcp crystals are seismically fast parallel to their c-axes (Bergman et al., 2000), a preferential alignment of the inner core's crystals with Earth's axis of rotation is widely accepted as responsible for the inner core anisotropy (e.g., Antonangeli et al., 2006;Sakai et al., 2011;Sayers, 1989;Stixrude & Cohen, 1995;Vočadlo et al., 2009). This explanation is experimentally supported by observations of strong c-axis aligned deformation textures in hcp metals, including ε-Fe (e.g., Antonangeli et al., 2006;Merkel et al., 2012Merkel et al., , 2013Miyagi et al., 2008;Wenk et al., 2000). It is worth noting that cubic Fe polymorphs cannot deform to produce a texture compatible with the observed anisotropy (Lincot et al., 2016).
There is considerable diversity amongst the mechanisms proposed to create this texture. One class of models relies on preferred orientation developing during crystallization (e.g., Karato, 1993;Bergman, 1997). However, if crystals instead solidify in random orientations, texture must develop by plastic flow in the solid aggregate. This can result from solid-state convection (e.g., Buffett, 2009;Deguen et al., 2013;Jeanloz & Wenk, 1988) or from west-to-east translation due to preferential solidification in one hemisphere (e.g., Alboussiere et al., 2010;Bergman et al., 2010;Monnereau et al., 2010). Flow may also be driven by forces external to the inner core, such as electromagnetism (e.g., Buffet & Wenk, 2001;Takehiro, 2011) or outer core convection (e.g., Aubert et al., 2008;Yoshida et al., 1996). Experimentally, Gleason & Mao (2013) found that Fe is very weak at ultra-high pressures. If this holds true for the inner core, post-solidification texturing could be driven by almost any source of differential stress, making it difficult to discriminate between various models.
An important caveat of previous studies is that they have investigated pure ε-Fe, but the core is known to also contain other elements. Ni is expected to be present at approximately 5 weight percent (wt%) (e.g., McDonough & Sun, 1995), but Fe-Ni is too dense and seismically fast to account for core properties, so a much lighter element must also be present (Birch, 1964). The identity of the light element(s) is one of the most controversial problems in deep Earth geophysics, but Si is a leading contender based on its ability to partition into the metallic phase during core formation (e.g., Siebert et al., 2013) and the isotopic (Georg et al., 2007) and major-element (Ringwood, 1961) ratios of terrestrial rocks. Estimates of inner core Si based on density range as high as 7 wt% (Fischer et al., 2014;Tateno et al., 2015), but this should be considered an upper bound since Si is unlikely to be the sole alloying light element (e.g., Edmund et al., 2019;Ozawa et al., 2016).
The presence of alloying elements may alter the deformation mechanisms and resultant texture of the inner core. Metallurgical studies have demonstrated that Ni (e.g., Ledbetter & Reed, 1973) and Si (e.g., Ros-Yanez et al., 2007) can significantly alter the mechanical properties of Fe. A previous experimental study (Reagan et al., 2018) on Fe-Ni alloys at high pressure did not characterize textures but found that addition of 5.5 wt% Ni could increase the strength (defined as the shear stress needed to induce plastic deformation) of Fe by 25% at inner core conditions. This increase alone is unlikely to change the dominant mechanism of inner core texturing, but a light element, possibly combined with strain hardening (Bergman et al., 2014), could further strengthen the alloy. Geodynamical models have difficulty reproducing seismic observations by imposing plausible flows on ε-Fe (Lasbleis & Deguen, 2015;Lincot et al., 2014), but this issue may disappear if alloy deformation textures are distinct from those of pure Fe. In this study, we measure both the strength and texture of deformed Fe-Ni-Si alloys to determine whether an alloying light element significantly changes the mode of plastic deformation or the interpreted strength of the inner core.

Methods
We investigated the properties of five Fe-Ni-Si alloys: Fe-9wt% Si (hereafter Fe-9Si) was a powder (Goodfellow FE166014); Fe-6Ni-8Si was synthesized in a piston-cylinder press at Harvard University (composition checked by energy dispersive spectroscopy); and Fe-5Si, Fe-5Ni-5Si, and Fe-5Ni-10Si were synthesized as foils by ultra-rapid quenching at the Institut de Chimie et des Matériaux de Paris-Est (details in Morard et al., 2011). Non-foil samples were loaded as powders and foil samples were laser-drilled into cylindrical slugs. All samples filled the sample chambers, which were 50 μm diameter cylindrical holes drilled into amorphous boron gaskets contained in Kapton supporting rings. This configuration allows X-rays to pass radially through the gasket and sample (Merkel & Yagi, 2005). A Pt flake was loaded along with each sample as a pressure standard (Fei et al., 2007). Samples were compressed between diamonds with 200 μm diameter culets in panoramic piston-cylinder type diamond anvil cells to initial pressures of 2-3 GPa. Additional compression was accomplished by an inflating gas membrane once the cell was on the beamline. We repeated the Fe-5Si experiment with double-sided laser heating during compression. For that experiment, gaskets were laser drilled from cubic boron nitride, and MgO flakes were used as thermal insulators and pressure standards (Speziale et al., 2001).
Experiments were conducted at Advanced Light Source beamline 12.2.2 (λ = 0.4613 or 0.4592 Å) with the diamond anvil cell mounted with its compression axis orthogonal to the Xray beam (Kunz et al., 2005). The detector was a mar345 positioned 385 or 386 mm from the sample, and the calibrant was CeO2 at ambient conditions. For each pressure step, a diffraction pattern was collected with a 120 second exposure. We used the program Dioptas (Prescher & Prakapenka, 2015) for live pattern integration, which allowed us to monitor phase changes and sample pressure during the experiment. Diffraction patterns were collected as the samples were compressed gradually to 50-60 GPa, near the maximum stability of the boron gaskets (Merkel & Yagi, 2005). Membrane pressure was then released over several hours while simultaneously collecting decompression patterns. Diffraction data were analyzed via Rietveld refinement (McCusker et al., 1999) in the crystallographic software package MAUD (Lutterotti et al., 1999). We analyzed several diffraction patterns for each experiment, including at least one pattern in the body-centered cubic (bcc) and the bcc-hcp transition regions, and at least four in the hcp region. Patterns were chosen based on their diffraction intensity, possession of sufficient Pt diffraction to refine an accurate pressure, and equally spaced coverage of the experiment's pressure range. These patterns were refined over a 2θ range of 11.5-25°and refined unit cell dimensions and elastic strains were compiled for the hcp phase of each sample (Supporting Information Table  S1). Textures were calculated using the Extended Williams-Imhof-Matthies-Vinel (E-WIMV) model (Chateigner et al., 2019) with fiber symmetry imposed about the axis of compression. Orientation density functions were refined at a resolution of 15°, exported to BEARTEX (Wenk et al., 1998), and smoothed with a 10° Gaussian filter prior to plotting as inverse pole figures (Section 3). The pressures of phase transitions in 300 K samples may not necessarily be the same as at equilibrium due to sluggish kinetics, starting material effects, and non-hydrostaticity, but the measured texture and strength of the phases present are still accurate.

Deformation Textures
As in pure Fe, our alloy samples transformed from the bcc (or bcc-like: Fischer et al., 2013) structure to the hcp structure upon compression. The deformation textures of our roomtemperature experiments are shown in Figure 1. The bcc textures are near-identical to those of pure Fe (e.g., Merkel et al., 2004;Miyagi et al., 2008), varying only in the strength of the 100 maximum. The similarity across compositions suggests that sufficient plastic flow occurred prior to the bcc-hcp transition to homogenize the textures (except for possibly in Fe-9Si, which may have led to its distinctive hcp texture). Once the transition to hcp is complete, the alloy textures show two distinct maxima: one near 112 � 0 and one near 0001. The non-foil samples (Fe-9Si and Fe-6Ni-8Si) are distinguished by the absence of the latter maximum; these were mechanically strong (Section 4) but weakly textured, perhaps indicating that plastic flow was impeded in these experiments. The non-foil textures are likely the result of compressive {21 � 1 � 2}〈21 � 1 � 3〉 twinning. This deformation mechanism is known to occur at large strains and low temperatures in ε-Fe (Kanitpanyacharoen et al., 2012;Merkel et al., 2004), so its presence at 300 K in our strongest alloys is reasonable. The foil experiments were more strongly textured (showing strong maxima at 112 � 0 and 0001), so they may be better representatives of texture development by dislocations (the type of deformation texturing proposed to occur in the inner core, see Section 4.4). Polycrystal plasticity simulations of hcp metals (e.g., Chapuis & Liu, 2015;Kanitpanyacharoen et al., 2012;Miyagi et al., 2008;Wenk et al., 2000) have shown that basal (0001)〈21 � 1 � 0〉 slip produces the 0001 maximum, while pyramidal {21 � 1 � 2}〈21 � 1 � 3〉 slip produces the 112 � 0 maximum. A large pyramidal contribution can result in a ~30° shift in the 0001 maximum (Miyagi et al., 2008;Merkel et al., 2013), as is seen in the Fe-5Ni-10Si textures. Together, these foil textures show evidence of significant pyramidal slip. Since basal slip is typically observed to dominate ε-Fe deformation at 300 K (e.g., Nishihara et al., 2018) we surmise that addition of Si increases the activity of pyramidal slip. Prismatic {101 � 0}〈1 � 21 � 0〉 slip is also likely to be active without producing a distinct textural signature (Merkel et al., 2004).

Figure 1.
Inverse pole figures (IPFs) showing alloy deformation textures at room temperature. The bcc textures were collected before the appearance of hcp, "transition" textures were collected in the bcc-hcp coexistence region, "compression" textures were collected during compression near peak sample pressure (in the hcp-only regime), and "decompression" textures were collected after the sample had relaxed by 5-10 GPa. Textural differences appear upon transformation to hcp and become more pronounced upon compression. Sample textures remained consistent upon decompression, in contrast to Miyagi et al. (2008). Asterisks indicate non-foil starting materials (Section 2). All IPFs are plotted on the same intensity scale, quantified as multiples of randomly distributed crystallographic orientations (mrd). Typical pressure uncertainties are ~5%.
Laser-heated deformation textures of a different Fe-5Si experiment are shown in Figure  2. Compared to the 300 K data, the high-temperature textures show a basal slip maximum focused in the 0001 direction and much stronger in intensity than the pyramidal maximum at 112 � 0. Consistent with the phase relations reported by Komabayashi et al. (2019), a few facecentered cubic (fcc) grains (< 2 volume % from the Rietveld refinement) crystalized above 1400 K (Supporting Information Figure S3), but they are likely too scarce to have influenced the texture; Miyagi et al. (2008) did not observe hcp textural changes even at higher fcc abundances. Focused 0001 textures previously observed in pure Fe have been attributed to basal slip (Nishihara et al., 2018;Wenk et al., 2000) and tensile twinning . Since heating disrupts twinning mechanisms, and there is no textural evidence for strong pyramidal slip (unlike at 300 K), it seems that deformation at high temperatures favors basal slip. This effect was previously suggested for ε-Fe by Steinle-Neumann et al. (2001) and could be associated with the high temperature breakdown of 〈 + 〉 deformation mechanisms observed in other hcp metals (Fan et al., 2017;Poirier & Langenhorst, 2002).

Figure 2.
IPFs of the hcp phase from diffraction images taken in the laser-heated Fe-5Si experiment. The 300 K pole figure was collected at 24 GPa (prior to any laser heating), and the others were taken consecutively during a cycle of simultaneous heating and compression at 43-46 GPa. The basal slip maximum at high temperatures is stronger and more focused on 0001 compared to 300 K textures, while a weaker local maximum appears to persist at 112 � 0, suggesting continued pyramidal slip activity. This texture persisted upon temperature quenching. Geometries and intensity scales are the same as in Figure 1. Typical temperature uncertainties are ~10%.
The relative dominance of basal and pyramidal slip systems may be significant to our understanding of the inner core. As mentioned above, most studies predict basal slip to dominate ε-Fe deformation at deep-Earth conditions since this mechanism produces strong 0001 textures analogous to the observed polar anisotropy (e.g., Wenk et al., 2000). We see this same texture in our heated experiment, so we cannot rule out this interpretation. However, some studies (e.g., Lincot et al., 2016;Steinle-Neumann et al., 2001) have suggested that pyramidal slip may be required to match seismic observations. In that case, our finding of significant pyramidal activity in Si-bearing alloys may help explain the discrepancy between the observed and expected modes of plastic deformation. Of course, it is also possible that the inner core texture is a solidification product and was not created by plastic deformation at all (Section 4.4).

Alloy Strength
Our experiments allow us to measure elastic stress in the sample and quantify its dependence on pressure and temperature. Calculating mechanical strength also requires knowledge of the alloys' shear moduli, the values of which were taken from previous studies. We extrapolate these parameters to inner core conditions to compare the mechanical strength implied by our Si-bearing experiments to previously-published estimates for Fe and Fe-Ni.

Strength calculations
The yield strength of a material is defined by the boundary between stresses low enough to be supported elastically and stresses high enough to induce plastic deformation (i.e., flow). In our experiments, the elastic stress resolved on a specific diffraction peak is determined by the dimensionless parameter (ℎ ), where h, k, and l are the Miller indices of the peak. This parameter is related to deviatoric (i.e., differential) stress by: where is the Voight approximation shear modulus, (ℎ ) is the Reuss approximation shear modulus of the peak (Anderson, 1965), α is a weighting factor, and is the deviatoric stress: = σ 1 − σ 3 (2) where σ 1 and σ 3 are the maximum and minimum stress components, respectively . Calculating (ℎ ) requires single-crystal compliance moduli ( ) for the sample material under conditions of hydrostatic pressure. Since these are seldom available, it is common to assume (ℎ ) = = , where G is the aggregate shear modulus, a measurable quantity. These assumptions reduce Equation 1 to: = 6 〈 (ℎ )〉 (3) where 〈 (ℎ )〉 (hereafter Q) is an average across all observed peaks in a diffraction pattern. If the sample's yield strength has been exceeded and it is plastically flowing, elastic deformation cannot accommodate any more strain, the peaks are expressing their maximum possible Q, and is equal to the material's yield strength (Hemley et al., 1997). Determining alloy strengths at conditions relevant to the inner core requires extrapolating both G and Q to high pressures (>330 GPa) and temperatures (>3000 K).

Extrapolation of Q
The pressure dependence of Q in our experiments is shown in Figure 3. Each experiment shows a positive linear relationship between P and Q, with four of the five experiments having similar slopes and Fe-5Ni-5Si exhibiting a shallower pressure dependence. There are no apparent experimental reasons for this deviation; Fe-5Ni-5Si is intermediate among the sample compositions, did not have large uncertainties in Q, had the same dimensions as the other samples, and the other two foil experiments did not exhibit shallow slopes. Extrapolating each trend to inner core pressures (Figure 3 inset) results in indistinguishable values for each experiment except Fe-5Ni-5Si. Since there does not appear to be a compositional trend in Q values, and we cannot rule out the shallower � , the following analysis will present results using both a fit to all the experimental alloys ( � = (1.1 ± 0.1) × 10 −4 GPa −1 ) and a fit to only Fe-5Ni-5Si ( � = (4.4 ± 0.9) × 10 −5 GPa −1 ). Both fits are steeper than literature values for Si-free alloys. For example, the highest value found by Regan et al. (2018) was � ≈ 3 × 10 −5 GPa −1 for Fe-20Ni.

Figure 3.
Pressure evolution of 〈 (ℎ )〉 for the hcp phase of each composition at 300 K. Relative strengths between the alloys do not appear to show significant compositional dependence, even within the three experiments (Fe-5Si, Fe-5Ni-5Si, Fe-5Ni-10Si) synthesized as foils. All these data were collected on compression. Fits to � for each composition (not shown) and to all data (solid line; dashed lines are 95% confidence intervals) were calculated by orthogonal distance regression considering the error bars of each point. Inset: These fits (and 95% confidence intervals) extrapolated to 329 GPa, the inner core boundary pressure. At this pressure, Fe-5Ni-5Si is noticeably weaker than the other alloys.
Our laser-heated Fe-5Si experiment allows us to estimate the effect of temperature on Q. As expected, high temperatures enhance plastic deformation, reducing the elastically-supported strain and resulting in Q reduction at a rate of � = (−1.60 ± 0.36) × 10 −6 K −1 (Supporting Information Figure S1). There are significant experimental limitations to this technique; the minimum power threshold for laser emission prevents measurements at intermediate temperatures, and laser heating generates large temperature gradients in the sample. However, Merkel et al. (2013) constrained the temperature effect on pure Fe by compiling data from several resistively-heated diamond anvil cell and multi-anvil experiments, and their reported alloy strengths imply � consistent with our value for reasonable (i.e., H. Mao et al., 1999)

values of G (Equation 3). Previous studies that extrapolated experimental values to
inner core conditions (Gleason & Mao, 2013;Regan et al., 2018) assumed � = 0 K −1 , meaning their calculated inner core strengths were upper bounds. In the following analysis, we recalculate their results using our estimated � value.

Extrapolation of G
In addition to Q, it is necessary to extrapolate shear modulus G to core-relevant conditions (Equation 3). Although Dubrovinsky et al. (2000) describes a technique to determine G values by Rietveld refinement, this requires high-quality and quasi-hydrostatic diffraction, restrictions incompatible with radial geometry experiments. Relatively few studies report shear modulus values, but from the relationships (where VS is shear velocity, VP is compressional velocity, ρ is density, and K is bulk modulus), G can be extracted from simultaneous density and velocity measurements. Figure 4 shows G versus P for available experimental data from the literature on Si-bearing hcp alloys. There is a clear difference between experimental techniques, with nuclear-resonant inelastic X-ray scattering (NRIXS) studies finding systematically lower G values than inelastic X-ray scattering (IXS) or acoustic methods. We will use NRIXS values for the following calculations since experimental uncertainties are sufficient to make extrapolated results of the techniques overlap at inner core conditions (Figure 4 inset). Note that while a linear G versus P relationship is experimentally supported at least to 220 GPa (H. Mao et al., 1999), a sublinear relationship could potentially reduce the shear modulus at inner core conditions.
Following the method of previous high-pressure strength studies (Gleason & Mao, 2013;Reagan et al., 2018), we extrapolated G using the formalism of Steinburg et al. (1980): where subscript 0 indicates ambient conditions (T = 300 K, P = 1 bar) and ρ is molar density. Values of 0 and � are taken from the NRIXS fit in Figure 4, ρ 0 is from equation of state literature (Fe-5Si from Edmund et al., 2019; Fe-5Ni-5Si from Edmund et al., 2020; other alloys calculated from these and the pure Fe values of Dewaele et al., 2006), ρ is the PREM inner core density, and G(P,T) is calculated from ρ and the PREM inner core shear velocity (Dziewonski & Anderson, 1981). Selecting locations in the inner core, namely the inner core boundary (ICB) and inner core center (ICC), allows � to be solved at the corresponding pressure and temperature conditions. This calculation implicitly assumes that the inner core has the same composition as the material used to derive the other parameters, which is most reasonable for compositions with Ni and Si. The calculation is also subject to the large (but difficult to quantify) uncertainty on the temperature of the inner core (e.g., Karato, 2008). Considering this, we elected to use the same inner core temperatures as Gleason & Mao (2013) and Reagan et al. (2018) (5500 K at the ICB, 6200 K at the ICC) to ensure that our results are directly comparable to their Si-free results; the differences between studies are not dependent on the specific pressure and temperature conditions. Calculated � values vary little between compositions; average values are -0.050 ± 0.008 GPa K -1 at the ICB and -0.043 ± 0.006 GPa K -1 at the ICC. These values are smaller than the corresponding values of Fe-12Ni (Reagan et al., 2018) and Fe (Gleason & Mao, 2013) by a factor of 1.25 and a factor of 2 respectively because of our smaller � term.

Figure 4.
Published experimental values of the shear moduli of hcp alloys in the Fe-Si and Fe-Ni-Si systems at 300 K, either explicitly reported or implied (Equations 4 and 5). The pink line is a fit to data collected by IXS and picosecond acoustic techniques (triangles and squares, respectively), excluding Sakairi et al. (2018). The blue line is a fit to NRIXS data (circles) and is corrected for 57 Fe enrichment. For context, a fit to experimental measurements of pure hcp-Fe would plot about halfway between these lines (e.g., H. Mao et al., 1999). Both fits (solid lines) are calculated irrespective of composition by orthogonal distance regression and are shown with 95% confidence intervals (dashed lines). Inset: These fits (and confidence intervals) extrapolated to 329 GPa. At this pressure, the techniques are indistinguishable within error.

Predicted inner core strength
Calculated alloy strength as a function of temperature is shown in Figure 5. Merkel et al. (2013) calculated trends directly as a function of t (rather than treating G and Q individually) based on several data spanning different studies and techniques. Extrapolating, their results imply stronger Fe than Gleason & Mao (2013) but still consistently weaker than our Si-bearing values. Even the Fe-5Ni-5Si experiment (with shallower � ) is stronger than all Si-free compositions. If solid state deformation occurs in the inner core, this enhanced strength may influence our interpretation of whether deformation is dominated by the flow of atoms towards concentrations of vacancy defects (diffusion creep) or the movement of lattice dislocations (power-law creep). Van Orman (2004) suggested the additional possibility of dislocation flow via "Harper-Dorn creep", but this mechanism is controversial (e.g., Kassner et al., 2015). Adjudicating between mechanisms is important because diffusion creep typically does not produce lattice-preferred orientation and thus would not generate anisotropy. Therefore, if the inner core is in a diffusion-dominated regime, this would tend to support anisotropy developing during crystallization rather than afterwards. Gleason & Mao (2013) concluded that their pure Fe data were consistent with dislocation-dominated (power-law) creep since mechanical weakness enhances dislocation mobility, allowing geophysically-plausible stresses to produce the required strain rates. Figure 6 (panels a and b) compares their results to those of Reagan et al. (2018) and this study. For a given differential stress, dislocation velocities and strain rates are about one order of magnitude reduced for Si-bearing alloys compared to pure Fe. An approximately equal depression is obtained (for both velocity and strain rate) by moving from the ICB to the ICC for a given alloy. Figure 6c shows a calculation of the boundary between diffusion-and dislocation-dominated regimes based on the parametrization of Reaman et al. (2011), which depends on ( , ) and ρ, but not explicitly on t. Here, the depressed G values of Si-bearing alloys slightly enlarge the dislocation-dominated field. It is important to note that our knowledge of inner core properties is incomplete. Estimates of the inner core stress field (τ) span at least five orders of magnitude (Buffet & Wenk, 2001;Koot & Dumberry, 2011;Yoshida et al., 1996). Likewise, estimates of the inner core's grain size vary widely, with constraints from earlier studies (e.g., Bergman, 1998) permitting crystallites as small as several centimeters while more recent studies (e.g., Yamazaki et al., 2017) prefer much larger grain sizes. These uncertainties make it impossible to say definitively which field the inner core falls in. Additionally, inner core viscosity, itself a poorly constrained parameter, determines the relationship between applied stress and strain rate. Gleason & Mao (2013) suggested a viscosity range of 10 15 -10 19 Pa s assuming a pure Fe inner core and simple isotropic shear; our "all alloys" fit would be one order of magnitude more viscous under the same conditions.

Figure 5.
Alloy strength in the inner core as a function of temperature. Curves from Merkel et al. (2013) use that study's parametrization while those for other studies have been recalculated using Equation 6 and including � . Curves from Reagan et al. (2018) assume that study's experimentally-determined strength ratio between Fe-5.5Ni and Fe-20Ni is constant with pressure and temperature. At inner core conditions, Si-bearing alloys are significantly stronger than alloys without light elements. The magnitude of this effect is reduced for the Fe-5Ni-5Si fit due to its smaller � .
The enhanced strength of Si-bearing Fe alloys is well known in the bcc regime (e.g., Ros-Yanez et al., 2007) and attributed to the impedance of lattice dislocations by the presence of solute atoms (Labusch, 1970). This "solid-solution hardening" effect is enhanced for alloying elements with atomic radii dissimilar to the solvent, hence Fe-Ni alloys (where solute and solvent have nearly identical radii) shows little or no strengthening (e.g., Kotan et al., 2013). Solid solution hardening is known to occur in other hcp metals (e.g., Toda-Caraballo et al., 2015) and reduced dislocation mobility may influence hcp deformation mechanisms (e.g., Yoo et al., 2002), so both the strengths and textures of inner core alloys may be functions of their solutes' atomic radii.

Figure 6.
Dislocation velocity (a), strain rate (b), and critical grain size (c) as functions of differential stress τ. Equations and constants used are from Gleason & Mao (2013) and Reaman et al. (2011) and are listed in the Supporting Information. For clarity, only the "all alloys" ICB line is plotted in (c); the other ICB lines would be similarly offset from their corresponding ICC lines. The shaded "thermal gradients" (Jeanloz & Wenk, 1988) and "Joule heating" (Takehiro, 2011) regions are representative examples of proposed inner core stress ranges (see main text).

Conclusions
We investigated hcp Fe-Ni-Si alloy deformation textures and strengths with highpressure radial X-ray diffraction experiments. These alloys plastically deform by the same principal mechanisms documented in other hcp metals, though possibly with greater pyramidal slip activity compared to pure ε-Fe at 300 K. Upon laser heating, we observed a change in the distribution of crystallographic preferred orientations which may suggest that basal slip is preferred at high temperatures. Experimental Fe-Ni-Si compositions are strong compared to previous Fe and Fe-Ni data and show significantly larger � values. Extrapolating to inner core conditions, Si-bearing alloys may be up to an order of magnitude stronger than pure Fe. This would reduce the mobility of anisotropy-favoring lattice dislocations by an equal factor. A mechanically strong inner core generally favors deformation by non-texturing diffusion creep and thus anisotropy as a crystallization (rather than a post-solidification) phenomenon. However, the large uncertainties on experimental and observational inner core parameters make it premature to conclude whether the strengthening effect of Si is significant in this regard.