Atomic-Scale Study of Intercrystalline (Mg,Fe)O in Planetary Mantles: Mechanics and Thermodynamics of Grain Boundaries Under Pressure

Abstract

Polycrystalline (Mg,Fe)O ferropericlase is the second most abundant mantle constituent of the Earth and possibly of super-Earth exoplanets. Its mechanical behavior is expected to accommodate substantial plastic deformation in Earth's lower mantle. While bulk properties of ferropericlase have been extensively studied, the thermodynamics of grain boundaries and their role on mechanical response remain largely unexplored. Here, we use density functional theory calculations to investigate mechanical behavior and thermodynamics of the {310}[001] grain boundary—a representative proxy for high-angle {hk0}[001] tilt grain boundaries—at relevant mantle pressures of the Earth and super-Earth exoplanets. Our results provide evidence that shear-coupled migration and grain boundary sliding are the dominant mechanisms of (Mg,Fe)O grain boundary mobility. We show that pressure-induced structural transformations of grain boundaries can trigger a change in the mechanism and direction of grain boundary motion. Significant mechanical weakening of the grain boundary is observed under multi-megabar pressures, caused by a change in the grain boundary transition state structure during motion. Our results identify grain boundary weakening in periclase as a potential mechanism for viscosity reductions in the mantle of super-Earths. We further demonstrate that structural grain boundary transitions control the spin crossover of Fe²⁺ in the grain boundaries. We model iron partitioning behavior between bulk and grain boundaries and predict equipartitioning to occur in μm size ferropericlase grains. Our findings suggest that the iron spin crossover pressure in ferropericlase may increase several tens of GPa by pressure-induced structural grain boundary transitions in dynamically active fine-grained lower mantle regions.

Key Points

• Pressure-induced transformations of grain boundary structures can trigger a change in the mechanism and direction of grain boundary motion

• Grain boundary weakening is identified as a mechanism for viscosity reductions at multi-megabar pressures in super-Earth exoplanets

• The spin crossover of ferrous iron in tilt grain boundaries at high pressure is likely controlled by structural grain boundary transitions

Plain Language Summary

Mantle convection in planets like the Earth is governed by the flow of rocks. This flow occurs by the motion of defects in the crystal lattices of minerals. The physical properties of these defects thus have profound implications on planetary dynamics. We use an atomic-scale modeling approach based on the fundamental grounds of quantum mechanics to examine the (a) mechanical behavior and (b) energetics of grain boundaries in (Mg,Fe)O, the second most abundant mantle mineral of the Earth and super-Earth exoplanets. (a) Our results on the mechanical behavior indicate that high pressure in planetary interiors has a strong effect on the mechanisms of grain boundary mobility which govern intercrystalline deformation. We show that mechanical weakening of grain boundaries is a potential mechanism for viscosity reductions in the mantle of super-Earth exoplanets. (b) Our thermodynamic modeling results predict that grain size is an important factor to control grain boundary segregation of iron. Further analyses suggest that the change in electronic spin state of iron in grain boundaries is governed by structural transformations of the grain boundaries at high pressure. This mechanism also likely influences the pressure condition of the iron spin crossover in (Mg,Fe)O with micrometer or smaller grain sizes.

1 Introduction

The Earth's lower mantle is composed of a mixture of ∼75% (Mg,Fe) (Si,Al)O₃ bridgmanite (Brg), ∼20% (Mg,Fe)O ferropericlase (Fp) and a few percent of CaSiO₃ calcium-silicate perovskite based on a pyrolitic compositional model (McDonough & Sun, 1995; Ringwood, 1991). Fp is widely known for its pronounced ductility compared to high-pressure silicates despite its smaller volume fraction (Cordier et al., 2012; Girard et al., 2016; Reali et al., 2017; Tromas et al., 1999). Therefore, even though a recent theoretical work reported that intracrystalline deformation of MgO periclase may be more sluggish than Brg under low mantle stresses (Cordier et al., 2023), Fp is expected to play an important role in the plastic deformation of the lower mantle. High stresses along the boundary of convection cells, for example, subducting slabs (Garel et al., 2014; Motoki & Ballmer, 2015), may affect the viscosity contrast between MgO and Brg (Boioli et al., 2017) toward what has been observed in deformation experiments (Girard et al., 2016; Tsujino et al., 2022). These deformation experiments on synthetic mantle aggregates demonstrate evidence of strain accumulation in Fp by grain elongation and comminution in absence of recrystallization processes (Girard et al., 2016; Kaercher et al., 2016; Nzogang et al., 2018). This suggests that Fp is able to accommodate most of the deformation. These findings are supported by a continuum mechanics study of Thielmann et al. (2020). Fp could therefore be responsible for strain localization in the Earth and planetary mantles, especially in regions of stress concentrations. Such localized deformation may have significant implications for the subduction behavior of slabs and the dynamics of plumes (e.g., Gülcher et al., 2022) and could affect the local degree of mantle stirring and explain possible survival of (primordial) geochemical reservoirs in unmixed mantle regions (Ballmer et al., 2017; Gülcher et al., 2021; Hofmann, 1997; van Keken et al., 2002).

Strain localization in Fp by grain elongation and comminution is intrinsically linked to an increase in the volume fraction of Fp-Fp interfaces. This raises the question about the role of Fp grain boundaries (GBs) in controlling the rheology of the lower mantle. The physical properties of GBs are known to influence polycrystalline mechanical behavior at elevated temperatures (Sutton & Balluffi, 1995), for instance through grain boundary motion during recrystallization and recovery processes (Urai et al., 1986), strain production by GB migration (Sun et al., 2017) and enhanced ionic diffusion along GBs (Dohmen & Milke, 2010; Hashimoto & Hama, 1971; Van Orman et al., 2003). GBs are also able to alter thermochemical properties of polycrystals since they may act as hosts for solutes and point defects (Karki et al., 2015), influencing elemental partitioning between intra- and intergranular regions (Hiraga et al., 2004; Mitamura et al., 1979).

GBs define the interface between misoriented crystal grains and are typically characterized by volumetric lattice distortions. The physical properties of GBs are, therefore, strongly pressure-dependent as confirmed by the early studies of Harris et al. (1996, 1999). However, little is yet known about GB properties of Fp across the pressure range of its stability field in terrestrial planets despite their potential to influence mantle convection patterns, mantle morphology, and geochemical (re)distribution. The above-mentioned experimental studies on GBs in MgO and those from high-resolution atomic imaging (e.g., Bean et al., 2017; Saito et al., 2013; Yan et al., 1998) are unfortunately restricted to low-pressure measurements due to technical limitations. On the other hand, quantum ab initio and atomistic classical molecular dynamics (MD) studies have progressively led to new insights into the effect of pressure on the microscopic structure, mobility and chemical segregation behavior of MgO GBs. It has been shown that pressure is able to induce structural GB transitions (Harris et al., 1996, 1999, 2001; Hirel et al., 2019; Verma & Karki, 2010; Yokoi & Yoshiya, 2018), that GB strength can vary with pressure (van Driel et al., 2020) and that GBs can be preferential sources and sinks for ions and their defective counterparts compared to the bulk lattice (Karki et al., 2015; McKenna & Shluger, 2009).

Here, we use a theoretical mineral physics approach based on the density functional theory to investigate detailed physical properties of the {310}[001] high-angle symmetrical tilt GB in the B1 (NaCl-type) phase of Fp. Earlier studies suggest that this is a representative high-angle [001] tilt GB of MgO in terms of atomistic structure and excess volume (Hirel et al., 2019; Verma & Karki, 2010), which has been experimentally observed in polycrystalline MgO samples (Bean et al., 2017; Saito et al., 2013). We focus on high-angle tilt GBs, predominantly composed of void-like structures, with energies typically higher than low-angle tilt GBs (Hirel et al., 2019). This class of GBs are therefore most likely involved in non-equilibrium processes as plastic deformation, recrystallization and grain growth.

In this work, we characterize the atomic-scale structure of the {310}[001] GB within and beyond the pressure range of Earth's lower mantle up to 400 GPa since Fp is also thought to be a major phase in the mantles of massive Earth-like exoplanets (e.g., Tsuchiya & Tsuchiya, 2011; Umemoto et al., 2006) and most stable in its B1 phase up to a pressure of ∼400 GPa along an extrapolated mantle adiabat (Cebulla & Redmer, 2014; Miyanishi et al., 2015). We then determine the critical shear stress (i.e., critical resolved shear stress at 0 K) of GB motion as a function of pressure (P) and investigate the most favorable mechanism of GB mobility. We show that variations in GB structure with increasing pressure influence both the mechanism of GB motion and preferred sliding direction, in contrast to what has been suggested in previous work (van Driel et al., 2020). Our novel results indicate significant grain boundary weakening with increasing depth in the pressure range of super-Earth mantles, potentially allowing for intercrystalline viscosity reductions with depth. Because MgO is expected to form a solid solution with FeO in terrestrial mantles, we also examine the influence of ferrous iron on the physical properties of the [001] GBs. It is well known that Fe²⁺ undergoes an electronic spin transition in Fp from high spin (HS) to low spin (LS) across the pressure range of Earth's lower mantle (Tsuchiya et al., 2006). But the influence of GBs on this spin transition in polycrystalline Fp remains unclear. To elucidate this, we systematically investigate the spin transition pressure of Fe²⁺ incorporated in the GB. Our results explain the critical role of pressure-induced structural transitions in the spin crossover of iron in GBs. Based on these results, we model iron segregation and partitioning behavior between the grain and GBs in the pressure range of Earth's lower mantle. We also quantify the effect of iron on GB strength. Last, we briefly discuss the geophysical implications derived from our modeling and their mutual influence on mantle rheology of the Earth and super-Earth exoplanets.

2 Methods

We quantify physical properties of the {310}[001] interfaces using a supercell approach. The initial simulation cells are constructed from two identical and optimized conventional (8-atom) cells of MgO at a given pressure and static temperature. After expanding both cells, each is rotated by an angle of ±θ/2, respectively along the [001] tilt axis and joined together to create a misorientation of θ = 36.8° with respect to the atomic planes of the {310} contact surfaces (Figure 1). The resulting orthogonal supercells along [001], [$$1\overline{3}0$$], and [310] contain a bicrystal composed of 224 atoms including two identical but oppositely oriented Σ5 grain boundaries in the coincidence site lattice (CSL) framework because of the periodic boundary condition adopted in our approach. In this set-up, both crystals comprise four tilted and stacked conventional MgO cells along [310], sandwiched between two consecutive GB structures. The MgO lattice parameter is pressure dependent and ranges from a₀ = 4.015 Å at 30 GPa to a₀ = 3.399 Å at 400 GPa at static temperature (T = 0 K). The lengths of the optimized defective simulation cells correspond to ∼3a₀/2 for [$$1\overline{3}0$$], ∼9a₀ for [310], and ∼2a₀ for [001].

Figure 1

Construction of the {310}[001] GB in a bicrystal. (Left) The system is built from two oppositely rotated MgO crystals (supercells) by an angle θ/2 along [001]. (Right) The translation vector r that yields the minimum energy of the system determines the most stable atomic structure of the GB at a given pressure.

Full structural optimization of lattice parameters and atomic positions are performed using the ab initio density functional self-consistent field method (Baroni et al., 1987) within the local density approximation applied to the exchange-correlation functional (Hohenberg & Kohn, 1964; Kohn & Sham, 1965). The Mg, O and Fe pseudopotentials with 3s²3p⁰, 2s²2p⁴, and 3s²3p⁶3d^6.54s¹4p⁰ pseudized electronic configurations were generated by the method of von Barth and Car (von Barth, 1984), Troullier and Martins (1991), and Vanderbilt (1990), respectively. The former two pseudopotentials are the same as those used in previous work on MgO of Ritterbex et al. (2018). The peudo-electronic wave functions are expanded by a plane-wave basis set with a kinetic energy cutoff of 70 Ry. Irreducible Brillouin zone sampling of the electronic states of defective MgO simulation cells are carried out on a 2 × 1 × 2 Monkhorst-Pack grid within the reference frame of Figure 1 (Monkhorst & Pack, 1976). For iron-bearing systems, we apply the LDA + U method (Anisimov et al., 1991; Cococcioni & de Gironcoli, 2005) to correct for the strong on-site Coulombic interactions in the Fe-O bonding. The Hubbard correction values U for iron-bearing MgO (U ∼ 6 eV for HS Fe²⁺ and U ∼ 5 eV for LS Fe²⁺) are determined within the internally consistent way (Tsuchiya et al., 2006; Wang et al., 2015). For a pressure range between 30 and 120 GPa and X_Fe ∼ 3%–19%, the U values of Fe²⁺ in (Mg_1−x,Fe_x)O varies only by ∆U ≈ ±0.5 eV. Therefore, the U values reported by Wang et al. (2015) are adopted between 30 and 120 GPa (Table S1 in Supporting Information S1). The damped variable cell shape molecular dynamics scheme (Wentzcovitch, 1991) as implemented in the Quantum Espresso code (Giannozzi et al., 2009) is applied to fully relax all structural parameters at static condition until the sum of residual forces became less than 1.0 × 10⁻⁵ Ry/a.u. in order to impose an internal energy convergence below 0.01 eV per formula unit and a pressure convergence within 0.1 GPa. The total energy of defective simulation cells is size sensitive due to the spurious self-interaction between the GBs of periodic replicas. We confirm that an increase in defective simulation cell size along [310] did not significantly affect the grain boundary formation energies.

3 Results

3.1 Grain Boundary Structures

Construction of the initial {310}[001] MgO GB (Methods) yields only one (i.e., r = 0, 0.5, 0) of the many possible interfacial configurations as shown in Figure 1. To search for the lowest energy atomic configuration of this GB at a given pressure, we propose an adapted version of the ab initio molecular dynamics (AIMD) approach as used by Verma and Karki (2010) and Yokoi and Yoshiya (2018). At each pressure, two initial GB structures with translation vectors (0, 0.5, 0) and (0.5, 0.5, 0.25) were structurally optimized corresponding to the local energy minima of the systems. To allow thermally activated interface translation toward the minimum energy configuration, AIMD simulations of both initial systems were conducted within the NVT canonical ensemble at elevated temperature T using the Verlet velocity algorithm to solve the classical equations of motion within a time step of 2 fs. The temperature is controlled by the kinetic energy rescaling method with a target temperature set to 4000 K. All simulations were run for a total time of ∼20 ps. At pressures between ∼30 and 400 GPa, we find both initial contact surfaces to shear across {310} within less than 3 ps to reach their stable configuration until the end of the simulations (Figure S1 in Supporting Information S1). After the AIMD runs, both GB structures were fully re-optimized at static temperature within the variable cell shape constant-pressure scheme to allow the simulation cell to adapt its lattice parameters and the ions to rearrange their stable positions at 0 K. At each pressure, both initial surfaces were found to shear toward the same final GB structure with identical translation vector r. This confirms that simulations are converged toward the minimum energy configuration of the interface. We have quantified the properties of the {310}[001] GB structures at static conditions in terms of the formation enthalpy ∆H_GB

$$${\increment}{H}_{\text{GB}}=\left({H}_{\theta }-{H}_{0}\right)/2S,$$$ (1)

where H_θ and H₀ correspond to the enthalpies of the defective and defect-free simulation cells (with the same number of atoms), respectively. 2S is the total area of the two GBs in the simulation cells. The GB excess volume ∆V_GB can be expressed as

$$${\increment}{V}_{\text{GB}}=\left({V}_{\theta }-{V}_{0}\right)/2S,$$$ (2)

where V_θ and V₀ are the volumes of the defective and defect-free simulation cells, respectively. Figure 2 shows the atomic structure, formation enthalpy, and excess volume of the lowest energy structures obtained by the above structure search process between 30 and 400 GPa. The most stable structure of the GB found between 30 and 80 GPa can be seen in Figure 2a. Some ions that initially belong to either of the grains shift to occupy the core of the interface after relaxation. This avoids the formation of Mg-Mg and O-O bonds and achieves optimal compaction of the void-like structure to minimize the work performed by pressure. This GB structure has been previously identified by the pioneering work of Verma and Karki (2010), but was first described in detail by van Driel et al. (2020). We obtain ∆H_GB = 3.47 J/m² at 30 GPa, consistent with ∆H_GB = 3.38 J/m² (LDA) at 31.8 GPa by van Driel et al. (2020), and slightly larger than ∆H_GB ∼ 2.9 J/m² found at 30 GPa based on classical MD with empirical model potentials (Hirel et al., 2019). This discrepancy can be likely attributed to the difference in methodology. At a pressure of ∼80 GPa, the GB undergoes a structural transition to a denser ground state configuration (i.e., drop in excess volume) as shown in Figures 2b and 2c. This structure is characterized by a rigid shear displacement between atomic planes across the interface along the [001] tilt axis. Under finite size and non-equilibrium thermodynamic conditions, it might be possible that this crystal shear could act as an ideal nucleation site for dislocations (i.e., Frank-Read sources). Both ground state structures and associated formation energies and excess volumes are found to be in good agreement with other studies up to a pressure of 120 GPa (van Driel et al., 2020; Verma & Karki, 2010), including the structural transition pressure determined at 90 GPa by van Driel et al. (2020) (Figure 2c). We find no further transition from the asymmetric ground state structure appearing at 80 GPa up to 400 GPa, close to the B1-B2 phase transition of MgO (Cebulla & Redmer, 2014; Miyanishi et al., 2015). The {310}[001] GB in the B1 phase of MgO might be thus characterized by two ground state structures within the pressure range of Earth's lower mantle and the mantle of super-Earths: a low-pressure symmetric structure (30–80 GPa) and a high-pressure asymmetric structure (80–400 GPa). Figure 2c shows that with increasing pressure ∆H_GB and ∆V_GB consistently increase and decrease, respectively, with a significant drop in ∆V_GB across the structural transition pressure as a result of GB compaction. The GB excess volume of 0.336 Å³/Ų at 30 GPa decreases significantly to 0.00932 Å³/Ų at 400 GPa. This demonstrates that the effect of increasing pressure tends to substantially reduce the void-like structural GB units by spatial rearrangements of the atomic GB sites. Previous studies (e.g., Hiraga et al., 2004) suggested that impurities may tend to segregate to GB structural units, thus suggesting them as an important factor for trace element storage. Our results indicate that the void-like structures of high-angle tilt GBs may become less attractive to impurities at high pressures of the mid to lower mantle and the mantle of exoplanets compared to the uppermost lower mantle. This implies that the tendency of impurity segregation in the open structural GB units, particularly those with larger ionic radii than the host anions and cations, may substantially decrease with increasing pressure in the lower mantle of the Earth and that the GB excess volume may not be the primary host for the storage of trace elements as Ca or radiogenic elements. A recent atomistic study of Hirel et al. (2019) investigated over 200 {hk0}[001] GBs and averaged the formation enthalpies and excess volumes over all high-angle GBs. Their results show similar trends of ∆H_GB and ∆V_GB with pressure with respect to our results. In addition, they demonstrate that up to 86.6% of all high-angle GBs become asymmetric at a pressure of 120 GPa. The similarities between our results and the average characteristics of {hk0}[001] GBs indicate that the {310}[001] GB can be considered as a representative proxy for high-angle {hk0}[001] tilt GBs at elevated pressures.

Figure 2

Optimized ground states, formation energies and excess volumes of the {310}[001] GB. (a) Ground state configuration between 30 and 80 GPa. Left hand side shows the structure viewed down the [001] tilt axis. The right-hand side shows the structure horizontally along [001]. Large yellow and small red spheres represent the Mg and O ions, respectively. Solid black lines indicate the GB unit cell. The GB is characterized by four distinct GB sites, labeled A–D (b) Ground state structure between 80 and 400 GPa. (c) GB formation enthalpy and excess volume of the GB ground state structure from this work and van Driel et al. (2020). Blue shaded color indicates the stability field of the low-pressure symmetric ground state structure and the yellow shaded color that of the high-pressure asymmetric one.

3.2 Mechanical Behavior

To investigate the mechanical behavior of the {310}[001] GB under pressure at static condition, we impose a simple shear to our simulation cells by increments of 0.1%–0.2% in all six possible directions: ε_yx, ε_yz, ε_xy, ε_xz, ε_zy, and ε_zx (Figure 1), and sample the evolution of the internal energy and stress by structural optimization after each loading increment. When a shear strain ε_yx of ∼0.2% is applied to the symmetric GB structure at 30 GPa, the interface starts to move by small and coherent rearrangements of the atomic positions after each loading step. The total atomic displacement after imposing full strain is small and remains below 0.74 Å for both cations and anions. We monitor the von Mises stress invariant (Li, 2003) in the simulation cell during shearing. Distribution of the stress invariant remains constrained to the atomic sites involved in GB motion. The bulk of the grains remain almost completely stress-free during simple shear. Figure 3b shows the corresponding stress-strain evolution. With increasing strain, stress (τ_yx) increases continuously toward a maximum (i.e., critical shear stress) followed by a decrease to a stress-free state (i.e., transition state). Applying both ±ε_yx allow us to track the complete reaction path for migration of the interface from its ground state structure to its saddle point configuration (Figure 4). Under ε_yx loading, we find the GB to move by shear-coupled migration (SCM) (Figure 3a): a combination of coherent sliding (in-plane motion) along [130] and migration (out-plane motion) along [310]. The maximum stress of the stress-strain curve corresponds to the critical shear stress τ_c = max (∂E/∂u) to trigger the complete motion of the GB over the migration energy barrier (MEB) E, where u is the GB displacement in terms of the imposed strain ε. The critical shear stress τ_c corresponds to 1.85 GPa. When a shear strain ε_ij is applied in the other five directions, we find the simulation cell to deform elastically (i.e., linear stress-stress curve) to stresses τ ≫ 2 GPa with no atomic rearrangements and preservation of structural integrity. This shows that SCM is the most likely mechanism by which the symmetric GB moves at 30 GPa. Similar results are obtained in the whole pressure range (30–80 GPa) of the symmetric ground state structure. At a pressure of 45 GPa, τ_c increases to 1.94 GPa after which it starts to decrease to 1.75 GPa at a pressure of 60 GPa. These results are consistent with previous work on SCM of the same GB (van Driel et al., 2020). They have shown a strengthening of the symmetric GB between a pressure of 30–41.4 GPa followed by a weakening until the structural transition pressure. This variation in mechanical strength is, however, relatively small. Van Driel et al. (2020) find τ_c to range from ∼1 to 2 GPa for the symmetric GB in the low-pressure regime in agreement with our results (Figure 3c). If we apply ε_yx to the asymmetric GB at 90 GPa (i.e., after the pressure-induced structural transition), the GB moves again by SCM. The critical shear stress has increased to 5.51 GPa and is significantly larger than for the symmetric GB in the low-pressure regime (τ_c < 2 GPa) (Figures 3b and 3c) in good agreement with van Driel et al. (2020). However, by applying a shear strain ε_yz (i.e., in the direction of the tilt axis), we find this GB to move with a pure translational motion along [001] at a substantially lower τ_c of 4.23 GPa. This mechanism of motion corresponds to GB sliding (Figure 3a). This implies that the GB strength is lower than previously reported and that the preferred mechanism of GB motion changes from SCM to GB sliding (GBS) across the structural transition to the asymmetric structure at 80 GPa (Figure 3a). During initial ε_yz loading, the simulation cell deforms elastically (i.e., linear stress-strain curve) up to a strain of ∼1% after which the GB starts to slide along [001] by atomic displacements on both sides of the interface to reach its transition state at a strain of ∼1.6%. The sliding process is conservative and the atoms remain at their respective side of the interface during GBS (i.e., no mass transfer across the interface). The transition state structure during GBS corresponds to the symmetrical equivalent of the asymmetric structure along [001] (Figure 4c). We were unable to determine other GB mobility mechanisms by imposing ε_xy, ε_xz, ε_zy, and ε_zx. Under these applied loads, mechanical failure of the interface occurred at significantly higher stresses than τ_c for SCM and GBM over the entire pressure range studied. The transition from SCM to GBS across the structural GB transformation is likely due to the presence of crystal shear along [001] in the asymmetric structure, causing the interface to weaken along [001] due to cation-cation and anion-anion interactions across the GB. Previous work of van Driel et al. (2020) suggested that the crystal shear displacements along [001] act as a significant barrier to interface mobility, but it is instead this structural feature that facilitates GBS and reduces τ_c of the interface compared to motion by SCM. GBS remains the favorable mechanism to move with the lowest τ_c until a pressure of 115 GPa. At this pressure, τ_c for GBS and SCM coincide. With increasing pressure, we find τ_c for GBS to increase monotonously, whereas for SCM it starts to decrease down to 3.28 GPa at a pressure of 400 GPa (Figure 3c). These results reveal yet another change in the preferred GB mobility mechanism from GBS back to SCM at 115 GPa. With increasing pressure up to 400 GPa, SCM remains the most favorable mechanism. Over the pressure range of 115–400 GPa, the interface undergoes significant weakening with a total decrease in τ_c of 5.96 GPa. The critical shear stress of the interface at 400 GPa is even lower than τ_c = 4.24 GPa at 90 GPa. The former indicates a significant decrease in the mechanical strength of the interface with increasing depth in the pressure range of the mantles of super-Earth exoplanets. To investigate the cause of this mechanical GB weakening above a pressure of 115 GPa, we modeled the MEB associated with SCM between 90 and 400 GPa. For this purpose, we reconstruct the energy barrier E for SCM from the evolution of the internal energy E_i as a function of imposed ith strain increment ε_i as

$$$E=\frac{1}{2S}\left[{E}_{i}-{E}_{0}-\frac{1}{2}{V}_{0}{\tau }_{ij}{\varepsilon }_{ij}\right],$$$ (3)

where E₀ is the internal energy of the unstrained simulation cell with the GB ground state structure. To obtain reliable results of the MEB, the following two features were handled with care. (a) Imposing simple shear implies a constant simulation cell volume V₀ during loading. Because of atomic displacements during SCM, confined pressure may vary if simulation cells are not sufficiently large, although PV₀ should be constant during GB motion. We confirm that pressure only varies (<0.05 GPa) below the pressure convergence criterion for the DFT simulations. (b) While strain is imposed, elastic energy is stored. Because this should not contribute to the MEB, we apply an elastic energy correction (i.e., third term on the right-hand side of Equation 3) (Varvenne et al., 2013). At 115 GPa, we find a total height of the energy migration barrier of ∼0.25 J/m², consistent with the result of van Driel et al. (2020) at 120 GPa, who used the nudged-elastic band (NEB) method. Results of the MEB for SCM between 90 and 400 GPa are shown in Figure 3d. Starting at a pressure of 90 GPa, the height of the barrier increases up to a pressure of 115 GPa. Further increase in pressure leads to a continuous decrease in the height of the energy barrier to ∼0.06 J/m². This is a significant decrease with ∆E ≈ 0.2 J/m² compared to the barrier at 115 GPa. This decrease in the height of the barrier flattens their slopes and lowers τ_c with increasing pressure. The turning point at 115 GPa coincides with a transition in saddle point configuration of the asymmetric GB during SCM to a more compact structure (Figure 4b). This likely affects the relative change in compressibility between the new transition state and unchanged ground state with pressure, inducing a decrease in MEB, similar to what has been observed by Yokoi and Yoshiya (2018). Mechanical weakening of the GB with increasing pressure between 115 and 400 GPa can therefore be attributed to a change in transition state structure of the asymmetric GB during SCM. Should this feature be intrinsic to asymmetric high-angle tilt GBs at high pressure, it can be expected to have significant effect on the mechanical behavior (i.e., intercrystalline viscosity reduction) of polycrystalline MgO with increasing depth in the mantle of super-Earth exoplanets.

Figure 3

Mobility and mechanical strength of the {310}[001] GB. (a) Illustration of the different GB mobility mechanisms inferred from this study: SCM and GBS. SCM occurs by both a sliding (u_x) and a migrational (u_y) displacement component of the GB. At 30 GPa, u_x = u_y = 1.25 Å for SCM, at 90 GPa, u_x = 1.21 Å for GBS and at 400 GPa, u_x = 1.05 Å and u_y = 1.12 Å for SCM. (b) Stress-strain evolutions during imposed-strain simulations between 30 and 90 GPa. (c) Critical shear stress of the GB for the inferred mechanisms of GB motion. The symmetric GB (blue shaded area) moves preferentially by SCM and the asymmetric GB (yellow area) by both GBS (P < 115 GPa) and SCM (P = 115–400 GPa). Between 115 and 400 GPa, the interface undergoes significant mechanical weakening. The inset shows details between 30 and 130 GPa. (d) Energy migration barrier (MEB) for SCM between 90 and 400 GPa. The height of the MEB decreases systematically between 115 and 400 GPa due to a transition in saddle point configuration of the interface at 115 GPa.

Figure 4

Ground (left) and saddle point (right) configurations of the GB. (a) Ground and transition states (SCM) of the symmetric GB between 30 and 80 GPa viewed down the [001] tilt axis. (b) Ground and transition states (SCM) of the asymmetric GB between 80 and 400 GPa viewed down the [001] tilt axis. A change in transition state structure occurs at a pressure of 115 GPa. (c) Ground and transition states (GBS) of the asymmetric GB between 80 and 115 GPa viewed horizontally along [001]. GBS becomes the dominant mechanism for GB motion between 80 and 115 GPa.

3.3 Iron—Grain Boundary Interactions

MgO forms a solid solution with FeO in the mantles of terrestrial planets as the Earth. It is well known that incorporation of ferrous iron in the B1 phase of MgO has a significant effect on its physical properties since Fe²⁺ undergoes an electronic HS to LS transition at high pressure in the lower mantle (Badro et al., 2003; Lin et al., 2005; Tsuchiya et al., 2006). Although the spin state of iron in bulk (Mg,Fe)O lattice under high pressure has been well studied, it remains unknown in GBs. The behavior of iron in GBs is directly related to the interaction between iron and GBs and affects in conjunction with the grain size to which extent GB segregation of iron occurs at lower mantle pressures.

3.3.1 Spin Transition in (Mg,Fe)O Grain Boundaries

We have started to investigate the favorable spin states of substitutional Fe²⁺ at the four available GB cation sites A–D of the [001] GB (Figure 2) over the pressure range of the lower mantle (30–120 GPa). For this purpose, we doubled our initial simulation cell along [$$1\overline{3}0$$] to reduce Fe-Fe interactions between periodic replica. The new supercells contain 448 atoms. Brillouin zone k-point sampling is performed on a 1 × 1 × 2 Monkhorst-Pack grid. Enthalpies of the HS and LS states of iron are computed at each GB cation site in the symmetric (30–80 GPa) and the asymmetric (80–120 GPa) ground state structures (Figures 2a and 2b), by replacing Mg²⁺ for one Fe²⁺ cation. This results in iron concentrations of $${X}_{\text{Fe}}^{\text{GB}}$$ = 3.571% and $${X}_{\text{Fe}}^{\text{GB}}$$ = 4.167% in our (Mg_1−x,Fe_x)O simulation cells for the symmetric and asymmetric GB, respectively. Different GB unit cells give rise to small differences in $${X}_{\text{Fe}}^{\text{GB}}$$. We perform spin-polarized simulations with the spin moment of Fe²⁺ free to vary, resulting for all iron configurations to a magnetic moment of 4μ_B (spin quantum number S = 2) and 0μ_B (S = 0) for HS and LS Fe²⁺, respectively. Figure 5 shows the enthalpy variation between the different iron sites A–D, where only the enthalpies corresponding to the energetically favorable spin state of Fe²⁺ at each GB site are shown. To benchmark our results, we tested values of U ± 0.5 eV at 30 GPa. The calculated results are found to be identical compared to those computed with the initial U values. At each pressure, we find two favorable iron sites with almost equivalent enthalpies, which are significantly lower than those of the other two sites: The B and D site for the symmetric structure and the B and C site for the asymmetric structure (highlighted by the black rectangle in Figure 5a). It can be seen that iron located at the B and D sites in the low-pressure symmetric structure systematically adopts the HS state whereas iron at the B and C sites in the high-pressure asymmetric structure prefers the LS state. These results indicate a correlation between the HS-LS transition of iron located in the most stable GB sites and the structural transition of the GB ground state at ∼80 GPa. This structural transition is characterized by a significant drop in its excess volume (Figure 2). Based on simple crystal field arguments (Burns, 1993) it can be expected that a decrease in GB excess volume intensifies the crystal or ligand field effects on iron at the interface. This correlation between excess volume and bonding environment is likely to trigger the HS-LS transition of iron in the GB. To better understand if this spin crossover in the GB is indeed caused by steric effects induced by the structural transformation of the GB, we have investigated the coordination environment of all GB cation sites compared to that of pristine MgO (Table S2 in Supporting Information S1). In bulk MgO we determined the static HS-LS transition pressure of Fe²⁺ at ∼30 GPa (Figure S2 in Supporting Information S1) similar to previous studies (e.g., Tsuchiya et al., 2006). The preferred GB sites (B and D) of iron in the symmetric structure are structurally undercoordinated (effective coordination number of ∼5.0 compared to 6.0 in the bulk). The crystal field might be weaker in this distorted coordination polyhedral than in the regular octahedral site. In addition, there exist more free spaces in the GB compared to the bulk cation sites related to the excess volume. These facts allow iron in the GB to remain in the HS state at pressures above 30 GPa. After the GB transformation into the asymmetric structure, iron preferentially resides at GB sites B and C. The common characteristic of these GB sites is that they exhibit similar average bond lengths and polyhedral volumes compared to bulk cation sites with small deviations of only 0.9%–3.4% and 2.4%–4.5%, respectively. It means that Fe²⁺ prefers those sites with coordination environment most similar to bulk cation sites in the asymmetric GB structure. Iron at its most stable GB sites thus adopts the LS state in the asymmetric GB since the structural transition (∼80 GPa) occurs at higher pressure than the static HS-LS transition in bulk (Mg,Fe)O (∼30 GPa). This indicates that the HS-LS transition of iron in the GB is a direct result of the transition of the ground state GB structure under high pressure. In case of the {310}[001] GB, the static spin crossover occurs at ∼80 GPa while it occurs at ∼30 GPa in bulk (Mg,Fe)O. Substantial excess volume of the symmetric GB structure allows iron to remain in the HS state at considerably higher pressures than in bulk (Mg,Fe)O. Our analysis suggests that (a) the HS-LS transition of ferrous iron in the studied GB is triggered by the pressure-induced transition of the GB structure to the asymmetric one as commonly observed in high-angle tilt GBs under high pressure (Hirel et al., 2019; Verma & Karki, 2010) and (b) that this spin crossover in the GBs is expected to occur at higher pressures compared to bulk (Mg,Fe)O due to significant excess volume of the void-rich structures in the symmetric GBs.

Figure 5

(a) Static enthalpy differences between the iron-bearing GB with Fe²⁺ located at the various GB sites A–D. Reference corresponds to a simulation cell with LS iron located in the bulk. Results at each GB site correspond to the spin state of iron with lowest enthalpy. Black rectangle shows the two most stable iron GB sites where the HS-LS transition coincides with the structural GB transition. (b) Segregation enthalpies of iron at the four GB sites. Black rectangle shows the segregation enthalpies of the two most stable GB sites of iron.

3.3.2 Grain Boundary Segregation of Iron

Despite the interesting behavior of the iron spin crossover in the high-angle tilt GBs, it only plays a role in Earth's lower mantle if GBs are able to host a substantial fraction of the total iron concentration. How much iron is incorporated in Fp GBs depends on the partitioning behavior between grain and GBs. Based on the results in Section 3.3.1, we determined the segregation enthalpies ∆H_seg of iron located at the different GB sites as

$$${\increment}{H}_{\text{seg}}=\left({H}_{\theta }^{\text{Fe}}+{H}_{0}\right)-\left({H}_{0}^{\text{Fe}}+{H}_{\theta }\right),$$$ (4)

where $${H}_{0}^{\text{Fe}}$$ and $${H}_{\theta }^{\text{Fe}}$$ are the enthalpies of an MgO system with one Fe²⁺ in the bulk and GB and H₀ and H_θ are that of the iron-free bulk and GB (Equation 1), respectively (Figure 5b). The spin states associated with $${H}_{\theta }^{\text{Fe}}$$ correspond to those shown in Figure 5a and that associated with $${H}_{0}^{\text{Fe}}$$ is HS below 30 GPa and LS above 30 GPa. Negative ∆H_seg indicates that the GB site is more stable than the bulk site. It can be seen that iron located at the two energetically favorable GB sites have negative segregation enthalpies ∼−0.5 eV over the pressure range of the lower mantle. This implies attractive interactions between iron and the GB, meaning that iron prefers these GB sites to the bulk. Figure 5 also indicates that these sites correspond to the sites at which iron undergoes a spin crossover across the structural transition of the GB. ∆H_seg of iron located in the other two GB sites is either close to zero or positive. Iron does not significantly favor these GB sites compared to the bulk sites. Although a previous atomic-scale study on [001] GBs (Hirel et al., 2019) suggested that GBs become less attractive to solutes with increasing pressure, our results predict only little variation in segregation enthalpy of substitutional iron located at the most stable GB sites.

At low temperature, it can be expected that iron in bulk (Mg,Fe)O preferentially segregates to the energetically favorable GB sites. However, in a polycrystalline system, the number of potential cation sites in the bulk and GB depend on the grain size. If grain size is relatively large, the number of bulk cation sites becomes much more significant than in the GB, leading to a large contribution of the iron configurational entropy to the free energy of the bulk. If grain size decrease, the relative volume fraction of GBs (i.e., number of GB sites) increase compared to the bulk, leading to suppression of this contribution. This size effect is expected to influence the partitioning behavior of iron between bulk and interface at high temperature T. We construct a simple model to include the entropy effect to GB segregration of iron, by taking into account both the configurational and magnetic entropy contributions, according to the following iron exchange reaction

$$${\text{MgO}}_{0}^{\text{Fe}}+{\text{MgO}}_{\theta }\rightleftharpoons {\text{MgO}}_{0}+{\text{MgO}}_{\theta }^{\text{Fe}},$$$ (5)

where $${\text{MgO}}_{0}^{\text{Fe}}$$ and $${\text{MgO}}_{\theta }^{\text{Fe}}$$ refer to polycrystalline MgO with Fe²⁺ in the bulk and GB and MgO₀ and MgO_θ refer to that without Fe²⁺ in the bulk and GB, respectively. The Gibbs free energy change of this reaction is approximated by the segregation free energy as

$$${\increment}{G}_{\text{seg}}=\left[\left({H}_{\theta }^{\text{Fe}}-{S}_{\text{conf}}^{\theta }T-{S}_{\text{mag}}^{\theta }T\right)+{H}_{0}\right]-\left[\left({H}_{0}^{\text{Fe}}-{S}_{\text{conf}}^{0}T-{S}_{\text{mag}}^{0}T\right)+{H}_{\theta }\right]={\increment}{H}_{\text{seg}}-\left({{\increment}S}_{\text{conf}}+{{\increment}S}_{\text{mag}}\right)T.$$$ (6)

$${S}_{\text{conf}}^{\theta }$$ and $${S}_{\text{conf}}^{0}$$ are the configurational entropy of iron in an ideal MgO-FeO solid solution of the GB and bulk in the dilute limit (i.e., no short-range Fe-Fe interactions) defined as

$$${S}_{\text{conf}}^{\theta /0}=-{k}_{B}\left[{X}_{\text{Fe}}^{\theta /0}\hspace*{.5em}\mathrm{ln}{X}_{\text{Fe}}^{\theta /0}+\left(1-{X}_{\text{Fe}}^{\theta /0}\right)\mathrm{ln}\left(1-{X}_{\text{Fe}}^{\theta /0}\right)\right],$$$ (7)

with iron concentration $${X}_{\text{Fe}}^{\theta }$$ and $${X}_{\text{Fe}}^{0}$$ in the GB and bulk, respectively. The magnetic contribution to the entropy $${S}_{\text{mag}}^{\theta }$$ and $${S}_{\text{mag}}^{0}$$ of iron in the GB and bulk, respectively is given by Tsuchiya et al. (2006)

$$${S}_{\text{mag}}^{\theta /0}={k}_{B}{X}_{\text{Fe}}^{\theta /0}\,\mathrm{ln}[m(2S+1)]\,.$$$ (8)

In this expression, m is the electronic configuration degeneracy with m = 3 and m = 1 for HS and LS Fe²⁺, respectively. Cubic (Mg,Fe)O grains are considered with edge length d. Iron concentrations $${X}_{\text{Fe}}^{\theta /0}$$ are determined from the total number of cation sites in the GB and bulk as a function of the grain size d. We assume that the bulk cation sites are equivalent and that iron only segregates to the two energetically favorable GB sites (Figure 5b). The equilibrium constant K_C of Reaction 5 can be expressed as

$$${K}_{C}=\frac{\left[{\text{MgO}}_{0}\right]\left[{\text{MgO}}_{\theta }^{\text{Fe}}\right]}{\left[{\text{MgO}}_{0}^{\text{Fe}}\right]\left[{\text{MgO}}_{\theta }\right]}=\mathrm{exp}\left(-\frac{{{\increment}G}_{\text{seg}}}{{k}_{B}T}\right),$$$ (9)

where [α] represents the fraction of α component in Equation 5 and k_B is the Boltzmann constant. The iron partition coefficient K_D can then be defined as

$$${K}_{D}=\frac{\left[{\text{MgO}}_{\theta }^{\text{Fe}}\right]}{\left[{\text{MgO}}_{0}^{\text{Fe}}\right]}=\frac{{x}_{d}^{\theta }}{{x}_{d}^{0\,}},$$$ (10)

where $${x}_{d}^{\theta }$$ and $${x}_{d}^{0}$$ are the equilibrium molar concentrations of iron in the GB and bulk, respectively. These quantities can vary as a function of grain size d. Equation 10 can be rewritten in terms of the total molar concentration of iron x_t in the system as $${x}_{d}^{\theta }=\left[{K}_{D}/\left(1+{K}_{D}\right)\right]\cdot {x}_{t}$$. Equipartitioning of iron between bulk and interface (i.e., same molar concentration of iron in bulk and GB) occurs when ∆G_seg = 0 at the critical grain size d_crit. Iron prefers to be incorporated in the GB when ∆G_seg < 0 and K_D > 1. Results of ∆G_seg can be seen in Figure S3 of the Supporting Information S1 and results of the partitioning behavior along a lower mantle adiabat are shown in Figure 6. The evolution of K_D and d_crit shows little variation across the P, T range of the lower mantle because of the small variation in segregation enthalpies ∆H_seg with pressure. The critical grain size at which iron equipartitioning occurs in the lower mantle typically lies between 0.1 and 1 μm. For smaller d, more than 50 wt% of iron will be incorporated in the GB. Previous scanning transmission electron microscopy study reported evidence of substantial GB segregation of iron in polycrystalline MgO with average grain size smaller than 2–3 μm at a temperature of ∼1800 K (Mitamura et al., 1979). Their experiments were conducted at ambient pressure. The void-like structure of high-angle tilt GBs generally exhibit larger excess volumes under low pressure which is expected to enhance GB segregation (Hirel et al., 2019; Verma & Karki, 2010). Nevertheless, the experimental findings of Mitamura et al. (1979) are consistent with our results which suggest significant GB segregation of iron in polycrystalline MgO with average μm grain sizes. It is worth to mention that Mitamura et al. (1979) observes a relation between GB segregation and iron concentration. They found the amount of segregated iron in GBs to increase with increasing iron content. Since our approach is based on a dilute segregation model, our findings may be interpreted as lower bounds.

Figure 6

Iron partitioning behavior between grain and GBs across the P, T range of Earth's lower mantle. (a) Evolution of K_D as a function of grain size d. (b) Variation of the critical grain size associated with iron equipartitioning with pressure. The temperature at each pressure corresponds to those of (a).

Although the average grain size of Fp in the lower mantle is poorly constrained, experimental works (Solomatov et al., 2002; Yamazaki et al., 1996) suggest a typical range from ∼0.1 mm to ∼1 cm. Our model results on the high-angle [001] tilt GB suggest that 99.9 wt% of iron is incorporated in the bulk at an average grain size of 1 mm. Therefore, it can be expected that intragranular (Mg,Fe)O hosts most iron at lower mantle temperatures. Nevertheless, substantial grain size reduction of Fp into the μm range has been observed in deformation experiments (Girard et al., 2016; Ledoux et al., 2022; Nzogang et al., 2018). Such fine-grained polycrystalline Fp may particularly develop in lower mantle regions subjected to localized deformation, for example, in and around down going slabs. These dynamically active regions are thus potential candidates where significant amounts of iron could partition in high-angle tilt GBs and influence the iron spin crossover by pressure-induced structural transformations across the depth range of the lower mantle.

3.3.3 Influence of Iron on the Mechanical Behavior

Interactions between solutes and GBs are known to influence the mechanical properties of polycrystalline materials (Sutton & Balluffi, 1995). The effect of the presence of solutes in GBs can either cause embrittlement or strengthening (e.g., solute-drag) of the interface. In the last two sections, we have seen that iron is able to be incorporated in the [001] MgO GBs. These iron-GB interactions may affect GB strength. We quantify the critical shear stress τ_c of iron-bearing GBs using the imposed-strain simulation method. We use the same simulation set-up as employed for the iron spin state calculations. For the dilute limit, only one iron is substituted at its most stable GB site (i.e., B site in the symmetric and C site in the asymmetric ground state structure) in both interfaces in the simulation cell. The symmetric GB has an iron concentration of $${X}_{\text{Fe}}^{\text{GB}}$$ = 3.571% and the asymmetric structure a concentration of $${X}_{\text{Fe}}^{\text{GB}}$$ = 4.167% in absence of short-range Fe-Fe interactions. At thermodynamic equilibrium, we find HS Fe²⁺ to be the most favorable spin state at the B site in the symmetric structure and LS Fe²⁺ most favorable at the C site in the asymmetric structure. However, the iron spin state is sensitive to the local coordination environment of the site (i.e., crystal field). The spin state of Fe²⁺ in the ground state structure could be affected by GB motion during the imposed-strain simulations. This in turn may affect the ideal shear strength of the GB. Therefore, we first investigate if GB motion is able to induce a spin transition. We calculated the enthalpy evolution of HS and LS iron in the GB during GB motion under imposed strain (Equation 3). Since atomic rearrangements are most pronounced during SCM, we only performed the calculations for SCM for the minimum (30 GPa) and maximum (120 GPa) pressure condition. Results are shown in Figure S4 of the Supporting Information S1. The enthalpy difference between HS and LS remains almost identical during SCM. HS is found to remain the favorable spin state of iron in the symmetric structure, whereas the LS state remains more stable in the asymmetric structure. GB motion is thus unlikely to cause a spin transition of iron in absence of direct Fe-Fe interactions, at least at pressures away from the spin crossover pressure at 80 GPa. We then compute the stress-strain evolution for iron-bearing GBs with iron in the lowest energy spin state at the preferred GB site (Figure S5 in Supporting Information S1). Results are summarized in Table 1. The results suggest that the low concentration iron-bearing GB tends to strengthen the interface by a maximum of ∼24% until a pressure of 90 GPa, independent on the mechanism of motion. We find that incorporation of iron in the GB does not induce a change in the preferred mechanism of GB motion. In contrast, at a pressure of 120 GPa, iron has a weakening effect of ∼19% on the GB strength. Although it is difficult at this point to assess the cause of this weakening, it is likely related to the change in transition state of the GB moving by SCM at 115 GPa which causes GB weakening with increasing pressure. The presence of iron might possibly enhance the mechanical weakening of the interfaces with increasing depth in the lowermost mantle D” region of the Earth and the mantle of super-Earths. But more in-detail studies of (Mg,Fe)O GBs, in particular at multi-megabar pressures for the latter, are needed to confirm this hypothesis.

Table 1. Difference in Critical Shear Stress Between Iron-Free (τ_c) and Iron-Bearing ($${\tau }_{c}^{\text{Fe}}$$) GBs

Pressure (GPa)	Mechanism of motion	τc (GPa)	$${\tau }_{c}^{\text{Fe}}$$ (GPa)	%∆τc
30	SCM	1.85	1.92	+3.8
90	SCM	5.51	5.88	+6.7
90	GBS	4.23	5.25	+24
120	SCM	8.80	7.17	−19

• Note. %∆τ_c corresponds to the relative deviation of $${\tau }_{c}^{\text{Fe}}$$ from τ_c.

Our results indicate that interactions between iron and the interface have a non-unique effect on the critical shear stress of the GB depending on its ground state structure, the mechanisms of GB motion and associated transition state which all vary with pressure (i.e., depth). On average, we find a moderate effect on τ_c of the investigated (Mg,Fe)O GB with low iron content of $${X}_{\text{Fe}}^{\text{GB}}$$ ∼ 4% up to a maximum variation of ∼25% compared to the iron-free MgO GB. The effect of Fe²⁺ on the mechanical behavior of the same GB may be different for higher iron concentrations, particularly when Fe-Fe interactions start to play a role, potentially important for (Mg,Fe)O with average grain size in the μm-range or smaller. It requires further investigation to unveil the mechanical behavior of high-concentration iron-bearing GBs.

4 Discussion

Based on a first-principles density functional theory approach, we modeled the physical properties of {310}[001] high-angle tilt GBs in Fp at pressures between 30 and 400 GPa. Pressure induces a structural transition of the GB ground state at ∼80 GPa from a symmetric to a more compact asymmetric structure. We investigated the mechanical behavior and resolved the minimum energy path for GB motion. Energetically favorable motion of the symmetric GB structure (stable between 30 and 80 GPa) occurs by SCM. The pressure-induced GB transition to the asymmetric structure changes the minimum energy path for GB motion from SCM to GBS. This leads to a change in preferred sliding direction from [$$1\overline{3}0$$] to [001]. It is well known that the critical shear stress for the onset of dislocation glide is strongly pressure sensitive (Amodeo et al., 2012). Here, we show similar behavior of high-angle tilt GBs under pressure. Pressure influences the critical shear stress for the onset of GB mobility and induces a change in the mechanism of GB motion and sliding direction. Therefore, the effect of pressure on the sliding direction might have implications for the development of lattice or shape preferred orientations during GB-assisted plasticity, recrystallization and grain growth. The change from SCM to GBS at the transition pressure is caused by a structural variation from the symmetric to the asymmetric GB structure which is characterized by shear displacements of atomic planes across the interface along [001]. Previous atomic-scale simulations on [001] GBs reveal that 73.2%–86.6% of all high-angle [001] GBs become asymmetric at pressures between 30 and 120 GPa (Hirel et al., 2019). Therefore, GBS is likely an important mechanism for GB motion of asymmetric GBs under high pressure. We find that pressure is also able to induce a transition in the saddle point structure of high-angle tilt GBs during their motion. Across the pressure range between 115 and 400 GPa, the easiest mechanism of GB motion turns back to SCM and the GB undergoes a monotonous decrease in its critical shear stress of up to ∼6 GPa. This mechanical weakening is caused by the change in transition state structure of the GB during SCM. If this pressure-induced transition is intrinsically related to the asymmetric structure of GBs, many other tilt GBs may undergo similar mechanical weakening at multi-megabar pressures since the majority of high-angle tilt GBs adopt an asymmetric structure at high pressure. This might have significant implications for the intercrystalline plasticity of MgO at ultrahigh pressures. More systematic investigation of the mechanical behavior of asymmetric GBs are needed to confirm this assumption. Generally, the viscosity of MgO is expected to increase with depth (Cordier, 2015) but some mechanisms of viscosity reductions have been proposed under multi-megabar conditions (Karato, 2011; Ritterbex et al., 2018) suggesting that mantle convection in super-Earths could be more vigorous than previous estimations by geodynamic studies which rely on simple extrapolations of viscosity (Stamenković et al., 2012; Tackley et al., 2013). Our results indicate that GB weakening in the B1 phase of MgO may yet be another potential mechanism for viscosity reductions with increasing depth in the mantles of super-Earths.

GB motion is widely considered as a process to reduce excess internal energy in polycrystalline materials across interfaces during recrystallization and recovery (e.g., GB bulging, etc.). Our imposed-strain simulations show that GB mobility in Fp can also act as a strain accommodating mechanism during deformation under high pressure as inferred from previous studies (e.g., Cahn et al., 2006; Molodov & Molodov, 2018). In fact, GB motion as a strain-producing mode of plastic deformation has been observed in many geologic materials (Means & Jessel, 1986). For example, SCM occurs in several other cubic, hexagonal or orthorhombic minerals such as rock salt (Guillope & Poirier, 1980), ice (Hondoh & Higashi, 1978; Pennock & Drury, 2020), and has been proposed in olivine (Cordier et al., 2014). GBS has been recently observed to accommodate MgO deformation at high P, T conditions (Ledoux et al., 2022). In this study, we quantified the static mechanical behavior for SCM and GBS in terms of the MEB (E) and its associated critical shear stress τ_c. An important limitation of our work is that we investigated the physical properties of static [001] GBs an do not account for the intrinsic effects of temperature. The thermally activated mechanisms of GB motion may vary from those observed at static conditions. At finite temperature, GBS typically operates in parallel with other elementary plasticity processes to prevent the formation of voids under pressure (Poirier, 1985). Also, SCM at finite T may depend on the nucleation of additional structural defects as kinks, terraces, disclinations and disconnections (Han et al., 2018; Rajabzadeh et al., 2013; Sun et al., 2017). The energetics of these thermally activated mobility processes may differ from the static motion of GBs presented in this work. Although the migration energy barriers (E) are independent of grain size, it is worth to mention that strain production by interfacial motion depends on grain boundary velocity, generally expressed as V_gb ∝ u_xν_D exp [−E(τ_ij)A/k_BT], where u_x is the GB sliding displacement, ν_D the Debye frequency, τ_ij the resolved shear stress, and A the grain boundary area subjected to motion (Ivanov & Mishin, 2008). When the total area of the grain boundary in a cubic material becomes mobile, A ≃ d², where d equals the grain size. Hence, the role of GB motion is typically expected to be important in fine-grained materials. If thermally activated GB motion is facilitated by the nucleation and propagation of other defects such as terraces or disconnections, only a fraction of the total interfacial area may become mobile and A ≪ d². Thus, GB-assisted creep may not be restricted to fine-grained polycrystalline materials as suggested by Hondoh and Higashi (1978) who observed SCM in deformed bicrystals of ice (Pennock & Drury, 2020). Therefore, thermally activated GB motion in Fp needs to be explicitly modeled to obtain a better understanding of the importance of GB mobility as a strain-producing mechanism in Fp under the P, T conditions of planetary mantles.

In the mantle of terrestrial planets including the Earth, MgO tends to form a solid solution with FeO. Substitutional iron in this solid solution undergoes a HS-LS transition at relevant pressure conditions. We have investigated the effect of the GB on this HS-LS transition. Iron at the energetically favorable GB sites undergoes a systematic spin crossover triggered by the pressure-induced GB transition to the more compact asymmetric structure at ∼80 GPa—as observed in most high-angle [001] tilt GBs at high pressure (Hirel et al., 2019). The static spin crossover in high-angle tilt GBs can generally be expected to occur at higher pressures compared to intragranular (Mg,Fe)O (∼30 GPa). We do not account for the effect of temperature on the spin crossover. Under static conditions the crossover takes place abruptly at a certain pressure, but at elevated mantle temperature it is expected to occur over a broad pressure interval due to the coexistence of mixed HS and LS states of iron (Tsuchiya et al., 2006). A similar broadening phenomenon of the spin transition pressure interval can be expected for GBs, especially because of the non-ideal GB site configurations of iron that are energetically allowed to exist at mantle temperatures. It might thus be expected that most iron in high-angle tilt GBs undergo an iron spin crossover at a relatively higher depth interval in the lower mantle with respect to the bulk, triggered by pressure-induced structural transitions of the GB to achieve optimal compaction. However, the influence of the GB on the spin crossover depends strongly on the concentration of iron in Fp GBs. We constructed a simple analytical model to constrain the partitioning behavior of iron between grain and GBs as a function of grain size. Our model predicts equipartition of ferrous iron between the bulk and interface in the μm-range over the pressure interval of the lower mantle. This small grain size might particularly develop by grain size reduction processes in Fp during enhanced plasticity of (Mg,Fe)O in lower mantle regions subjected to localized deformation. Strongly deformed mantle regions are thus potential candidates where the iron spin crossover could be influenced by the presence of GBs. Our results suggest that dynamically active regions might experience elevated iron spin crossover pressures compared to thermodynamically stable regions in the lower mantle.

5 Conclusions

Based on first-principles electronic structure calculations we investigated structural properties, thermodynamics and the mechanical behavior of high-angle {310}[001] tilt grain boundaries at pressures corresponding to Earth's lower mantle and the mantle of super-Earth exoplanets. Our results suggest that pressure-induced structural transformations of high-angle GBs strongly affect their dynamical behavior and mechanical properties. Abrupt and gradual variations of mechanical hardening and weakening of the GB could be observed with increasing pressure. In particular, significant mechanical weakening of the asymmetric GB structure is found with increasing pressures in the multi-megabar range in the mantle of super-Earth exoplanets. Results show that structural transitions are able to change the energetically favorable mechanism of GB mobility and preferred sliding direction. These structural transitions under high pressure also control the spin crossover of iron in tilt GBs and may potentially alter the HS-LS transition of iron in lower mantle regions where ferropericlase exhibit micrometer-sized grains. Interaction between iron and the interface in the dilute limit are shown to have non-unique and moderate effects on the mechanical strength of the GB. Since our study focusses on a single [001] high-angle tilt GB, more systematic data from experiments and theoretical modeling are necessary to achieve better understanding of the collective effects of tilt GBs on the thermodynamic properties and rheology of polycrystalline Fp at P, T conditions of planetary mantles.

Acknowledgments

This work was supported by JSPS KAKENHI Grant JP20K14580 and the High Performance Computing Infrastructure of Japan (HPCI) Grant hp200019 awared to S. Ritterbex. O. Plümper was supported by the European Research Council starting Grant 852069-nanoEARTH. Calculations are performed on the parallel computation systems at the Geodynamics Research Center, Ehime University, Japan, the Information Technology Center, Nagoya University, Japan and the Utrecht Geosciences Computation Facility Eejit, Utrecht University, the Netherlands. We thank the two anonymous reviewers for their constructive comments.

Open Research

Data Availability Statement

This work does not contain empirical data. It is purely based on a theoretical mineral physics approach. The modeling methods and all information necessary to reproduce the results of this work are described in detail in the manuscript and Supporting Information S1. All atomistic modeling has been performed with the ab initio density functional self-consistent field method implemented in the open-source Quantum Espresso code package (Giannozzi et al., 2009) available at https://www.quantum-espresso.org.