P Bower

P Carrez

Cordier

Riccardo Reali
email: riccardo.reali@ed.univ-lille1.fr

J M Jackson

J Van Orman

D J Bower

P Cordier

Modeling viscosity of (mg,fe)O at lowermost mantle conditions

Keywords: (Mg,Fe)O, 2.5D Dislocation Dynamics, Viscosity, Core-Mantle Boundary, Ultralow-Velocity Zones

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2

Abstract

The viscosity of the lower mantle results from the rheological behavior of its two main constituent minerals, aluminous (Mg,Fe)SiO 3 bridgmanite and (Mg,Fe)O ferropericlase.

Understanding the transport properties of lower mantle aggregates is of primary importance in geophysics and it is a challenging task, due to the extreme time-varying conditions to which such aggregates are subjected. In particular, viscosity is a crucial transport property that can vary over several orders of magnitude. It thus has a first-order control on the structure and dynamics of the mantle.

Here we focus on the creep behavior of (Mg,Fe)O at the bottom of the lower mantle, where the presence of thermo-chemical anomalies such as ultralow-velocity zones (ULVZ) may significantly alter the viscosity contrast characterizing this region. Two different iron concentrations of (Mg 1-x Fe x )O are considered: one mirroring the average composition of ferropericlase throughout most of the lower mantle (x = 0.20) and another representing a candidate magnesiowüstite component of ULVZs near the base of the mantle (x = 0.84). The investigated pressure-temperature conditions span from 120 GPa and 2800 K, corresponding to the average geotherm at this depth, to core-mantle boundary conditions of 135 GPa and 3800 K.

In this study, dislocation creep of (Mg,Fe)O is investigated by dislocation dynamics (DD) simulations, a modeling tool which considers the collective motion and interactions of dislocations. To model their behavior, a 2.5 dimensional dislocation dynamics approach is employed. Within this method, both glide and climb mechanisms can be taken into account, and the interplay of these features results in a steady-state condition. This allows the retrieval of the creep strain rates at different temperatures, pressures, applied stresses and iron concentrations across the (Mg,Fe)O solid solution, providing information on the viscosity for these materials.

Introduction

The Earth's core-mantle boundary (CMB) is a major thermochemical boundary layer within the Earth, exerting a primary influence on the evolution of the planet. Seismic observations indicate a significant complexity in this boundary region, including structures of varying length scales and velocity perturbations. In particular, patches near the CMB have been identified, ranging from a few to tens of km thick and ~100 km across, in which the seismic wave velocities are reduced by about 5 to 30% (e.g., [START_REF] Garnero | A very slow basal layer underlying large-scale lowvelocity anomalies in the lower mantle beneath the pacific -evidence from core phases[END_REF][START_REF] Revenaugh | Seismic evidence of partial melt within a possibly ubiquitous low-velocity layer at the base of the mantle[END_REF][START_REF] Rost | Fine-scale ultralow-velocity zone structure from high-frequency seismic array data[END_REF][START_REF] Hutko | Localized double-array stacking analysis of PcP: D″ and ULVZ structure beneath the Cocos plate, Mexico, central Pacific, and north Pacific[END_REF][START_REF] Rost | Deep Earth: Core-mantle boundary landscapes[END_REF][START_REF] Sun | Rolling hills on the core-mantle boundary[END_REF][START_REF] Thorne | Mega ultra low velocity zone and mantle flow[END_REF][START_REF] Gassner | Probing the core-mantle boundary beneath Europe and Western Eurasia: A detailed study using PcP[END_REF][START_REF] Frost | Dynamical links between small-and large-scale mantle heterogeneity: Seismological evidence[END_REF][START_REF] Zhang | Constraints on smallscale heterogeneity in the lowermost mantle from observations of near podal PcP precursors[END_REF]; see [START_REF] Yu | Ultra-low velocity zone locations: A global assessment[END_REF] for a recent review). These ultralow-velocity zones (ULVZs) have been interpreted as aggregates of partially molten material or as solid, iron-enriched assemblages, typically based on proposed sources of velocity reduction (Knittle and Jeanloz, 1991a;[START_REF] Williams | Seismic evidence for partial melt at the base of the mantle[END_REF][START_REF] Mao | Ferromagnesium postperovskite silicates in the D″ layer of the Earth[END_REF][START_REF] Labrosse | A crystallizing dense magma ocean at the base of the Earth's mantle[END_REF][START_REF] Sakai | Fe-Mg partitioning between post-perovskite and ferropericlase in the lowermost mantle[END_REF][START_REF] Wicks | Very low sound velocities in iron-rich (Mg,Fe)O: Implications for the core-mantle boundary region[END_REF][START_REF] Dorfman | Effect of Fe-enrichment on seismic properties of perovskite and post-perovskite in the deep lower mantle[END_REF][START_REF] Muir | Elastic properties of ferropericlase at lower mantle conditions and its relevance to ULVZs[END_REF][START_REF] Liu | Hydrogen-bearing iron peroxide and the origin of ultralow-velocity zones[END_REF].

It is suggested that more extreme chemical heterogeneity, such as Fe-rich (Mg,Fe)O or magnesiowüstite residuum from large-scale melting events in Earth's history [START_REF] Labrosse | A crystallizing dense magma ocean at the base of the Earth's mantle[END_REF]) may be at least partially responsible for explaining ULVZs [START_REF] Wicks | Very low sound velocities in iron-rich (Mg,Fe)O: Implications for the core-mantle boundary region[END_REF][START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF][START_REF] Muir | Elastic properties of ferropericlase at lower mantle conditions and its relevance to ULVZs[END_REF][START_REF] Wicks | Sound velocity and density of magnesiowüstites: Implications for ultralow-velocity zone topography[END_REF]. Interpretation of these features and the dynamic processes that shape them requires knowledge of their rheological properties. In particular, [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF] presented a geodynamic and mineral physics model of a solid-state ULVZ by exploring the steady-state morphology of a magnesiowüstite-bearing layer at the base of the mantle. Implemented as a chemical density anomaly defined by variable proportions of magnesiowüstite co-existing with bridgmanite, the layer was relatively well constrained in terms of its equation of state. However, one of the least constrained variables in dynamic modeling is the viscosity of the system. In [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF], the viscosity of the system was prescribed only as a function of temperature. While recent progress has been made in estimating the rheological properties of MgO and magnesium-rich (Mg,Fe)O under lower mantle conditions (e.g., [START_REF] Yamazaki | Some mineral physics constraints on the rheology and geothermal structure of Earth's lower mantle[END_REF][START_REF] Van Orman | Diffusion in MgO at high pressures: Constraints on deformation mechanisms and chemical transport at the core-mantle boundary[END_REF][START_REF] Lin | Deformation of lower-mantle ferropericlase (Mg,Fe)O across the electronic spin transition[END_REF][START_REF] Cordier | Modelling the rheology of MgO under Earth's mantle pressure, temperature and strain rates[END_REF][START_REF] Marquardt | Slab stagnation in the shallow lower mantle linked to an increase in mantle viscosity[END_REF][START_REF] Deng | Viscosity jump in the lower mantle inferred from melting curves of ferropericlase[END_REF][START_REF] Immoor | Evidence for {100}<011> slip in ferropericlase in Earth's lower mantle from high-pressure/high-temperature experiments[END_REF], no such studies exist for magnesiowüstite.

In this study, we apply the recent development of numerical modeling of dislocation creep based on 2.5D dislocation dynamics to the rheology of iron-rich (Mg,Fe)O magnesiowüstite at the PT conditions of the CMB. Since high-temperature creep significantly depends on diffusion, we first present a critical assessment and evaluation of this parameter for (Mg,Fe)O under the conditions of the CMB. Then dislocation dynamics simulations of creep are performed. Using our results of strain rates as a function of the applied stresses, we obtain the effective viscosity of magnesiowüstite. For comparison, we also perform these simulations for ferropericlase. Our results point to very low values of viscosity for magnesiowüstite. These findings motivate the final section of this paper, where we assess the influence of low viscosity magnesiowüstite on the dynamics of the lowermost mantle by re-visiting the framework presented in [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF].

Method: 2.5D dislocation dynamics

2.5-dimensional (2.5D) dislocation dynamics (DD) is now a well-established numerical tool which considers the collective motions and interactions of dislocations at the mesoscale. It has been first developed to model creep in aluminum [START_REF] Keralavarma | Power-law creep from discrete dislocation dynamics[END_REF] and then applied to olivine (Boioli et al., 2015a,b) and MgO [START_REF] Reali | Modeling plasticity of MgO by 2.5D dislocation dynamics simulations[END_REF]. It is therefore a viable instrument to model creep of (Mg,Fe)O at high pressures and temperatures.

A full description of the 2.5D-DD simulation technique adopted in this study along with a comparison with other methods employing dislocation dynamics is provided in previous work [START_REF] Reali | Modeling plasticity of MgO by 2.5D dislocation dynamics simulations[END_REF]. There, the model was benchmarked for MgO at ambient conditions with respect to experimental and other numerical data, demonstrating its validity in a variety of thermal regimes. In particular, at high temperatures, by combining dislocation glide together with dislocation climb, this approach could satisfactorily reproduce the existing creep data on MgO. In fact, when at high temperatures diffusion is active, dislocations can escape their glide planes by climb. Dislocations in a dipole configuration may consequently approach each other until they reach a given critical distance and annihilate. Hence, the dislocation density is controlled by the interplay between dislocation multiplication and dislocation annihilation promoted by climb. This leads to a steady-state regime where, with respect to time, the dislocation density fluctuates around an equilibrium value and the plastic strain increases linearly. In this condition, the rate of creep strain as a function of applied stress and temperature can be described using a power law creep equation, in good agreement with experimental data.

Here we focus on this high temperature regime (above approximately half of the melting temperature) at lowermost mantle conditions. In this regime, temperatures are high enough to favor diffusion and consequently render dislocation climb, which is diffusion-dependent, a viable recovery mechanism. The main features of the model in this regime are outlined below.

Dislocation dynamics is based on continuum elasticity theory, which describes the elastic field induced by dislocations in a crystal, and the interactions of dislocations with each other and with the stress field resulting from an external, deviatoric loading. The 2.5D-DD formulation for this work considers a simplified 2D framework in which local rules are taken into account in order to mimic relevant 3D mechanisms, such as dislocation multiplication, annihilation and junction formation (Benzerga et al., 2004;[START_REF] Gómez-García | Dislocation patterns and the similitude principle: 2.5D mesoscale simulations[END_REF][START_REF] Durinck | Modeling the plastic deformation of olivine by dislocation dynamics simulations[END_REF].

Subsequently, these features were tested on olivine by Boioli et al. (2015a,b) and on MgO by [START_REF] Reali | Modeling plasticity of MgO by 2.5D dislocation dynamics simulations[END_REF], which provides more details on the adjustment of these parameters. This 2.5D formulation allows the construction of a simple model that captures the most important 3D dislocation mechanisms and readily includes the interplay between glide and climb in high temperature, high pressure creep conditions.

Throughout most of the lower mantle, the magnesium-iron oxide in equilibrium with calcium silicate perovskite and bridgmanite is thought to be magnesium-rich (Mg,Fe)O ferropericlase, which exhibits the B1 rock-salt structure with space group 3 . Ferrous iron in ferropericlase undergoes a gradual spin pairing transition, which is completed by the PT conditions of the base of the mantle [START_REF] Sturhahn | The spin state of iron in minerals of Earth's lower mantle[END_REF][START_REF] Tsuchiya | Spin transition in magnesiowüstite in Earth's lower mantle[END_REF]. In addition to the temperature dependence, the pressure of the spin pairing transition, as well as the width (pressure range), increases with increasing iron concentration (e.g., [START_REF] Solomatova | Equation of state and spin crossover of (Mg, Fe)O at high pressure, with implications for explaining topographic relief at the core-mantle boundary[END_REF].

Although studies indicate that the iron cations in iron-rich (Mg,Fe)O begin to pair at pressures of around 120 GPa and 300 K (e.g., [START_REF] Speziale | Iron spin transition in Earth's mantle[END_REF][START_REF] Wicks | Very low sound velocities in iron-rich (Mg,Fe)O: Implications for the core-mantle boundary region[END_REF][START_REF] Fischer | Equation of state and phase diagram of FeO[END_REF][START_REF] Ozawa | Spin crossover, structural change, and metallization in NiAs-type FeO at high pressure[END_REF], the presence of magnesium widens the pressure-temperature stability field of the B1 structure compared to FeO [START_REF] Wicks | The thermal equation of state and stability of iron-rich (Mg,Fe)[END_REF]. Thus, thermal effects and the presence of a small amount of magnesium would likely prohibit a spin crossover in iron-rich (Mg,Fe)O at CMB conditions. Therefore, in this study, we consider high-spin magnesiowüstite (Mg 0.16 Fe 0.84 )O and low-spin ferropericlase (Mg 0.8 Fe 0.2 )O to represent possible end-member cases of (Mg,Fe)O at lowermost mantle conditions. For these compositions, and for a sake of comparison, we consider two PT conditions: one resembling the geotherm at 120 GPa and 2800 K and the other representing CMB conditions suitable for solid-state ULVZs, at 135 GPa and 3800 K. Additionally, in order to have a first estimate of the creep activation enthalpy for magnesiowüstite at CMB conditions, another set of calculations were run at 135 GPa and 3200 K.

The elastic properties for high-spin magnesiowüstite are taken from experimental results

of [START_REF] Wicks | The thermal equation of state and stability of iron-rich (Mg,Fe)[END_REF][START_REF] Wicks | Sound velocity and density of magnesiowüstites: Implications for ultralow-velocity zone topography[END_REF], which constrained the thermal equation of state and shear elastic properties for this phase under lowermost mantle conditions using powder X-ray diffraction and inelastic X-ray scattering techniques. The elastic properties for low-spin ferropericlase are taken from a theoretical study that investigated the thermoelastic effects related to the spin crossover up to pressures of 130 GPa and 4000 K [START_REF] Wu | Spin crossover in ferropericlase and velocity heterogeneities in the lower mantle[END_REF]. We provide the elastic properties used for these two end-member cases along with their lattice constant and burgers vector magnitude in Table 1.

In our simulations, 200 to 400 dislocations having Burgers vectors b = ½<110> are introduced in a simulation cell having edges of ~10-60 µm. These dislocations are allowed to glide in two slip systems belonging to the easiest slip family -i.e. ½<110>{110}. The cell is a square of size Lx = Ly and dislocation lines are introduced as straight segments of constant length L = 1 µm perpendicular to the reference plane, where both glide and climb directions lie. All dislocation lines have edge character, which is the limiting feature concerning dislocation climb.

The dislocation climb direction is perpendicular to the glide direction. Deviatoric stress is applied in order to symmetrically load the two slip systems, resulting in an equal Schmid factor of 0.433 for both of them [START_REF] Reali | Modeling plasticity of MgO by 2.5D dislocation dynamics simulations[END_REF]. Dislocations gliding in different planes may cross each other and form junctions which are oriented along the direction of intersection between the two planes. Here, we consider two glide planes that form an angle of 120 degrees with each other, allowing for the formation of stable, sessile junctions [START_REF] Carrez | Dislocation reactions and junctions in MgO[END_REF].

For all the tested PT conditions, a constant creep stress σ ranging from 10 to 50 MPa was applied to the simulated system. Furthermore, in the case of (Mg 0.8 Fe 0.2 )O ferropericlase, some tests were run also at higher stresses, up to 500 MPa, in order to better anchor the fit for the strain rate as a function of stress. In fact, since climb in (Mg 0.8 Fe 0.2 )O ferropericlase is very slow for many of the PT conditions considered, it was possible to achieve a minimum acceptable strain of around 1% in practical times only by raising the applied stress. At steady-state, the equilibrium dislocation density, equal to the number of dislocations in the system divided by its area, rises with the applied, constant creep stress. Therefore, the cell size was adjusted in order to have a representative and comparable number of dislocations at equilibrium for all the tested stresses. Equilibrium dislocation densities in this study varied between 10 11 and 10 13 m -2 , requiring cell dimensions L x = L y from 8 to 60 µm.

A key point in modeling dislocation creep is the description of the interplay between a strain-producing mechanism and a recovery mechanism, which controls the time scale at which the deformation is taking place. Here, athermal dislocation glide is the strain producing mechanism. The recovery mechanism (dipole annihilation) is controlled by dislocation climb, which allows the dislocation to move outside its glide plane through the absorption or emission of a formula unit to or from the dislocation line.

At the temperatures considered here, thermal activation is sufficient to overcome lattice friction. Thus, plastic shear is athermal and constrained by the interactions between dislocations.

In this regime, dislocations move by glide at a free flight velocity vath described by a viscous drag equation:

= (1)
where b is the Burgers vector, τ is the effective resolved shear stress acting on the dislocation and is a viscous drag coefficient that varies linearly with temperature [START_REF] Hirth | Theory of Dislocations[END_REF].

In particular, τ = τ app + τ int is calculated at each dislocation position and represents the sum of the applied stress τ app and the elastic interaction stress τ int induced by all the other dislocations, both projected along the slip direction. An experimental estimate of for (Mg,Fe)O is lacking, therefore a value measured on other fcc oxides at 300 K is taken from the literature [START_REF] Singh | Acoustical dissipation due to phonon-phonon interaction, thermoelastic loss and dislocation damping in MnO and CoO[END_REF] and is scaled to the considered temperatures (i.e. from 2800 to 3800 K). The free-flight mobility law is not expected to play a relevant role in simulations describing plasticity in this regime [START_REF] Amodeo | Dislocation reactions, plastic anisotropy and forest strengthening in MgO at high temperature[END_REF] because the average velocity of dislocations in the microstructure is limited by their forest interactions (i.e. dislocation-dislocation contact reactions). For this reason, the forest obstacles control the plastic flow and in the present model they are represented by dislocation junctions, which are formed when two attractive dislocations gliding in different planes come to a distance smaller than the average length of junctions under stress, which is estimated from 3D simulations [START_REF] Devincre | Physical analyses of crystal plasticity by DD simulations[END_REF]. In this case, dislocation positions are locked and the interaction stress between them is set to zero. The junction will remain stable as long as the stress on each dislocation composing it, evaluated similarly to τ in eq. ( 1), is lower than a critical value. If this value is exceeded, the dislocations composing the junction will be unlocked and made free to move. The local rule to break a junction in a 2.5D-DD model is defined by [START_REF] Gómez-García | Dislocation patterns and the similitude principle: 2.5D mesoscale simulations[END_REF] and was tuned for MgO by [START_REF] Reali | Modeling plasticity of MgO by 2.5D dislocation dynamics simulations[END_REF].

The other important mobility law is the one describing dislocation climb, which allows dislocations stuck in a minimum energy configuration to escape their glide plane and eventually interact and annihilate with other segments. Climb is the recovery mechanism which counterbalances the athermal, forest dislocation glide mechanism (in turn responsible for dislocation multiplication) and brings the system to the steady-state creep regime. Climb takes place through the self-diffusion of defects, mainly vacancies, which are adsorbed or emitted by the dislocation line. Climb controls the dynamics of the entire creep process, under the assumption that the dislocation lines are saturated with jogs (i.e., steps on the dislocation line in the climb direction). Therefore, under these conditions, vacancies are adsorbed or emitted instantaneously. At the steady-state, the dislocation climb velocity vc can be expressed following the [START_REF] Hirth | Theory of Dislocations[END_REF] and [START_REF] Caillard | Thermally Activated Mechanisms in Crystal Plasticity 8[END_REF] formulation:

= -! " ( 2
)
where τc is the climb stress, calculated similarly to τ but resolved along the climb direction, T is the temperature and kB the Boltzmann constant. = 2$/ln (*/+ ) is a geometrical factor that describes the cylindrical geometry of the vacancy flux field around the dislocation line, where R and r c represent the two radii of the cylindrical surfaces through which the vacancy flux is calculated. r c is the dislocation core radius and R is taken as a fraction of the average dislocation distance. Being within the logarithmic term, the R/r c ratio does not significantly affect the climb velocity values and here is taken constant and equal to 100 (Boioli et al., 2015a,b).is the formation volume of vacancies in the cation and O sites and is calculated from the unit cell volume of (Mg,Fe)O -i.e. -= a 3 / Z where Z = 4 is the number of formula units per unit cell. . / and . 0 are the vacancy concentrations far from the dislocation and the equilibrium vacancy concentration in the bulk volume, respectively. Following Boioli et al. (2015a), we assume that far away from the dislocations the vacancy concentration is constant and equal to the equilibrium concentration in the bulk volume (. / = . 0 ). Finally, 1 23 is the vacancy self-diffusion coefficient which controls the flow of atomic species from and to the climbing dislocation. Thus, this parameter is of considerable importance, since it is the one that ultimately controls the strainrate. Self-diffusion is described by an Arrhenius-type formula:

1 23 = 1 45 = 1 0 - 67 (3)
where 1 0 is a pre-exponential factor and 89 23 is the activation enthalpy for oxygen selfdiffusion which is the sum of the vacancy formation and migration enthalpy. As will subsequently be shown, the climb-limiting species in (Mg,Fe)O is oxygen, and therefore it is necessary to properly estimate 1 45 for (Mg,Fe)O as a function of composition at CMB conditions.

Evaluation of high pressure and high temperature self-diffusion coefficients of (Mg,Fe)O

Self-diffusion of MgO has been extensively studied over the past decades, for both the cation and anion sublattices. However, a full description of this phenomenon as a function of pressure, temperature and composition for lowermost mantle conditions is lacking. Therefore, it is necessary to develop a method to conveniently estimate diffusion coefficients to be introduced into the 2.5D dislocation dynamics model.

Concerning the magnesium sublattice in periclase, it can be stated that extrinsic cation vacancies control Mg self-diffusion at any temperature up to the melting point. This is because aliovalent positively charged solutes like Fe 3+ , Al 3+ and Cr 3+ tend to be present at concentrations much higher than the intrinsic vacancy concentrations, even in nominally pure synthetic crystals.

The presence of aliovalent solutes leads to the formation of cation vacancies, to maintain charge balance, and hinders the formation of oxygen vacancies, which also have a net positive charge.

For this reason, oxygen self-diffusion coefficients are at least 2 to 3 orders of magnitude smaller than those of magnesium at any investigated temperature (Van Orman and Crispin, 2010 and references therein). Oxygen self-diffusion 1 45 is thus considered as the rate-limiting mechanism for dislocation climb.

A review of the literature on oxygen self-diffusion coefficients in MgO at ambient conditions [START_REF] Van Orman | Diffusion in oxides[END_REF] shows that at high temperatures an activation energy resembling that of the intrinsic regime is observed in some cases, while a phenomenon with smaller temperature dependence is often observed at lower temperatures. Self-diffusion in the latter case appears to be strongly influenced by structural defects, but not by chemical impurities (e.g. [START_REF] Yang | Intrinsic diffusion properties of an oxide: MgO[END_REF]1996), and is consistent with fast transport along dislocation cores [START_REF] Van Orman | Diffusion in oxides[END_REF].

Although fast transport via dislocations may have an important role in the bulk diffusivity of oxygen in both experimental and natural samples, this process is not relevant to dislocation climb. The latter refers to the net flux of vacancies from and to the dislocation line across a cylindrical surface, which excludes the core-related effects and therefore does not consider the aforementioned, structure-sensitive factors affecting diffusion (cf. the definition of the geometrical parameter α in the climb velocity law, eq. 2). Therefore, the starting point for the evaluation of diffusion at lower mantle conditions is the assumption that oxygen diffusion is intrinsic.

Experimental measurements indicate that oxygen diffusion coefficients are insensitive to the concentration of aliovalent dopants, either in the intrinsic regime [START_REF] Yang | Intrinsic diffusion properties of an oxide: MgO[END_REF]1996) or in the extrinsic regime controlled by structural defects [START_REF] Ando | Oxygen self-diffusion in Fe-doped MgO single crystals[END_REF].

At high temperatures, as mentioned before, higher activation enthalpies for diffusion are generally observed [START_REF] Yang | Intrinsic diffusion properties of an oxide: MgO[END_REF]1996;[START_REF] Ando | Oxygen self-diffusion in Fe-doped MgO single crystals[END_REF]. In particular, [START_REF] Yang | Intrinsic diffusion properties of an oxide: MgO[END_REF] found an activation enthalpy in good agreement with the theoretical value for intrinsic oxygen diffusion. Furthermore, the absolute values of the diffusion coefficients determined experimentally by these authors were found to be in good agreement with the theoretical values. As it provides strong experimental evidence for intrinsic diffusion, [START_REF] Yang | Intrinsic diffusion properties of an oxide: MgO[END_REF] will be considered as a benchmark at ambient conditions for the calculated high pressure, high temperature diffusion coefficients.

Intrinsic diffusion in MgO under lower mantle conditions was calculated by [START_REF] Ita | Effects of pressure on diffusion and vacancy formation in MgO from nonempirical free-energy integrations[END_REF], who used molecular dynamics simulations and found that oxygen-magnesium vacancy pairs would predominate over isolated oxygen vacancies (Schottky defects) at low temperatures, while the opposite was found for temperatures exceeding 1000 K. Nevertheless, we expect small differences between the cases where oxygen diffusion occurs through isolated oxygen vacancies or vacancy pairs, as discussed by [START_REF] Yang | Intrinsic diffusion properties of an oxide: MgO[END_REF]. In fact, values yielded by [START_REF] Ita | Effects of pressure on diffusion and vacancy formation in MgO from nonempirical free-energy integrations[END_REF] are in close agreement with the ones of [START_REF] Yang | Intrinsic diffusion properties of an oxide: MgO[END_REF].

Ita and Cohen (1997) also found that the homologous temperature relation works well to describe intrinsic diffusion in MgO:

1 45 = 1 0 * (-; * < = < ⁄ ) (4)
where T m is the melting temperature and 1 0 and g are empirical constants. This relation also describes the pressure dependence of diffusivity in crystalline metals, as [START_REF] Reaman | Interdiffusion of Earth's core materials to 65 GPa and 2200[END_REF] showed using diamond anvil cell experiments at pressures up to about 70 GPa. As shown below, the homologous temperature relation also provides a remarkably good description of intrinsic anion diffusion in ionic crystals, with anion diffusion data for a wide range of crystals having the rock salt structure all following a common homologous temperature scaling.

As 1). For a solid solution there is no single value of T m at a given pressure, and in particular there is a wide gap between the solidus and liquidus curves for (Mg,Fe)O [START_REF] Wu | Critical evaluation and optimization of the thermodynamic properties and phase diagrams of the CaO-FeO, CaO-MgO, CaO-MnO, FeO-MgO, FeO-MnO, and MgO-MnO systems[END_REF][START_REF] Zhang | Melting behavior of (Mg,Fe)O solid solutions at high pressure[END_REF]. Our estimates for the values of T m to be used in the homologous temperature relation were calculated as follows. First, we use the melting curve of MgO computed by [START_REF] Cohen | The melting curve and premelting of MgO[END_REF] up to 100 GPa, because it was determined self-consistently with the diffusion model implemented here [START_REF] Ita | Effects of pressure on diffusion and vacancy formation in MgO from nonempirical free-energy integrations[END_REF]. The MgO melting curve at higher pressures has been constrained using recent shock measurements that places a lower bound melting temperature of 9000 K at 240 GPa (Fat'yanov and Asimow, 2014). Second, we use the FeO end-member melting curve defined by the combined ex situ determinations by [START_REF] Fischer | High-pressure melting of wüstite[END_REF] to 77 GPa and Kato et al.

(2016) to 131 GPa. Third, we assume that the effective T m values for magnesiowüstite and ferropericlase vary linearly between these two end-members. This assumption is a reasonable first-order approximation, given (1) the large spread in reported high pressure melting temperatures of FeO [START_REF] Knittle | The high-pressure phase diagram of Fe 0.94 O: A possible constituent of the Earth's core[END_REF][START_REF] Shen | Melting of wüstite and iron up to pressures of 600 kbar[END_REF][START_REF] Fischer | High-pressure melting of wüstite[END_REF][START_REF] Komabayashi | Thermodynamics of melting relations in the system Fe-FeO at high pressure: Implications for oxygen in the Earth's core[END_REF][START_REF] Kato | Melting in the FeO-SiO 2 system to deep lower-mantle pressures: Implications for subducted Banded Iron Formations[END_REF] and MgO (see Fat'yanov and Asimow, 2014 for a review); (2) the large spread in the pressure dependence of the solidus and liquidus curves for the (Mg,Fe)O solid solution [START_REF] Zerr | Constraints on the melting temperature of the lower mantle from high-pressure experiments on MgO and magnesiowüstite[END_REF][START_REF] Zhang | Melting behavior of (Mg,Fe)O solid solutions at high pressure[END_REF][START_REF] Du | High-pressure melting of MgO from (Mg,Fe)O solid solutions[END_REF][START_REF] Deng | Viscosity jump in the lower mantle inferred from melting curves of ferropericlase[END_REF][START_REF] Fu | Melting behavior of the lower-mantle ferropericlase across the spin crossover: Implication for the ultra-low velocity zones at the lowermost mantle[END_REF]; and (3) the lack of empirical evidence that the solidus or liquidus temperature of a solid solution is a better approximation for the effective melting temperature used in the homologous temperature scaling than a linear interpolation of the end-member melting temperatures.

The self-diffusion coefficients to be implemented in the 2.5D-DD code for this study were evaluated starting from the results of [START_REF] Ita | Effects of pressure on diffusion and vacancy formation in MgO from nonempirical free-energy integrations[END_REF], which provide an empirical equation expressing the self-diffusion coefficients of Mg and O as a function of pressure and temperature:

?@1 23 (A, <) = ?@(C D E) + G ! HIG ! J - K ! HIL ! HI M L ! J ( 5
)
where P is pressure in GPa and T is temperature in K. The other parameters are taken from oxygen self-diffusion data since the anion diffusion represent the slowest mechanism in MgO, and are listed in Table 2.

Starting from this equation it is possible to calculate the oxygen self-diffusion coefficient as a function of temperature for a fixed pressure. This was made for pressures ranging from 100 to 140 GPa and temperatures between 1500 and 4000 K. Each curve was sampled every 100 K and the retrieved values were plotted as a function of the homologous temperature in order to obtain a dataset to be fit with the homologous temperature relation highlighted in eq. ( 4).

The obtained dataset along with the relative fit using the homologous temperature relation is showed in Figure 1, together with other experimental data on intrinsic anion selfdiffusion for oxides and halides from the literature. The agreement between the scaling law adopted in this study and the ambient condition points is remarkable, and indicates that ionic crystals with the rock-salt structure all follow a common homologous temperature relationship with respect to intrinsic diffusion on the anion sublattice. The retrieved values for g and 1 0 are 21.5 and 7.2•10 -4 m 2 /s, respectively.

Substituting T m of MgO with values for (Mg 0.80 Fe 0.20 )O and (Mg 0.16 Fe 0.84 )O at 120 and 135 GPa allows to calculate the intrinsic oxygen self-diffusion coefficient at a given temperature for the two different compositions. The diffusion coefficients are then evaluated in terms of the more canonical Arrhenius type of relation described in eq (3). Thus, the self-diffusion activation enthalpy is equal to:

89 23 = ; * < = * N (6)
while the pre-exponential factor 1 0 is the same for eq. ( 3) and ( 4).

The values obtained from this approach and employed in the present study are shown in Table 3.

Dislocation dynamics creep simulations

Results

The strain evolution as a function of time for magnesiowüstite at the three tested PT conditions is shown in Figure 2 for all of the considered applied stresses. All the curves correspond to a linear behavior that allows for the determination of the creep strain rate as a function of stress. To further verify that the steady state condition is achieved, we plot the dislocation density as a function of time. This is done in Figure 3a where the results obtained at 135 GPa and 3800 K are shown. The dislocation densities oscillate around a constant value, demonstrating that dislocation annihilation and multiplication are counterbalancing each other.

This means that the coupling of the two mechanisms introduced in the code -i.e. dislocation glide and climb -leads to a steady state equilibrium condition and therefore reproduces well the process of creep. Figure 3b displays the individual contributions of these two mechanisms to the resulting creep strain rate: it is evident that the vast majority of deformation is due to dislocation glide (ε glide ), which is a very efficient strain-producing mechanism. Climb, on the other hand, is diffusion-dependent, and therefore has a major control on the evolution of strain with time -i.e.

it defines the time scale at which the deformation is taking place. In Figure 2a,b,c this is evident from the different orders of magnitude on the time axes. Given that in this regime dislocation glide is athermal and therefore is not affected by temperature variations, a comparable deformation of 1-2% is achieved for every PT condition. On the other hand, the different temperatures affect the diffusion coefficient, on which the climb velocity is directly dependent (cf. eq. 2), changing the time control of the dynamics and therefore the rate of deformation.

The results found in this study confirm that this kind of interplay works well to describe creep of (Mg,Fe)O after other studies had confirmed its validity by applying the same method at ambient conditions to olivine [START_REF] Boioli | Modeling the creep properties of olivine by 2.5-D dislocation dynamics simulations[END_REF] and MgO [START_REF] Reali | Modeling plasticity of MgO by 2.5D dislocation dynamics simulations[END_REF].

The results for strain as a function of time for ferropericlase (Mg 0.8 Fe 0.2 )O are shown in Figure 4. As stated above, in this case several tests were run at higher applied stresses -up to 500 MPa -in order to obtain reliable deformations in practical simulation times. Nonetheless, a steady-state condition at the usual stresses from 10 to 50 MPa was verified also for each tested

PT condition for ferropericlase.

It is therefore possible to evaluate the creep strain rates from the obtained curves and characterize them using the well established power law type of equation:

OP = O 0 P Q R S - T (7)
where O 0 P is a fitting parameter, σ the applied stress, µ the shear modulus, @ the power law stress exponent and U the creep activation enthalpy.

The strain rates as a function of stress for magnesiowüstite and ferropericlase are shown in Figure 5. The retrieved average stress exponents are 1.6 for magnesiowüstite and 2.8 for ferropericlase. Regarding the evaluation of the creep activation enthalpy U only a first attempt could be done for the case of magnesiowüstite at 135 GPa. This is the only case, in fact, where the strain rates could be compared at the same pressure and different temperatures. The obtained average value for U in this case was 8.0 eV, in fair agreement with the oxygen self-diffusion activation enthalpy 89 23 previously evaluated for this case (that is, 8.2 eV, cf. Table 3). A similar result was also found by [START_REF] Reali | Modeling plasticity of MgO by 2.5D dislocation dynamics simulations[END_REF] for MgO. Therefore, this finding further confirms how dislocation climb, which is diffusion-dependent, is the mechanism controlling the rate of deformation and therefore creep.

20

Knowing the strain rates it is possible to deduce the viscosity of (Mg,Fe)O. This is done by considering the effective viscosity η:

V = Q WP (8)
The obtained viscosities for magnesiowüstite and ferropericlase at CMB conditions are shown in Figure 6.

Discussion of uncertainties

The major result is the prediction that magnesiowüstite has a viscosity that is reduced by about 10 orders of magnitude relative to ferropericlase at CMB conditions. In support of this general conclusion, we consider some of the uncertainties involved in the methods presented here. One source of uncertainty is the estimate for T m . A recent study on recovered samples from laserheated diamond-anvil cell experiments up to 80 GPa suggests significantly low liquidus and solidus temperatures for ferropericlase [START_REF] Deng | Viscosity jump in the lower mantle inferred from melting curves of ferropericlase[END_REF]. If this phenomenon occurs at CMB conditions, this could explain why at upper D″ conditions simulated here (120 GPa and 2800 K), ferropericlase exhibits smaller strain rates (10 -19 s -1 , resulting in higher viscosities) than the expected values for this region (i.e. 10 -16 -10 -14 s -1 ), although the value we obtain at lowermost mantle conditions of 135 GPa and 3800 K is comparable (10 -14 s -1 ).

Other sources of uncertainty are the elastic properties of (Mg,Fe)O at CMB conditions. In the case of magnesiowüstite, we investigate the influence of increasing the shear modulus by 10% on our results at 135 GPa. By evaluating a stiffer material, we indirectly address two effects: 1) a relatively cooler CMB, 2) uncertainty of the T m value for Mw. By simulating a stiffer material (cooler CMB), for example, the conditions remain below the solidus of Mw. We find smaller strain rates for the stiffer Mw, varying rather uniformly between 20-30%, thus of the same order of magnitude. These results are shown in Figure 7 and the numerical values of the different strain rates in Table 4.

It is becoming more likely that some ULVZs may represent chemically dense solid-state assemblages and others may indicate the presence of partial melts (e.g. [START_REF] Hernlund | Some dynamical consequences of partial melting in Earth's deep mantle[END_REF][START_REF] Hernlund | Dynamics and structure of a stirred partially molten ultralow-velocity zone[END_REF][START_REF] Li | Compositionally-distinct ultra-low velocity zones on Earth's core-mantle boundary[END_REF]. The stable phase assemblage in any particular region is dictated, in part, by phase equilibria and the relationship of the assemblage's solidus to the thermal profile, all of which have relatively large uncertainties in the D″ region (e.g. [START_REF] Lay | Core-mantle boundary heat flow[END_REF][START_REF] Zhang | Melting behavior of (Mg,Fe)O solid solutions at high pressure[END_REF][START_REF] Fiquet | Melting of peridotite to 140 Gigapascals[END_REF][START_REF] Fischer | High-pressure melting of wüstite[END_REF][START_REF] Andrault | Melting of subducted basalt at the core-mantle boundary[END_REF][START_REF] Du | High-pressure melting of MgO from (Mg,Fe)O solid solutions[END_REF][START_REF] Komabayashi | Thermodynamics of melting relations in the system Fe-FeO at high pressure: Implications for oxygen in the Earth's core[END_REF][START_REF] Nomura | Low Core-Mantle Boundary Temperature Inferred from the Solidus of Pyrolite[END_REF][START_REF] Tateno | Melting experiments on peridotite to lowermost mantle conditions[END_REF][START_REF] Pradhan | Melting of MORB at core-mantle boundary[END_REF][START_REF] Zhang | Temperature of Earth's core constrained from melting of Fe and Fe 0.9 Ni 0.1 at high pressures[END_REF][START_REF] Deng | Viscosity jump in the lower mantle inferred from melting curves of ferropericlase[END_REF]. Combined with previous reports on the sound velocities of magnesiowüstite that implicate its presence in ULVZs [START_REF] Wicks | Sound velocity and density of magnesiowüstites: Implications for ultralow-velocity zone topography[END_REF], the presence of such a low viscosity solid phase in localized patches above the CMB should influence the dynamics of the region.

Implications

The major result of our DD calculations and diffusion estimates is the prediction that iron-rich (Mg,Fe)O (magnesiowüstite) has a very low viscosity relative to iron-poor (Mg,Fe)O

(ferropericlase) at CMB conditions (Figure 6). Therefore, to assess the influence of low viscosity magnesiowüstite on the dynamics of the lowermost mantle we revisit the proposal that the sound velocities of an assemblage containing magnesiowüstite can explain a suite of seismic observations [START_REF] Wicks | Very low sound velocities in iron-rich (Mg,Fe)O: Implications for the core-mantle boundary region[END_REF]2017) and topography [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF] of ULVZs.

Geodynamic model

The geodynamic model is described in detail in [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF]. We use CitcomS [START_REF] Moresi | Citcoms v3.3.1 [software[END_REF] to solve the equations of mass, momentum, and energy conservation in cylindrical geometry for a small region near the CMB (23 degrees by 488 km). The model includes a chemically-distinct material that has a larger intrinsic density than ambient mantle to model the iron-rich assemblage that forms the ULVZ. The larger intrinsic density is defined by a fraction of magnesiowüstite coexisting with the silicate mantle which can explain seismic wave speed reductions [START_REF] Wicks | Sound velocity and density of magnesiowüstites: Implications for ultralow-velocity zone topography[END_REF]. We now account for the viscosity reduction of Mw-bearing assemblages by reducing the intrinsic viscosity of the chemically-distinct material (ULVZ)

relative to ambient mantle. The viscosity formulation therefore includes a compositional prefactor η C in comparison to eq. 8 in [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF]:

V(X, <) = V Y Z-?@(10 ] )< ^_ ( 9
)
where η is viscosity, η C compositional prefactor, q order of magnitude viscosity contrast due to temperature (relates to activation energy), and Tη the non-dimensional temperature.

We select η C = 1 for ambient mantle (C = 0) and η C = 0.1 for ULVZ material (C = 1), which ensures a discernible effect of composition versus temperature on the flow pattern and hence the resulting geometry of the ULVZ. Since the ULVZ is not composed exclusively of (Mg 0.16 ,Fe 0.84 )O [START_REF] Wicks | Very low sound velocities in iron-rich (Mg,Fe)O: Implications for the core-mantle boundary region[END_REF][START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF]2011;[START_REF] Wicks | Sound velocity and density of magnesiowüstites: Implications for ultralow-velocity zone topography[END_REF], it would be both physically unreasonable and numerically intractable to prescribe the total 10 orders of magnitude viscosity contrast between magnesiowüstite (component of ULVZ material) and ferropericlase (component of ambient mantle) (Figure 6). Rather, the compositional prefactor extends the total viscosity range of the model to 4 orders of magnitude since q = 3. Other model parameters are the same as [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF] to enable a direct comparison of results.

Our new set of calculations prescribe an initial chemical layer thickness d ch =4 km at the base of the model, which was previously found to produce ULVZs that have reliefs within the mid-range of seismic predictions (Fig. 8C, 9C in [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF]. We apply an isothermal and free-slip top boundary condition (referred to as "impermeable cases" in [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF].

The buoyancy number B is non-dimensional and quantifies the intrinsic density contrast of the chemically-distinct material (ULVZ) relative to ambient mantle (i.e., Δρ cd ρ ⁄ ). We choose B = 0.75, 1, 1.25, 2, 4, and 6 (Table 5).

Morphology of ULVZs with low viscosity magnesiowüstite

We quantify the morphology of the ULVZ at steady-state by the relief H, defined as the 95% percentile of its height above the CMB, and half-width W ½ , defined as half of its maximum (angular) footprint on the CMB. Table 5 shows the results for cases computed with (this study) and without [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF] an intrinsic viscosity reduction for ULVZ material.

An intrinsic viscosity reduction produces flatter ULVZs than predicted in models that do not consider the rheology of iron-rich (Mg,Fe)O; heights decrease by 10-20% and half-widths increase by 10-18% which results in aspect ratios decreasing by 14-26% (Table 5). This is because the intrinsic reduction in the viscosity of the ULVZ further decouples the ULVZ from the exterior flow in addition to the temperature-dependence of viscosity (Figure 8). The tractions on the boundaries of the ULVZ are reduced, which means the exterior flow is less dominant in supporting the intrinsically dense ULVZ above the CMB. Rather, the ULVZ slumps to a greater extent due to its intrinsic density contrast.

Hence, an implication of comparatively weak magnesiowüstite is that less iron-rich (Mg,Fe)O needs to be incorporated into the ULVZ assemblage to account for the relief of a particular ULVZ. This is because both reducing the viscosity of the ULVZ material and increasing its intrinsic density contrast results in a flatter ULVZ that is characterized by both reduced height and greater width. For example, the B = 1 scenario (Δρ cd ρ ⁄ = 1.5%) corresponds to the inclusion of around 10 to 15 vol% magnesiowüstite into the mantle assemblage at CMB conditions, with the range depending upon the actual phase assemblage and iron partitioning (e.g., [START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF][START_REF] Wicks | Sound velocity and density of magnesiowüstites: Implications for ultralow-velocity zone topography[END_REF]. Alternatively, another class of lowermost mantle seismic features could be explained by the presence of low viscosity magnesiowüstite. Kanamori (1976) and [START_REF] Buchbinder | Problems related to PcP and the core-mantle boundary illustrated by two nuclear events[END_REF] suggest that a thin core-mantle transition zone (CMTZ) could produce additional waveform complexity rather than a discontinuity, and further seismic studies indicate that a CMTZ of less than 4km thick could be an alternative explanation for some ULVZs (e.g., Vidale and Benz 1992).

Conclusions

Dislocation dynamics simulations have been used to explore the dislocation creep behavior of (Mg,Fe)O ferropericlase and magnesiowüstite at D″ conditions. Upon retrieving the creep strain rates at different temperatures, pressures and applied stresses, we find that the viscosity of magnesiowüstite is reduced by about 10 orders of magnitude at the core-mantle boundary. By exploring the effects of an estimated viscosity contrast of a magnesiowüstite-bearing layer at the CMB, lower concentrations of magnesiowüstite would be required to explain the topographic relief reported in some ULVZ locations. However, thinner seismic features, such as the core-mantle transition zone, could host magnesiowüstite. In either of these cases, the recent findings that magnesiowüstite may have twice the shear wave anisotropy of postperovskite [START_REF] Finkelstein | Strongly anisotropic magnesiowüstite in Earth's lower mantle[END_REF] suggests that the presence of only small amounts of magnesiowüstite may significantly alter the characteristics of the core-mantle boundary. [START_REF] Ita | Effects of pressure on diffusion and vacancy formation in MgO from nonempirical free-energy integrations[END_REF] (red dots) are fit using the homologous temperature relation described in eq. (4) (black line), which is in good agreement also with other intrinsic, oxygen self-diffusion experimental data of oxides and halides retrieved from literature. F66: [START_REF] Fuller | Diffusion of the chlorine ion in potassium chloride[END_REF], B65: [START_REF] Barr | Anion diffusion in crystals of NaCl[END_REF], CJ69: [START_REF] Chen | Oxygen self-diffusion in undoped and doped cobaltous oxide[END_REF], YS82: [START_REF] Yamaguchi | The tracer diffusivity of oxygen in Wüstite and Cobaltous oxide[END_REF], YF94: [START_REF] Yang | Intrinsic diffusion properties of an oxide: MgO[END_REF], IC97: [START_REF] Ita | Effects of pressure on diffusion and vacancy formation in MgO from nonempirical free-energy integrations[END_REF].

Figure captions Figure 1 .

 captions1 Figure captionsFigure 1. Oxygen self-diffusion coefficients 1 45 as a function of the homologous temperature for different oxides and halides. The high pressure, high temperature numerical data

Figure 2 .

 2 Figure 2. (Mg 0.16 Fe 0.84 )O magnesiowüstite: plastic strain as a function of time at 2800 K

Figure 3 .

 3 Figure 3. (Mg 0.16 Fe 0.84 )O magnesiowüstite: evolution of the dislocation density as a

Figure 4 .

 4 Figure 4. (Mg 0.80 Fe 0.20 )O ferropericlase: plastic strain as a function of time at 2800 K and

Figure 5 .

 5 Figure 5. Creep strain rates as a function of stress (over the shear modulus) of

Figure 6 .

 6 Figure 6. Viscosities as a function of stress (over shear modulus) for (Mg 0.16 Fe 0.84 )O

Figure 7 .

 7 Figure 7. Strain as a function of time for (Mg 0.16 Fe 0.84 )O magnesiowüstite at 135 GPa

Figure 8 .

 8 Figure 8. Case 2 (B = 1) shows that a ULVZ containing iron-rich (Mg,Fe)O has lower

Fig

  Fig. 1.

  noted above, aliovalent impurities, including trace levels of iron, have no influence on oxygen diffusion in MgO. However, iron in (Mg,Fe)O has a strong influence on the bond strength, influencing dramatically the melting temperature and, according to the homologous

	temperature relation, the oxygen diffusion coefficient. Here, we use the homologous temperature
	relation, fit to anion diffusion data for MgO and other rock-salt structured ionic crystals, to
	calculate oxygen diffusion coefficients in (Mg,Fe)O as a function of composition, pressure and
	temperature. To use this relation for (Mg,Fe)O, it is necessary to estimate reasonable values of
	T

m for magnesiowüstite and ferropericlase (cf. Table

Table captions Table 1 .

 captions1 Lattice constant, ½<110> Burgers vector magnitude, elastic properties and estimated liquidus temperatures for (Mg 0.16 Fe 0.84 )O magnesiowüstite and (Mg 0.8 Fe 0.2 )O ferropericlase at the pressure and temperature conditions of interest. a Wicks et al. (2015, 2017), b Wu and Wentzcovitch (2014) extrapolated to 135 GPa, c Values are taken as a linear interpolation between the melting points of MgO and FeO[START_REF] Cohen | The melting curve and premelting of MgO[END_REF][START_REF] Fat'yanov | MgO melting curve constraints from shock temperature and rarefaction overtake measurements in samples preheated to 2300[END_REF][START_REF] Fischer | High-pressure melting of wüstite[END_REF]. See text for details.

Table 2 .

 2 Oxygen self-diffusion parameters for MgO global fit as a function of pressure and temperature (from Ita and Cohen 1997).

Table 3 .

 3 Numerical values for 1 45 , 89 23 and T m employed in this study for (Mg,Fe)O.

Table 4 .

 4 Strain rates of magnesiowüstite at 135 GPa and at 3200 K and 3800 K for different applied stresses σ. The results obtained by raising the shear modulus µ of 10% are also shown, together with the calculated difference with respect to the original ones. a In this case µ is equal to 100 and 96 GPa for the cases at 3200 and 3800 K, respectively (cf. Table1), while b the values of µ here are raised by 10% (resulting in 110 and 105.6 GPa, respectively).

Table 5 .

 5 Relief H (km), half-width W ½ (km), and aspect ratio (A = H / W ½ ) for cases with (this study) and without[START_REF] Bower | A geodynamic and mineral physics model of a solid-state ultra-low velocity zone[END_REF] an intrinsic viscosity reduction for ULVZ material. Differences (%) are computed for the case with the ULVZ viscosity reduction relative to the case without the reduction. B is buoyancy number and the initial chemical layer thickness for all cases is d ch = 4 km.

				Tab. 1.
			(Mg 0.16 Fe 0.84 )O a		(Mg 0.8 Fe 0.2 )O b
			120 GPa,	135 GPa,	135 GPa,	120 GPa,	135 GPa,
			2800 K	3200 K		3800 K	2800 K	3800 K
	Lattice constant a (Å)	3.83	3.81		3.82	3.86	3.87
	Burgers vector b (Å)	2.71	2.69		2.70	2.73	2.74
	Density (g/cm 3 )		7.90	8.03		7.97	5.38	5.36
	V S (km/s)		3.50	3.52		3.47	7.16	7.09
	V P (km/s)		8.20	8.19		8.14	13.58	13.79
	Poisson ratio		0.39	0.39		0.39	0.31	0.32
	Shear Modulus µ (GPa)	97	100		96	276	270
	Bulk Modulus (GPa)	402	406		401	625	660
	Liquidus (K) c		4300	4440		4440	7220	7410
				Tab. 2.
		Equation parameter	Symbol	Value
	cubic cell parameter attempt frequency activation entropy (P=0) i 0 pressure derivative activation energy activation volume (P=0) l 0 pressure derivative		C E i 0 j i 0 k 0 l 0 l 0 j	4 Å 5.2 THz 4N 0.02N /GPa 9.4•10 -19 J 16.7 Å 3 -0.038 Å 3 /GPa
				Tab. 3.
			(Mg 0.16 Fe 0.84 )O			(Mg 0.8 Fe 0.2 )O
		120 GPa,	135 GPa	135 GPa,	120 GPa,	135 GPa,
		2800 K	3200 K	3800 K	2800 K	3800 K
	1 45 (m 2 /s) 89 23 (eV)	3.4•10 -18 8.0	8.2•10 -17 8.2	9.1•10 -15 8.2	6.2•10 -28 13.4	4.6•10 -22 13.7
	T m (K)	4300	4440		4440	7224	7414
	T / T m	0.7	0.7		0.9	0.4	0.5

Table 5

 5 and text). The ULVZ is bounded at the CMB by the black dashed line.

Acknowledgment