The role of iron on the solid solution of CaSiO3 and MgSiO3 perovskites in the lower mantle

The solid solution of CaSiO3 and MgSiO3 perovskites is an important control on the properties of the lower mantle but the effect of one of the largest defective elements (iron) on this solution is largely unknown. Using density functional theory (DFT), ferrous iron's influence on the reciprocal solubility of MgSiO3 and CaSiO3 perovskite (forming a single mixed perovskite phase) was calculated under pressures and temperatures of 25 125 GPa and 0 3000 K, respectively. Except in iron-rich conditions, iron preferentially partitions into the mixed perovskite phase over bridgmanite. This is a small effect (KD~0.25-1), however, when compared to the partitioning of ferrous iron to ferropericlase which rules out phase mixing as a mechanism for creating iron-rich regions. Iron increases the miscibility of the two perovskite phases/reduces the temperature at which the two phases mix but this effect is highly nonlinear. We find that for pyrolytic mantle (Ca%=12.5 where Ca%=Ca/ (Ca + Mg)) a perovskite iron concentration of ~13% leads to the lowest mixing temperature/highest miscibility. With this composition, 1% iron in the pyrolytic solution would lead to mixing at ~120 GPa along the geothermal gradient and 6.25% iron leads to mixing at ~115 GPa. At high iron concentrations Fe starts to impair miscibility, with 25% iron leading to mixing at ~120 GPa. Thus, in normal pyrolytic mantle iron should induce a small amount of mixing near the D'' layer. Extremely iron rich parts of the lower mantle such as potentially at the CMB and in ULVZs are, however, not a likely source of phase mixed perovskites due to the nonlinear effect of ferrous iron on phase mixing.

As the lower mantle is more Mg rich than Ca rich, we shall focus on the solubility of Ca in Mg-pv. Experimentally Irifune et al. (1989) found that the solid solubility of Ca in Mg-pv was limited to 2% or even lower at 25 GPa and ~1800 K. Armstrong et al. (2012) found that at 2000 K, the solubility of Ca in Mg-pv is less than 5 mol % at 30 GPa and at least 10 mol % at 55 GPa. With increasing pressure, the solubility thus increases. Theoretically Jung and Schmidt (2010) found that the solubility of Ca in Mg-pv was 0.5% at 25 GPa and 2000 K, but 5% at 25 GPa and 3000 K and Vitos et al. (2006) claimed that under the temperature and pressure conditions of the upper mantle and transition zone, the solid solubility of Ca in Mgpv is 4-6%. Jung and Schmidt (2010) and Vitos et al. (2006) found that pressure decreased the solid solubility of Ca in Mg-pv in contrast with experimental findings (Armstrong et al. 2012). This was rectified by Muir and Zhang (2020) who found that the explicit inclusion of vibrational entropy caused pressure to increase miscibility. In this study they found that above the D'' pyrolytic compositions of Mg-pv and Ca-pv will not mix. Thus, all studies agree that pure Ca-pv and Mg-pv will not mix in pyrolytic compositions in lower mantle compositions except possibly in deep hot parts of the lower mantle.
Defect elements, such as Fe, Al, and Ti may, however, play important roles in the mixing of these phases. Armstrong et al. (2012) have clearly pointed out that the addition of titanium will enlarge the single phase domain of MgSiO3-CaSiO3. Muir and Zhang (2020) established a simple conceptual model to explore the influence of different impurities on the miscibility of Ca-pv and Mg-pv and found that large amounts of any defect element (>~1%) were required to significantly affect miscibility. Iron is the most prominent defect element in pyrolytic compositions and thus is a key candidate for potentially changing the dynamics of this phase mixing. In the lower mantle iron concentration in bridgmanite is around ~10% (Lin et al. 2018), and the iron concentration may be even higher near the Core Mantle Boundary (CMB). Fujino et al. (2004) explored the effect of iron on the perovskite two-phase mixing using laser heated DAC and TEM. They found that the solid solubility of Mg in Ca-pv was 4% at ~30 GPa and 1930 K but increased to ~18% at 30 GPa and 1800 K when around 9% iron was added which shows clearly that iron promotes miscibility at least to some degree. There was not enough P, T and Fe% points in that study, however, to explore systematically the effect of iron on the miscibility of the two perovskites. In this study, therefore, we conducted theoretical calculations to examine the effect of iron concentration, pressure (P) and temperature (T) on the mutual solubility of Ca-pv and Mg-pv in the conditions of the lower mantle.
A unit cell of 40 atoms was used in static calculations and one with 80 atoms (2x2x1) in Molecular dynamics (MD) simulations. In our calculations, MgxFe1-xSiO3 and Mg1-x-yCayFexSiO3 were modelled with an orthorhombic (pbnm) unit cell, for Mg1-x-yCayFexSiO3 i4mcm and pm3m structures were found to be less stable at all P and T conditions. For CaSiO3 both i4mcm and pm3m phases were modelled with pm3m favoured at high temperatures. Calculations were done at 25,75 and 125 GPa (all pressures are uncorrected) and at 0, 1000, 2000 and 3000 K. Static simulations (~0 K) were calculated with a (3x5x5) kpoint mesh, and molecular dynamics runs were conducted at the gamma point only. Both static and MD calculations had an energy cutoff of 500 eV and were converged to within 10 −5 eV.

Mixing Thermodynamics
In order to find the solubility of CaSiO3 and (Mg, Fe) SiO3, we examined the following reaction: The iron is placed in Mg-pv before mixing is simulated. Although some iron must thermodynamically enter into Ca-pv as shown in the text this amount is generally negligible and can be ignored. To determine mixing we calculate the free energy of Reaction 1 using = − Equation 1 where = 0 represents the mixing boundary and mixing occurs when it is negative. TMix is the mixing temperature defined as the T which makes = 0 (ie. the solvus temperature).
is the mixing enthalpy. Determining SMix (the mixing entropy) is complex; in our case we have defined it as the sum of two parts-configurational and vibrational entropies. where is Boltzmann's constant, T is temperature and Ei is the relative energy of each atom unit cells when these atoms are confined to relaxed A lattice sites. In addition, we calculate the "perfect" entropy which is the entropy if all arrangements had the same energythe Boltzmann entropy. This is done via Sconfig = lnZ where Z is the total number of arrangements of all atoms as outlined above. We found that the difference between the two calculation methods is very small as outlined in the paper. Therefore, our configuration entropy for mixing in this study is calculated by the formula Sconfig = lnZ.
For vibrational entropy (Svib), we obtain the velocity autocorrelation function through molecular dynamics calculations. The vibrational entropy is then determined by: where N is the number of atoms in the system, ma is the mass of atom a, is the spectral density of atom a in the direction k (x=1, y=2, z=3) and V is the velocity.

Compositions
In this work, we examine how varying the Ca and the Fe content of bridgmanite and Calcium silicate perovskite (Ca-pv) mixtures varies their solubilities. We shall thus define two terms Ca% which is Ca / (Mg + Ca + Fe) x 100%, and Fe%, which is Fe / (Mg + Ca + Fe)

Fe Partitioning
It is important to know into which of the three phases, Ca-pv, Mg-pv and the mixed phase, that iron likes to partition. The partitioning coefficient of Fe between MgSiO3 and CaSiO3 is defined as ( 1 = − ∕ − ) (Table 1), and between Mg1-xCaxSiO3 and MgSiO3 is defined as Table 2). Between Mg-pv and Capv we find that Fe has strongly preferential partitioning into MgSiO3. Between Mg-pv and the mixed phase we find that except for very large amounts of iron at low pressures (where mixing is not expected to occur), iron always favors the mixed phase over Mg-pv and this preference increases with pressure. This preference is not particularly strong with all KD values being above 0.25 except with extreme iron contents or in a basaltic mixture. The amount of iron in the system has an effect on the partitioning but it this small as shown in Figure 1. In no cases at high pressure (where mixing is likely to occur) does iron favor bridgmanite over the mixed phase.
In the lower mantle a third phase is present, that of ferropericlase. Iron could potentially partition from bridgmanite either to ferropericlase or to the mixed phase. Muir and Brodholt (2016) claimed that there is a strong partitioning of iron from bridgmanite to ferropericlase with a KD of ~0.32 at 30 GPa dropping to ~0.06 at 120 GPa at 2000 K and dropping further with increased temperature. This is a much stronger preference of iron into ferropericlase than is seen with the mixed phase and so the preference of ferrous iron at deep mantle pressures and temperatures where mixing occurs (see Figure 5) is ferropericlase > the mixed phase > bridgmanite in that order. Thus, the formation of a perovskite mixed phase does not outcompete ferropericlase as an iron-sink and should not substantially alter the distribution of iron in the deep lower mantle. Our concentrations of iron in the rest of the paper shall refer to the concentration of ferrous iron in the perovskite phases (bridgmanite and the mixed phase).
In a real lower mantle with ferropericlase this concentration will be lower than the concentration of iron in the system and could be up to 20 times lower as we approach the CMB (Muir and Brodholt (2016)).

HMix
We determined the effect of iron on the mixing enthalpy (HMix) using static DFT calculations as shown in Figure 2. The mixing enthalpy has a nonlinear trend with iron content. HMix is always positive showing that Ca-pv and Mg-pv are naturally immiscible and temperature is required to mix them. With increasing pressure HMix increases which will lead to less mixing. Iron in general decreases HMix and thus promotes phase mixing. Initially with an increasing concentration of iron, HMix decreases but then after a point HMix increases with increasing Fe concentration. The reason for this nonlinearity can be seen in Figure S1,

SConfig
We examined the effect of different iron arrangements, as explained in the method, to estimate the effect of iron on configurational entropy (SConfig). In a perfect system (where all arrangements of atoms are energetically equivalent) the presence of iron does not cause an increase to SConfig. This is because iron exists on the A site where mixing between Mg and Ca occurs in the perfect system and because iron is primarily partitioned to a single phase before mixing. As shown in Table S1, the configurational entropy of iron, Mg and Ca is near the perfect Boltzmann entropy limit. At all conditions, the difference between a perfect configurational entropy and our actual configurational entropy is < ~ 4 meV/atom. This is a very small energy term and is much smaller than the HMix term. This suggests that all arrangements Mg, Fe and Ca on the A sites in both Mg-pv and the mixed phase are effectively equivalent and that iron does not cause large structural rearrangements in the Mgpv or the mixed phase. Considering a system with 12.5% Fe and 12.5% Ca changing between perfect and non-perfect entropy changes the mixing temperature by ~50 K. Thus, the primary effect on iron on mixing should be enthalpic and effect of iron on Sconfig can be largely ignored.

SVib
Vibrational entropy (SVib) depends on long-range phonons and is likely unaffected by small additions but could be strongly affected by defects present in large quantities like Fe.
SVib is essential to calculating mixing parameters of these systems (Muir and Zhang 2020) but ΔSVib-Fe (the change caused to SVib by iron) is not. Iron makes no large structural rearrangements to the system as indicated by Sconfig and thus likely also has small effects on long range entropy. As shown in Table 3, the change in the vibrational entropy term from replacing a Mg atom with an iron atom is extremely small particularly at high pressures where mixing occurs. Therefore, in this work we shall include SVib but ignore the effects of ΔSvib-Fe.

Mixing
In Figure 3, we plot the Tmix of a 1:7 (pyrolytic) mixture of Ca-pv and Bridgmanite as a function of iron at pressures of 25, 75 and 125 GPa. When iron is added into the system, initially the mixing temperature decreases by about 80 K per 1% Fe. This Tmix decrease in a pyrolytic mixture is not universal with concentration, however. At ~13% iron we see a maximum decrease in Tmix and beyond this increasing the iron concentration causes Tmix to rise. As shown in Figure 2 Fe decreases in HMix are largest at ~13% and increase on either side and these nonlinear Hmix effects are the origin of this non-linear Tmix behaviour. This means that while a small amount of iron (6.25%) causes a Tmix decrease of 500 K a large amount of iron (25%) causes a decrease in Tmix of less than 100 K.
Unlike the behavior of the pyrolytic mixture, in a 1:1 mixture of Ca-pv and Mg-pv (Ca%=50, basaltic) Tmix decreases with iron concentration continually up until 25%. A likely explanation for these differing behaviors of pyrolytic and basaltic compositions is due to iron partitioning preferences. The iron partition coefficient ( 1 ) between CaSiO3 and MgSiO3 is very large, with strong partitioning of Fe into the Mg end-member. The iron partition coefficient ( 2 ) between bridgmanite and the mixed phase is smaller and closer to 1, particularly at lower pressures. Thus, there is a strong dislike of Fe going into Ca sites rather than Mg sites which makes sense as Fe 2+ is much closer in size to Mg 2+ than Ca 2+ (~78/72/100 pm respectively). At low iron concentrations iron partitions into the mixed phase and thus reduces the mixing temperature. As iron concentration increases relative to Ca concentration increasingly iron will partition into the Mg-end member and not take part in the miscibility process. Thus, it will cost energy to put the surplus iron back into the mixture (Tmix increase).
When Ca% is 12.5% (pyrolytic mixture), the iron-driven two-phase solid solution Tmix minimum is around 13% Fe, when Ca% is 50% (basaltic mixture) this Tmix minimum increases to ~30% Fe which has not been plotted in the Figure 3.
In the deep lower mantle, the CaSiO3 and MgSiO3 two-phase mixing temperature at 125 GPa is about 2550 K in the Fe free system for both a pyrolytic mixture (Ca%=12.5) and a basaltic mixture (Ca%=50) (Figure 4). With the introduction of 6.25%/12.5% iron the mixture temperatures reduce to ~2280 K/~2180 K for the pyrolytic mixture and ~2290 K/2040 K for the basaltic mixture. With a large amount of iron (25% Fe) the mixing temperature is very different in pyrolytic (~2520 K) and basaltic (~1850 K) mixtures for the reasons discussed above. Direct comparison with Fujino et al. (2004) is difficult due to the high levels of Fe involved in that study (with Fe nearly equivalent to Mg) and its focus on the Ca-pv side of the solubility diagram. As demonstrated above the behavior of high iron concentrations is not easy to predict and the concentrations of iron in that study are beyond the scope of our calculations. Regardless our study is consistent with the findings of Fujino et al. (2004) in that iron can promote the miscibility of two phases. Figure 5 presents the mixing temperature of a pyrolytic mixture (Ca%=12.5) as a function of depth and iron concentration. Regardless of the amount of iron no mixing is seen at the top of lower mantle. With the amount of iron that causes the maximum mixing (~13%) perovskite phase mixing can be seen to occur at ~70 GPa in the hottest parts of the mantle. As the iron concentration increases or decreases or the mantle cools then mixing occurs at deeper parts of the mantle. 13% would be an extremely iron rich part of the mantle, especially when considering the effect of ferropericlase, and more reasonable iron concentrations lie between 0-6.25%. With these iron concentrations and the "standard" geotherm mixing is found to occur between 115-125 GPa depending upon the iron concentration. Thus, in pyrolytic mantle iron causes only very small changes to the depth at which phase mixing occurs and is not a large control on this process. Similar observations can be made with the basaltic phase ( Figure S2) but in this case the higher Ca% means no phase mixing is observed under any conditions. As shown in Figure 4 the Ca% of the pyrolytic mixture has little effect on the mixing temperature across the observed range of pyrolite and at most causes Tmix by vary by ~50 K.
Iron-rich regions present an interesting case. Various regions, particularly those near the CMB, have been speculated to be iron rich and these could present quite different behaviour from the rest of the mantle. We find however that large amounts of iron decrease the stability of the mixed phase and do not promote its formation. Thus, these regions would not have different phase characteristics from the rest of the more iron-poor mantle. The potential formation of a mixed phase also does not provide a mechanism for forming iron rich regions.
Strong partitioning of iron from bridgmanite to the mixed phase could provide a physical mechanism for dynamically separating iron from the overall mantle across the physical barrier of a phase transition but the partitioning coefficient is small (Table 2) and moves closer to 1 with increasing iron (Figure 1).
So far, we have considered only univariant mixing but the introduction of iron will lead to a phase loop. Calculating the exact dimensions of this phase loop is challenging. The similar trends of formation enthalpies vs Fe% for Mg-pv and the mixed phase ( Figure S1), and their highly variable relationship to each other means that constraining the phase loop requires very high accuracy in both the number of Fe concentrations that are measured and the precision of those points. This is potentially important future work but the phase loop is unlikely to be large or important simply due to the partitioning of Fe between Mg-pv and the mixed phase being near 1. A wide phase loop would require strong partitioning of the iron to either Mg-pv or the mixed phase. Using a common tangent method, we calculate that at 125 GPa the width of the phase loop is < 200 K at 6.25% Fe and < 50 K at 12.5% Fe (both with Al could also affect mixing dynamics. The effect of these elements was studied with a simple enthalpic model in Muir and Zhang (2020) where it was concluded that Al raises the mixing temperature and Fe 3+ -Al 3+ causes the mixing temperature to remain largely static. While this was a simplistic model it correctly predicts the trends seen for Fe 2+ at low Fe concentrations.
It did not predict the high concentration effects of Fe 2+ on mixing seen in this study because it assumed a linear effect of Fe on enthalpy which is not the case (Figure 2). Thus, this model likely captures the broad trend of Ferric iron and aluminum in that they have little effect or slightly raise the mixing temperature. Thus, these elements further reduce the miscibility of Ca-pv and Mg-pv and adding in Fe-Al or Al to our model likely will not lead to phase mixing.
Therefore, our predictions here represent the most favorable conditions for mixing in the lower mantle and adding in Al and/or converting iron to Ferric iron will not lead to increased mixing.

Conclusion
The effect of iron on Ca-pv and Mg-pv perovskite phase mixing also been investigated in this study. Iron reduces the mixing temperature/increases miscibility but in highly nonlinear ways. Low iron contents promote the mixing of these two phases but only to a small degree whereas high iron contents have very little effect on the miscibility of these phases and can even hinder mixing. We find that Ca-pv and Mg-pv exist as independent phases in the lower mantle but starting at 75 GPa in iron-rich hot mantle they can form a single phase. As iron content decreases or the mantle cools the pressure of this transition deepens to around 120 GPa for a mantle at normal temperatures with a small amount of iron (~1% Fe in the perovskite). There is weak partitioning of iron from bridgmanite to the mixed phase but this is less favourable than partitioning Fe from bridgmanite to MgO and thus is not expected to have strong dynamical consequences.

Acknowledgement
This work was supported by National Natural Science Foundation of China (41773057)   12.5% Fe (purple line) bearing systems. the normal mantle geothermal gradient and the cold subduction slab geothermal is also presented for guidance (Ohtani et al. 2018)