The Effect of Ti on Ca-pv and Mg-pv phase stability

Magnesium silicate perovskite in the form of bridgmanite (bdg) and Calcium silicate perovskite (Capv) have similar chemical structures and may mix into a single perovskite phase in the lower mantle which would have profound effects on many seismic properties. While we have previously found that this is unlikely to occur in pure bdg and ca-pv in this paper we examine whether phase mixing can be induced by Titanium. We predict that even small amounts of Ti can cause significant increases in mixing of the two phases. Miscibility of the phases has a strong dependence upon how Ti is partitioned between the two phases before mixing and thus the source and history of introduced Ti is important in determining miscibility. We predict basalts, even those with heavy Ti enrichment (10%), will not form a single phase in subducted slabs as their mixing temperatures remain above 2500 K for most compositions throughout lower mantle pressures. In pyrolytic mantle it is predicted that at shallow depths large amounts of Ti are needed to induce phase mixing (~40% Ti at 25 GPa and geotherm temperatures) but less Ti is needed to induce mixing with depth (~1% Ti at 125 GPa and geotherm temperatures). Thus we predict that enriched Ti regions will see perovskite mixing near the bottom of the lower mantle. These mixed perovskite regions partition Ti out of unmixed regions and thus provide a mechanism for Ti enriched regions to form in the deep lower mantle. Both ferrous iron and the Ca:Mg ratio are predicted to have a larger control on the mixing temperature of pyrolytic systems than Ti, however. For Ti and Ca rich pyroxene megacrysts we find that they should become a single phase along a lower mantle geotherm at around 40-115 GPa depending heavily upon Ti and Ca concentration.

ilmenite megacrysts were found to be likely to exist as single phases below 45 GPa. Subducted Ocean Island Basalt (OIB) was predicted to convert into a single phase around 80-100 GPa whereas MORB and primitive mantle with lower Ti concentrations was predicted to remain as two phases all the way down to the D'' layer.
This previous work does not explore the effect of temperature above or below 2000 K which is important in a thermally hetereogenous lower mantle and does not reach the pressures of the D'' layer. Thus in this work we shall seek to establish the miscibility of Ca-pv and bdg as a function of Ti concentration and temperature using Density Functional Theory (DFT). We shall examine two pressures-25 GPa as the top of the lower mantle and for comparison with Armstrong et al. (2012) and 125 GPa as an extension of this work down to the D'' layer. Once this has been established, we shall then use our results combined with a simple model to see how other non-Ti elements such as Al and Fe affect this miscibility and then speculate on the phase structure of Ti-rich parts of the lower mantle.

General Method
To determine whether two phases mix we simply need to determine the energy of the following reaction: The energy of this mixing reaction can be represented by: where Hmix is the enthalpy of mixing, T is the temperature, Smix is the entropy of mixing and Gmix is the free energy of mixing. Mixing will occur when Gmix is negative. SMix will be broken into two components-Svib a component representing vibrational entropy and SConfig a component representing configurational entropy. Gmix will then be determined in two parts. Hmix and Svib will be determined through molecular dynamics (MD) calculations and SConfig will be determined through static calculations.
To determine mixing over an array of conditions we ran calculations at various points and extrapolated between them. For the molecular dynamics portions we ran calculations at 25, and 125 GPa and at 1000, 2000 and 3000 K. All pressures are uncorrected. Energies were determined at Ca%=0, 25, 50 and 100 and Ti%=0, 25, 50 and 100. To calculate Gmix at any arbitrary T, P, Ti% and Ca% we then used the following scheme. First at each pressure point (25 and 125 GPa) and each Ti% and Ca% we calculated G of the products and the reactants as a function of T. We then fit polynomials as a function of T and determined the G of the products and the reactants at the T of interest. We then fit polynomials as a function of Ca% and then pressure and calculated Gmix at the appropriate Ti%, Ca% and pressure in this order.
The fits across P and T are relatively linear and are likely reliable. As shown in Figure S1 fits across Ca% are also likely reliable. As shown in Muir and Zhang 2020 fitting across Ca% with a solid solution model does not vary results significantly and statistical errors in the molecular dynamics are more important.
These calculations give us 2 pressure points which we use to calculate the effect of pressure at the top and bottom of our range. To explore a pressure range we use some assumptions which will be discussed in the results section of the manuscript.

Computational Details
For these calculations we used the VASP code (Version 5.4.4) (Kresse andFurthmuller, 1996b, Kresse andFurthmuller, 1996a). This is a density functional theory approach where planewave pseudopotentials are used to simulate supercells which represent infinite crystals. The PBE (Perdew et al., 1996) exchange correlation functional was used alongside the included VASP PAW potentials (Kresse and Joubert, 1999 (Monkhorst and Pack, 1976). Energies were relaxed to within 10 -5 eV and forces between atoms were relaxed to below 10 -4 eV/Å. For molecular dynamic runs the gamma point was used with cutoffs of 600 eV and relaxed to within 10 -4 eV. 80 atom unit cells were used (2x2x1) except for the configurational entropy as noted below. Vibrational entropy was determined by applying a Velocity-Autocorrelation function to molecular dynamics runs while configurational entropy was calculated by determined the relative enthalpy of all arrangements of Ca and Mg (on the A site) and Ti and Si (on the B) in 40 atom unit cells and then calculating the Gibbs entropy. Further details on this are given in the supplementary methods.

Phases:
Multiple different structures are possible in this system. MgSiO3 is usually in the orthorhombic pbnm spacegroup (Zhang et al., 2013) while CaSiO3 is in the cubic pm3m or the tetragonal i4mcm spacegroup (Stixrude et al., 2007, Sun et al., 2014. All systems (end members and mixtures) were calculated in all 3 of these possible structures. All extrapolations across Ca%, Ti% and temperature were done for all 3 symmetry structures and then at any specific composition and temperature point the lowest energy structures was chosen. We find that Ti does not change the preferences seen for Ti-free systems (Muir et al., 2020)-ie Mg end members and mixed phases exist as pbnm structures, Ca end members as i4mcm and pm3m structures with pm3m structres favoured by high temperatures. To determine phase loops we plotted the energy of the unmixed and mixed phases as a function of Ti either between Ti% 0-0.5 or between 0.5-1, fit them to polynomials and then found the common tangent between them.

Ti partitioning
When mixing two phases that have a defect element it is important to know how that defect element is distributed in the two phases before mixing. We first consider the thermodynamic partitioning of Ti between our two separate phases of Calcium Silicate perovskite (Ca-pv) and bridgmanite (bdg) as shown in Figure S2-S3. This can be defined by a partitioning coefficient which we shall define as: At low pressure (25 GPa) Ti is preferentially partitioned into Ca-pv when the Ti% is below ~40% with lower temperatures favouring this sense of partitioning more. At higher pressure (125 GPa) the reverse is the case with Ti favoured very strongly in the bdg phase. This makes sense as CaTiO3 is a low pressure phase but at lower mantle pressures small amounts of MgTiO3 tend to stabilise bridgmanite (Matrosova et al., 2020). Thus with increasing pressure Ti moves from Ca-pv to bdg. We shall refer to these distributions as the "equilibrated" cases as Ti is spread into its thermodynamic equilibrium before mixing is attempted. It should be noted that we do not consider the solubility of Ti as a whole and thus in some of our cases a separate TiO2 phase is likely stable and this is an important future step to consider.
There is another possibility however. If Ti diffusion is very slow then chemical mixing may occur at substantially faster timescales than Ti partitioning. This would have the effect of kinetically promoting the mixed phase as once it forms it is unlikely to convert to two heavily partitioned separate phases even if these are thermodynamically more stable. This case will have quite different mixing dynamics to the equilibrated case and will always have higher miscibility. We are not aware of any studies on Ti diffusion in bdg or Ca-pv but Si diffusion is very slow (10 -19 to 10 -20 m/s (Xu et al., 2011)). Ti exists on the Si site and thus likely diffuses via a similar mechanism and at a similar rate to the Si. In the case of a Si vacancy mechanism unless Ti diffuses considerably faster than Si, Ti diffusion will be slower as it relies upon the product of Ti and Si vacancy concentrations. This is seen in olivine where Ti diffusion is around an order of magnitude slower than Si diffusion (Cherniak and Liang, 2014). In Armstrong et al. (2012) the two perovskite phases were mixed in ~1 hour with grains that were ground to mostly sub-micro sizes during which time Ti would diffuse a maximum of a few femtometres using bdg Si diffusion rates from Xu et al (2011). Such a number suggests that chemical mixing is likely much faster than Mg or Ti diffusion in bridgmanite leading to the possibility of non-equilibrium kinetics.
In this "non-equilibrated" case the distribution of Ti depends upon the source and initial distribution of Ti. There are multiple possible cases and we shall consider a few of them to establish the range of such an effect. We shall consider a CaTiO3 source (all the Ti resides initially in Ca-pv, K>1000), a MgTiO3 source (all the Ti resides in bdg K<0.0001) and a source that places an equal concentration of Ti in each phase before mixing such as when a large quantity of Titanium is introduced simultaneously to a Capv and bdg interface (K=1). This last case shall be referred to as the "distributed" case as Ti is equally distributed across two phases.

Enthalpy and Entropy
There are 3 key terms to mixing, Hmix, Sconfig and Svib. For a review of these values in a Ti free systems see Muir et al. (2020). The effect of Ti on Hmix is presented in Figure S4 and on Svib in Table S1 and Sconfig in Table S2-S3.
On the addition of Ti, Sconfig increases significantly peaking at Ti%=50 while Svib and Hmix have more complex effects. Hmix decreases significantly when Ti is distributed equally to the two phases but can both increase and decrease when partitioning is considered premixing. The former effect is because the mixed phase is less dense than a mixture of the two unmixed phases (Table 1) and is thus able to incorporate larger Ti atoms more effectively. Svib typically decreases slightly with Ti concentration but this is a small effect and can typically be ignored. Unlike in the pure case of MgSiO3-CaSiO3 (Muir et al., 2020) where the Sconfig term can be reasonably approximated with that derived from perfect mixing, in the Ti-containing case Sconfig is significantly non-perfect. In some cases Sconfig being non-perfect raises Tmix by > 200 K when compared to its perfect equivalent. Overall, the addition of Ti causes a significant decrease in Tmix and this decrease is primarily driven by the increase in Sconfig from adding in Ti with a small secondary effect coming from changes to Hmix.

Mixing of CaTixSi1-xO3 and MgTixSi1-xO3s
We shall consider mixing at two pressures, 25 GPa and 125 GPa. These represent roughly the top of the lower mantle and the top of the D'' layer. While not a full sampling of pressure this shall allow us to examine the maximum effect that pressure can have on this system.

GPa
The solubility of Ca in bdg is shown in Figure 1 and for Mg in Ca-pv in Figure S5 but as the mantle has more Mg than Ca we shall focus on the former case. As shown in Figure 1 adding Ti substantially reduces Tmix for the reasons discussed above. This can clearly be seen at high solubilities where the Tmix value plateaus (as configurational entropy dominates the system) and this plateau temperature changes from ~3200 K with no Ti to ~2750 K with Ti%=10. With no Ti a pyrolytic mixture of Ca and Mg (Ca%=10) mixes at ~3100 K, with Ti%=1 this drops to ~3000 K and with Ti%=10 this drops to ~2500 K. For a more MORB like composition with Ca%=50 the relative values are ~3160, ~3120 and 2780 K respectively.
As also shown in Figure 1 in general the solubility at this pressure has little dependence on the partitioning of Ti before mixing. An equilibrated and distributed sample have very similar Tmix values with maximum differences in Tmix of <50 K or ~4% of Tmix. Significant differences to solubility are only found with a CaTiO3 source of Ti. With a CaTiO3 source the solubility of Mg in Ca-pv is essentially unchanged from the equilibrated sample but the solubility of Ca in bdg is substantially increased.
The addition of Ti allows a phase loop to form. This is pictured in Figure 2. As shown the phase loop causes the onset of mixing to drop by up to 200 K compared to the univariant case for both pyrolytic ( Figure 2) and basaltic ( Figure S6) compositions and causes a maximum phase loop width of around 400 K. These values are small when considering the changes to mixing temperature induced by geological variability in Ca%, Ti% and other elements (see for example Figures 7-9). In the real mantle the phase widths will be even smaller than calculated here. Our calculated values are for the thermodynamic maximum width of the phase loop but various kinetic effects will likely narrow the phase loop in a real system. We consider two cases.
First we consider a case where Ti diffusion is fast enough that equilibration of Ti occurs. In this case the generally large value of the partitioning coefficient between the two phase and one phase system means that the effective width of the phase transition seen by seismic waves is likely considerably smaller (Stixrude, 1997).
Second we consider a case where Ti diffusion is sluggish. In the limit of very slow Ti diffusion a univariant phase transition would be obtained. As Ti diffusion becomes slower the phase transition thus narrows towards the univariant. As explained above Ti diffusion is likely slow and thus trends towards the univariant case. Thus we shall present univariant results from now on but the phase loop could lower the onset of mixing by up to 400 K across the Ti% range. Figure 3 shows a comparison between our calculated data and points determined in Armstrong et al. (2012) for pressures 21-30 GPa and temperatures 1800-2200 K. In this we plot both the equilibrated case and the distributed case which are near identical as was also seen in Figure 1. It is difficult to know the exact dynamics of the experiment but due to its short nature (by mantle timescales) it is possible Ti was not fully equilibrated before mixing occured. If the Ti was primarily in MgTiO3 then it would have essentially the same trace as our equilibrated case. The CaTiO3 source case is also plotted but has very substantial differences to the experiment and so is unlikely to have occurred.
The majority of the phase determinations in Armstrong et al. (2012) fit our phase boundary with some notable exceptions. At high Ca% one composition that was observed to be 1 phase is inside our 2 phase region but is within the error caused by temperature fluctuations. At Ca%=60 and Ti%=50 a 2 phase mixture was observed in the experiment at 26 GPa and 2000-2100 K. Our calculations predict that a single phase should form above 1955 K at 25 GPa. This difference could be related to errors in our projection across Ca% where we only have points at 0, 25, 50 and 100. Alternatively the mixed phase could be somewhat sluggish in forming at such middling values of Ti% and thus was not formed in the experimental timeframe or exsolution of TiO2 could cause the Ti% value in the mixture to be lower that predicted here.
Our calculated phase boundary in Figure 3 at 2000 K is similar to that determined experimentally by Armstrong et al. (2012) at low Ti% but varies significantly with high Ti%. We find that mixing/solubility increases rapidly as Ti% approaches 50 due to SConfig which leads to our different high Ti% behaviour.
It is possible that exsolution of TiO2-which is not considered in our model-causes some raising of the phase boundary curve at high Ti% in real samples as exsolution of TiO2 must necessarily raise Tmix.
Alternatively the high Ti% trend speculated in Armstrong is between our predicted high Ti% trends predicted for 1600 and 2000 K and thus the Armstrong phase relations being near the lower end of their temperature range (~1800 K) may also explain these results. This plot has a few interesting features. On the low Ti% side there is a large patch of high Tmix (>3000 K) stretching across the entire Ca% range. This shows that regardless of composition very large amounts of Ti are required to drop Tmix to ~2000 K, around 30% at a pyrolytic composition (Ca%=10) and above 45% with a basaltic composition (Ca%=50). It also shows that thermal fluctuations are relatively unimportant at this pressure. At ~60% Ti there is a band of extremely low Tmix across all Ca% values which relates to the maximum of the Sconfig and Smix terms. Compositions with high Ti% behave similarly to those with low Ti% but we have no measure of Ti solubility in these calculations and so these structures likely break down at some point. Figure 5 shows the Ca solubility in bdg at 125 GPa with different partitioning values ( Figure S10 shows the Mg solubility in Ca-pv). We find that partitioning of the Ti before mixing leads to larger differences in Tmix than was seen for 25 GPa due to the larger difference in enthalpy between CaTiO3 and MgTiO3 at this pressure. The equilibrated and MgTiO3 cases are near identical and are the two most likely possibilities in the lower mantle. The distributed case is less likely and has Tmix values that are somewhat lower (up to 200 K lower at 50% Ca). A CaTiO3 source has much lower Tmix values but is unlikely to occur.

GPa
As shown in Figure 2 the phase loop is similar at 125 GPa as at 25 GPa but less wide due to the large energy differences between the phases. Similar arguments regarding the phase width and the relevance of this phase loop apply at these pressures as at 25 GPa. This is due to the stability of Ti in the bdg phase at this pressure and this somewhat cancels out the large increase in SConfig seen with ~50% Ti. Thus Tmix does decrease with increasing Ti% at 125 GPa but this decrease is much smaller than is seen at 25 GPa and mixing at this pressure is more robust to varying Ti concentrations.  Figure 3). We also never approach the maximum solubility seen in Armstrong et al. (2012) even with a CaTiO3 source-at 97 GPa they observed solubility of 45% (Ca-pv into bdg) with only 5% Ti and this solubility should be even larger at 125 GPa). Thus the differences between our theoretical results and those measured by Armstrong et al (2012) is some fundamental difference in how pressure derivatives are calculated and are not simply related to how

Mixing as a function of pressure
Ti is distributed in the sample before mixing. In Ti-free samples we also predicted a smaller pressure derivative of solubility than was observed experimentally (Muir et al., 2020).
There are a few possible causes of this discrepancy. The most likely reason for the discrepancy is that our model fails to capture some aspect of the dissolution. Our model does not include TiO2 exsolution but this can only lower solubility and so it cannot increase our agreement with experimental data which predict higher solubilities than we obtain.
One possible aspect that we do not calculate in our model is pressure non-linearity. We consider only the points 25 and 125 GPa whereas the experimental data runs from 20-96 GPa. If pressure varied highly non-linearly between 25 and 125 GPa this would explain these differences. We consider this to be unlikely as the component energies of the phases have near-linear relationships with pressure.
There are two major components to the mixing energies: Sconfig and Hmix. Sconfig has only a small dependence on pressure (see Table S2 and S3). Hmix has a strong non-linear dependence on pressure but this pressure dependence is related to partitioning-if we remove the partitioning component Hmix is largely linear with only some small deviations near 125 GPa ( Figure S4). We can enforce near linear behaviour in Hmix therefore by fixing both 25 and 125 GPa samples to the same partitioning regime of a CaTiO3 source (K>1000). By doing so we have a regime that should be highly linear with pressure and as shown in Table 2 this still does not reproduce the experimental results and predicts lower solubility at high pressures than Armstrong et al. (2012). Thus this is highly unlikely to be the source of our discrepancy.
Contamination of the experiment is unlikely to be important as we find in the next section that very large concentrations of defects are required to shift Tmix values. To shift Tmix by a few hundred degrees as would be required to explain our observed discrepancies would require defects with a concentration on the order of 10% which is far higher than experimental contamination levels.
The most likely explanation is that we do not consider in our model the kinetics of dissolution, we only consider thermodynamics and how to minimise the overall energy. There may be macroscopic kinetic effects which increase the propensity for either global or local mixing. Regions with heterogeneously high concentrations of Ti may induce local and then global mixing which would explain why solubilities are higher in experiment than in our theoretical prediction. Alternatively there may be some other aspect of pressure which our model does not consider. Regardless our results predict the thermodynamic minimum and thus are possibly more robust in the long-time scales of the mantle where thermodynamic equilibriums should be obtained.

Partitioning
In the lower mantle the equilibrated case is much more likely to occur. Bdg and Ca-pv are produced as two separate phases in different transitions in the mantle and at the top of the lower mantle (when bdg is first produced) we predict them to exist as two unmixed phases (Fig 5 and Muir et al. (2020)).
While Ti diffusion is likely slow, over the long timescales of the mantle this should not matter and Ti could equilibrate across the two phases before conditions which induce mixing are reached. Nonequilibrated mixing would only occur if large amounts of Ti were introduced suddenly deep in the mantle where mixing is more favoured (Fig 7). Non-equilibrated mixing is likely, however, to be extremely important in experimental measurements over short timescales and thus should inform experimental design. We shall only discuss equilibrated cases for the rest of this paper however as those the most likely to be relevant in the Earth.

Mixing in the Lower mantle
To examine mixing in the lower mantle we built a small model. Ti-free values at 25, 75 and 125 were taken from Muir et al. (2020) and Ti values at 25 and 125 GPa. These were then extrapolated over pressure to predict the mixing of different compositions in the lower mantle.

Basalts
First we will consider basaltic mixtures. Basalt was previously found not to mix in the absence of Ti (Muir et al., 2020)in geotherm conditions. Figure 7 show the mixing of a basalt enriched in 10% Ti as is possible in OIB compositions, a more MORB like basalt with 1% Ti is shown in Figure S14 but Tmix values are ~400 K larger in the 1% MORB compared to the 10% OIB. We find that even with the maximum amount of Ti the mixing temperature of basalt remains far above likely slab temperatures thus ruling out perovskite phase mixing in descending slabs. At ~115 GPa highly enriched OIB basalts reach the temperature of the geotherm and thus mixing could occur at the outer edge of descending slabs or in basaltic regions of the lower mantle. Lesser enriched MORB basalts never reach such temperatures and thus never mix in descending slabs.
In Figure 7 we show the effect of a plausible range of Ca% (30-60%) values but this has very low effect on Tmix as these values are all in the plateau region of solubility ( Figure 1) and have similar solubilities (variations in Tmix <100 K). Thus variation in Ca% is not a significant factor for basalts.
Other elements are present in basalts and we address these using a simple model described in Muir et al. (2020) and shown in Table S5-S6. We find extremely similar results to the Ti-free system-notably that large amounts of any defect (~1%) are needed to make substantial changes to Tmix and that of the likely elements in these concentrations Fe(II) decreases Tmix and Al and Fe(III) increase it. These are plotted in Table 3 and visualised for basalts in Figure 8 (other compositions are shown in Figure  Thus we conclude that basaltic compositions in descending slabs will remain as two phases even if they descend right to the D'' layer and even if they have the maximum amount of Ti (10%) speculated to be in basaltic compositions. Small amounts of mixing will be possible on the edges of slabs as they approach lower mantle temperatures but this will be limited only to extreme depths and small portions of the slab. No seismic anomalies should thus occur from this phase change in descending slabs. Basalt that is present at lower mantle temperatures, from unmixed pyrolite for example, will mix but only in the deep lower mantle (>~115 GPa) and with high concentrations of Ti (~10%).

Pyrolytic Mantle
Finally we shall consider the effect of Ti on pyrolytic mantle (Figure 7). Increasing the amount of Ti in pyrolytic mantle decreases the pressure at which it reaches the geotherm ( Figure S19). With no Ti the pressure at which phase mixing is seen along the geotherm is ~126 GPa and drops to ~104 GPa with 10% Ti. At 25 GPa ~40% Ti induces mixing at the geotherm. At 125 GPa ~1% Ti induces mixing at the geotherm. Temperature fluctuations will only have small effects on these numbers. Thus in regions enriched in Ti pyrolytic mantle will undergo mixing near the bottom of the lower mantle.
Phase mixing also provides a method for producing Ti rich regions in the lower mantle. As shown in

Megacrysts
Next we shall consider the case of megacrysts. The phase of megacrysts is important as to determining their origin. One suggested origin is that they began as single phase perovskites under lower mantle conditions (Collerson 2004(Collerson , 2005. Iron and aluminium free clinopyroxene megacrysts (~Ca% 15-35% Ti% ~15-40%) were found by Armstrong et al. (2012) to convert to a single phase around 50-80 GPa and around ~65 GPa for an average composition. We show a sample CMC (Ca% 30 Ti% 25) in Figure 9 and find that in the absence of other elements it mixes at around 85 GPa along a geotherm. This is higher than the value derived in Armstrong et al. (2012) due to our different pressure derivatives.
Within the geological variation of CMC (Ca%=20-40, Ti%=15-30) we find that this value can vary between ~65-115 GPa with Ti% being the strongest control on this depth. OMC have lower levels of Ca% than CMC and thus have higher miscibility and are observed to mix (in the absence of other elements) at around 30-65 GPa along a geotherm. Moderate amounts of iron increase mixing even more ( Figure S19-S20). Thus single phase ilmenite pyroxene megacrysts can be found at depths ~1000 km with the actual depth depending upon the exact composition which can cause large variations in Tmix (~700 K) and the depth at which single phase perovskites are favoured.

Conclusion
We find that while Ti has large effects on miscibility of bdg and Ca-pv it should not induce mixing in basalts in descending slabs and thus there should no seismic signals from phase mixing in these slabs.
While Ti can induce mixing in pyrolytic compositions this will only occur near the bottom of the lower mantle where seismic signals are complicated by the presence of the D'' layer and the CMB.
Additionally Ca% is likely to be a stronger control on the miscibility. The main effect of Ti is converting pyroxene ilmenite megacrysts into single phases at depths of greater than 1000 km with strong variability dependent upon Ca% and Ti% ratios. This is evidence promoting the single phase origin hypothesis in the literature.
These speculations all assume equilibrium chemistry. In non-equilibrium chemistry as may occur in experiments or dynamic parts of the mantle the effect of Ti on inducing mixing can be much larger.
Thus it is important to constrain the dynamics of this mixing and of cationic diffusion in bdg and Ca-pv in future works to fully account for this effect.      forms Ferrous iron or Al-Al pairs. Mixing depths above 1800 km and below 2800 km have been truncated to these values to follow the stability field of bdg. Tmix always remains well above the coldest slab adiabat (see Figure S15).      forms Ferrous iron or Al-Al pairs. Mixing depths above 1800 km and below 2800 km have been truncated to these values to follow the stability field of bdg. Tmix always remains well above the coldest slab adiabat (see Figure S15).     Table S7 and have the similar values.

Supplementary Information
This document contains supplementary figures and tables for the manuscript "The Effect of Ti on Capv and Mg-pv phase stability". Presented within are additional details of our method and additional graphs mostly plotting data for different Ca% and Ti% values.

Supplementary Methods: Molecular Dynamics:
To determine the vibrational entropy we used a Velocity-Autocorrelation Function (VACF) method.
More accurate methods such as thermodynamic integration are possible but as Gmix values are fairly large the extreme accuracy of these methods is likely unnecessary.
Vibrational entropy determination requires the vibrational density of states function, S(ν), which represents the distribution of normal modes (ν) in the system. This can be represented as: where N is the total number of atoms in the system, mi is the mass of atom i, is the spectral density of atom i in the direction k (x=1, y=2, z=3), T is the temperature and kb is the Boltzmann constant. In our case is obtained by taking the fourier transform of the VACF. Entropy is then obtained by = ∫ ( ) ∞ 0

Equation S2
Entropies and enthalpies were determined from molecular dynamics runs with a length of 2.5 ps.
The error of the energies obtained from molecular dynamics were calculated for each individual run using the method of Flyvbjerg and Petersen (Flyvbjerg and Petersen, 1989) and were less than 1.5 meV/atom in all cases. Propagating these errors leads to the error in mixing temperatures < 5 K ( σ) for all mixtures. As shown in Figure 7-9 compositional variation causes larger changes than our likely errors.

Configurational Entropy:
To determine the configurational entropy of mixed phases we calculated the enthalpy of different configurations of Mg and Ca on the Mg site and Ti and Si on the Si site in the unit cell. For CaxMg1-xSiyTi1-yO3 we calculated the configurational entropy for all combinations of x and y=0, 0.125,0.25,0.375,0.5,0.75,0.875,1 at 25 and 125 GPa.
As the number of configurations increase with N factorial, 80 atom unit cells proved too large to obtain a workable number of configurations. The energy difference of different configurations is independent of unit cell size, however, and the disadvantage of using a small unit cell in these calculations is simply that some configurations may not be appropriately sampled. To this end for CaxMg1-xSiyTi1-yO3 we determined configurational entropy in a 40 atom unit cell (2x1x1) when either x or y were between 0.25 and 0.75 or they were both 0.125/0.875 and we used 80 atoms when x or y=0.125 or 0.875 and the other value was 0 or 1. For each of these systems we determined all unique configurations of Ca, Mg, Si and Ti and calculated their static enthalpy. To determine Sconfig we then used the Gibbs entropy formulation: where Z is the partition function, i is each configuration including degenerate copies of each configuration, Ei is the energy of that configuration (in this case we used the enthalpy) and pi is the probability that it occurs. If every configuration has equivalent energy these calculations reduce to the Boltzmann entropy formula: where W is the number of possible configurations. This is equivalent to ideal mixing of Mg,Ca, Si and Ti provides the maximum possible configurational entropy in this system. Sconfig was interpolated across Ca% and Ti% using a polynomial and incorporated in the equation for ΔGmix (Equation 1). As all possible configurations (in 40 or 80 atom simulation cells) were evaluated the uncertainties in these estimates of Sconfig are limited to missed configurations that may occur at high temperature or in simulation cells with an increased number of atoms that are more like infinite crystals.  . Values below 1000 K and above 3000 K were truncated to these values respectively. Figure S8: As Figure S7 but with the CaTiO3 source case (K=1000). The black region requires an additional source of Ti and thus cannot be properly shown in this case. This is similar to the equilibrated case in Figure 4 but with slightly lower mixing temperatures.  quil Figure S11 Heatmap of univariant Tmix as a function of Ti% and Ca% at 125 GPa with the distributed case K=1. Values below 1000 K and above 3000 K were truncated to these values respectively. This has a much smoother distribution than the equilibrated case which has large distortions due to the partitioning of Ti. At low Ti% a large band of high Tmix stretches across the Ca% range. There is no decrease of Tmix at 50% and instead it steadily decreases across the Tmix range due to the nature of Hmix at this pressure.
Figure S12: As Figure S11 but the CaTiO3 source case (K=1000). The black region requires an additional source of Ti and thus cannot be properly shown in this case. This is similar to the distributed case in Figure S11 but with much lower Tmix values due to the instability of CaTiO3 at this pressure.
Figure S13: As Figure S11 but the MgTiO3 source case (K=0.001). The black region requires an additional source of Ti and thus cannot be properly shown in this case. This is similar to the equilibrated case in Figure 6 but with lower Tmix values. asalt Figure S15 Plot of Tmix of a sample OIB configuration (Ca=50%, Ti=10%, equilibrated Ti, univariant) as a function of pressure and with different elements added in. These compositions are somewhat arbitrary and illustrate how different ranges of elements can affect the miscibility. In the labels Fe=ferrous iron, Al= Al-Al pairs and FeAl= Fe-Al pairs, perfect represents a mixture with no added iron or aluminium. The method for constructing this graph is explained in the text. The black lines represent various temperature profiles through the lower mantle-that of standard geotherm (Ono, 2008), that of the coldest possible slab adiabat (Eberle et al., 2002) and an artificial "hot" geotherm which is 500 K hotter to show how temperature fluctuations could affect things. Even with extremely large amounts of ferrous iron Tmix for this basalt remains well above the slab adiabat. e l Figure S16 Depth at which Tmix crosses the geotherm for a clinopyroxene ilmenite-megacryst (Ca%=30, Ti%=25, equilibrated Ti, univariant) mixture with various amounts of Fe and Al as determined via the model outlined in the text. For this model the formation of Fe-Al was prioritised such that Fe-Al forms first and then leftover Fe or Al forms Ferrous iron or Al-Al pairs. Mixing depths above 1800 km and below 2800 km have been truncated to these values to follow the stability field of bdg.
Figure S17: As Figure S16 but for a low Ti basalt mixture more representative of MORB (Ca%=50, Ti%=1). The mixing temperature here is always strongly above a slab adiabat.
Figure S19: As Figure S16 but for a pyrolytic mixture containining Ti (Ca%=10, Ti%=10). Figure S20: Pressure at which a pyrolytic mixture (Ca%=10) with either Fe or Ti reaches the geotherm as a function of Fe or Ti%. Ti data is projected from our calculations in this paper, Fe data is calculated using our simple model.    Table S5: Effect of various elements on Tmix at 25 GPa with Ca%=50 or Ca%=10 and either Ti%=1 or 10. Columns are name of the element, site at which that element was placed (A=Mg site, B= Si site, AB= 1 element at each, Int=interstitial), the change in ∆Hmix in eV from placing one defect element, proportion of this element in the Ca-pv before mixing (1 is all in Ca-pv, 0 is all in bdg), change in Tmix (K) with various amounts of element (in atomic % of bridgmanite). All elements are non-spin polarised except those labelled HS which were run with their standard high spin configuration. 2H represents a water molecule where a Mg has been replaced with 2 Hydrogens in the vacancy. Fe-Al represents a high spin ferric iron replacing a Mg and an Al replacing a Si.