The diffusion of titanium along with Ni, Co and Mn in pure synthetic forsterite has been studied as a function of temperature, oxygen fugacity, crystallographic orientation and chemical potentials in the four-component system MgO–SiO2–TiO2–TiO1·5. In over 100 different experimental conditions, no Ti diffusion profiles with the classical shape of the error function were observed. Instead, the profiles vary between ‘stepped’ at high fO2 (probably owing to diffusion on octahedral sites and trapping in tetrahedral sites), to hockey-stick shapes at low fO2 owing to concentration-dependent diffusion. The profile shapes vary systematically between these end-members according to oxygen fugacity. These profile shapes are also reproduced in experiments using natural San Carlos olivine. The change in profile shape is interpreted to be a function of valence-state change from Ti4+ to Ti3+, shown by a sigmoidal log fO2–diffusivity relationship. To distinguish between Ti4+ and Ti3+ in forsterite, a diffusion profile was also investigated using ‘hydroxylation spectroscopy’ whereby a crystal from a diffusion experiment was hydroxylated by annealing in an H2O-rich fluid at moderate pressure and temperature. The point defects associated with the added structural OH were then determined by infrared spectroscopy, providing evidence for the presence of Ti4+ and Ti3+ at intermediate oxygen fugacity. The transition from Ti4+ to Ti3+ occurs at considerably higher fO2 in these diffusion experiments than in equilibrium experiments in similar systems. Titanium diffusion, qualified using a mobility parameter (M), is fastest along the c-axis, at high activity of silica, low oxygen fugacity and high temperature. The rate of Ti diffusion is broadly similar to those of Mn, Ni and Co, and closer to published rates of Mg self-diffusion in olivine than Si self-diffusion. The observed Ti3+ and Ti4+ diffusion occurs on the M-sites. From statistical examination of the large Mn, Ni and Co diffusivity dataset, we determine that there is a 102/3 dependence of diffusion on aSiO2; the cations diffuse more rapidly at high aSiO2. In addition, diffusivity has a 10–1/8 dependence on fO2, suggested to be the result of increased concentration of loosely bound Ti3+–vacancy pairs, which enhance the mobility of other cations. This behaviour is likely to be present in natural systems where trivalent (e.g. Fe3+, Al3+, Cr3+) and divalent cations diffuse together. Consideration should be given to diffusive interference by highly charged, fast-moving cations when extracting time scales from frozen diffusion profiles.

## INTRODUCTION

Frozen diffusion profiles in natural olivine xenocrysts have been used to yield information regarding their residence time in, or ascent rate of, magmas, (e.g. Chakraborty, 2008; Costa et al., 2008; Qian et al., 2010; Ruprecht & Plank, 2013), and diffusion rates can help to elucidate mantle rheological behaviour (e.g. Bejina et al., 1999; Fei et al., 2012, 2013). Therefore, it is imperative that the diffusive process itself is well understood. Studies of diffusion are generally split into those that attempt to recreate complex natural systems to yield geologically meaningful results and those that investigate simple systems with few components to allow experimental variables to be controlled individually, as far as possible. Although the former strategy may seem the direct way to calibrate geologically useful mineral ‘geospeedometers’, the latter method is mandatory if the fundamentals of diffusion are to be understood. Without such knowledge, the information from diffusion profiles in natural minerals may be misinterpreted. Whereas considerable experimental and theoretical progress has been made in olivine in the Mg–Si–O ± Fe systems (e.g. Smyth & Stocker, 1975; Stocker, 1978; Nakamura & Schmalzried, 1984; Dohmen & Chakraborty, 2007), there are relatively few thermodynamically well-constrained experimental data for systems with components that are not major components of olivine.

The rates of trace and major element diffusion in olivine generally increase with temperature. This is expressed through Arrhenius relationships with the form D = D0exp(−Ea/RT) where the diffusion coefficient (D) at a given temperature (T) is determined using a pre-exponential factor (D0), an activation energy (EA) and the gas constant (R). However, temperature is only one of several variables affecting diffusion. Various studies have shown that diffusion coefficients in olivine may be affected by, for example, oxygen fugacity (Dohmen et al., 2007), crystallographic orientation (Chakraborty et al., 1994; Ito & Ganguly, 2006), chemical activity (Jollands et al., 2014; Zhukova et al., 2014), pressure (Bejina et al., 1999; Holzapfel et al., 2007), hydrogen availability (Wang et al., 2004; Hier-Majumder et al., 2005; Costa & Chakraborty, 2008) and mineral composition (Chakraborty, 1997; Morioka, 1981). Even after constraining these variables we generally assume that diffusivity is independent of both concentration and time, but even this may sometimes be invalid (e.g. Van Orman et al., 2009).

Titanium has the ability to substitute in both tetrahedral and octahedral coordination in olivine (i.e. for silicon or magnesium, respectively; Hermann et al., 2005; Berry et al., 2007b); Ti4+ substituting directly for Si4+ is the main high-temperature equilibrium substitution mechanism under anhydrous conditions. Therefore, diffusion on this site should allow this defect to propagate at around the same rate as that of Si self-diffusion, and therefore help elucidate (if it can be measured) the rate of Si self-diffusion, which is generally agreed to be very slow (e.g. Houlier et al., 1990; Bejina et al., 1999; Dohmen et al., 2002; Costa & Chakraborty, 2008; Fei et al., 2012, 2013). When substituting into the octahedral site (i.e. for Mg2+ in Mg2SiO4, forsterite) as either Ti3+ or Ti4+, a charge-balancing agent must be involved. Where H is available, the substitution of Ti4+ for Mg2+ may be charge balanced by two protons (2H+) replacing Si4+, creating the Ti-clinohumite point defect (Berry et al., 2005, 2007b; Walker et al., 2007). This may be an important site for water storage in mantle olivine (e.g. Berry et al., 2005). In the anhydrous setting, Ti in either valence state may replace Mg2+ and be balanced by accompanying vacancies on an adjacent metal site. The transition from predominantly Ti4+ to predominantly Ti3+ is expected to occur at very reducing conditions, around QFM – 4 to QFM – 6 (where QFM is the quartz–fayalite–magnetite buffer) at 1300 °C (Mallmann & O’Neill, 2009). In natural systems other charge-balancing agents may be involved; these may include, but are not limited to, Na+, Li+, Al3+, and Fe3+.

Relatively few studies have examined rates and mechanism of Ti diffusion in olivine. Spandler et al. (2007) and Spandler & O’Neill (2010) measured element diffusion profiles between natural olivine and silicate melt at 1300 °C and 1 bar using a scanning laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) technique, and found Ti diffusion to be anisotropic and to occur at approximately comparable rates to Fe and Mg diffusion. In contrast, Cherniak & Liang (2014) measured Ti diffusivities in forsterite and natural olivine to be around three orders of magnitude slower, based on profiles of a few tens of nanometres in length, measured by Rutherford backscattering spectroscopy (RBS). Cherniak & Liang also found no appreciable effect of fO2 or crystal orientation on diffusivity. A limitation of the existing experiment data on Ti diffusion in olivine is that the effects of parameters such as chemical activity, fO2 and temperatures have not been comprehensively studied.

In this study, we examine Ti diffusion in forsterite and San Carlos (∼Fo90) olivine over a large range of fO2 conditions and temperatures under conditions where the chemical activities of the major components are buffered. In our experiments we use the minimum number of components; that is, MgO and SiO2, to give the magnesian olivine forsterite Mg2SiO4, and TiO2 to act as a diffusant source. Small quantities of divalent cation dopants, namely Mn, Ni and Co, are added to determine if local defects created by Ti may be exploited by other cations, and hence affect the diffusion of these other cations. By fully controlling the chemical potentials of components it is possible to determine which substitution point defect mechanisms dominate in the diffusion experiments, as demonstrated by Jollands et al. (2014) and Zhukova et al. (2014). In addition, by varying the oxygen fugacity, the change in substitution mechanism as a function of the potential valence change between Ti4+ and Ti3+ may also be investigated. This is also facilitated by the experimental and analytical methods used in this study, which allow large datasets to be rapidly collected; in this study over 100 different conditions were investigated and around 600 LA-ICP-MS traverses were conducted (∼500 000 data points).

## METHODS

### Experimental design

#### Diffusion experiments in pure forsterite

In a three-component (MgO–SiO2–TiO2) system, three phases would be necessary to fully buffer chemical activity according to Gibbs’ phase rule. However, given the redox variability of titanium there are actually four components (MgO–SiO2–TiO2–TiO1·5) such that another variable, the oxygen fugacity, must be specified to fully control the activities of all components. In this study, four three-phase assemblages were synthesized to buffer the chemical activities of the solid phases with the fO2 being imposed by the gas composition in the furnace. The three-phase assemblages were, in order of increasing activity of silica (aSiO2) (decreasing activity of magnesia), forsterite–periclase–qandilite, forsterite–qandilite–geikielite, forsterite–geikielite–karooite and forsterite–protoenstatite–karooite (qandilite, geikielite and karooite are the Mg end-members of ulvöspinel, ilmenite and armalcolite, respectively) (Table 1; Fig. 1). Hereafter, the abbreviations listed in Table 1 will be used. Assemblages were synthesized such that target mineral proportions were all approximately 1:1:1 (molar). In addition, all four buffers were doped with ∼0·1 mol % of CoO, NiO and MnO. The use of these divalent cations as tracers allows comparisons with other studies (e.g. Morioka, 1980, 1981; Jurewicz & Watson, 1988b; Ito et al., 1999; Holzapfel et al., 2007; Spandler & O’Neill, 2010; Zhukova et al., 2014) and allows diffusive interference between Ti and these cations to be investigated. Mn, Ni and Co are expected to substitute into forsterite only as M2+ over the range of experimental conditions, although Co and especially Ni will be effectively lost from the system at low fO2 because of reduction to the metal (O’Neill & Pownceby, 1993a). It is also possible that small amounts of trivalent Mn will be present in the most oxidized experiments; at 1300–1500 °C the fO2 of air (10–0·68 bars) is 0–1 log units above the MnO–Mn3O4 transition (O’Neill & Pownceby, 1993b).

Table 1

Abbreviations used within this study

Abbreviation Unabbreviated term Formula
fo forsterite* Mg2SiO4
ol olivine (Mg,Fe)2SiO4
prEn protoenstatite* Mg2Si2O6
opx orthopyroxene (Mg,Fe)2Si2O6
per periclase* MgO
fpr ferropericlase (Mg,Fe)O
qan qandilite* Mg2TiO4
gei geikielite* MgTiO3
kar karooite* MgTi2O5
ulv ulvöspinel (Mg,Fe)2TiO4
ilm ilmenite (Mg,Fe)TiO3
arm armalcolite (Mg,Fe)Ti2O5
Abbreviation Unabbreviated term Formula
fo forsterite* Mg2SiO4
ol olivine (Mg,Fe)2SiO4
prEn protoenstatite* Mg2Si2O6
opx orthopyroxene (Mg,Fe)2Si2O6
per periclase* MgO
fpr ferropericlase (Mg,Fe)O
qan qandilite* Mg2TiO4
gei geikielite* MgTiO3
kar karooite* MgTi2O5
ulv ulvöspinel (Mg,Fe)2TiO4
ilm ilmenite (Mg,Fe)TiO3
arm armalcolite (Mg,Fe)Ti2O5

*Represents end-member composition of solid solution (Mg–Fe) phases.

Fig. 1

Phase diagram (mol %) in the MgO–SiO2–TiO2 system [after Massazza & Sirchia (1958) and MacGregor (1969)] showing the relevant phases and abbreviations. Coloured triangles represent the four three-phase Ti source buffers.

Fig. 1

Phase diagram (mol %) in the MgO–SiO2–TiO2 system [after Massazza & Sirchia (1958) and MacGregor (1969)] showing the relevant phases and abbreviations. Coloured triangles represent the four three-phase Ti source buffers.

The assemblages were synthesized from oxide powders that were weighed, ground under acetone in an agate mortar, dried, pressed into pellets in a tungsten carbide dye, sintered at 1300 °C in air for 48 h, and phases were verified by X-ray diffraction (XRD). The pellets were then crushed and reground in an agate mortar. Cubes of synthetic forsterite (2–3 mm in each dimension) were cut from a large, oriented single crystal grown by the Solix corporation (Belarus) with very low contaminant concentration (20–30 µg g–1 Al and Co; <3 µg g–1 Sc, Fe and Mn; see Zhukova et al., 2014) and mounted in epoxy, then polished with p1200 SiC paper followed sequentially by 6, 3 and 1 µm diamond paste on cloth laps. The cubes were then removed from the epoxy by heating and cleaned using acetone. The powders were coupled to the crystals using polyethylene oxide–water glue, and residual water was removed by leaving the charges in a 100 °C drying oven for 2–24 h until dry. These were then placed into small (∼12 mm × 2 mm) Pt boxes and annealed in Gero gas-mixing furnaces for 2–43 days. The preparation of crystals, buffer assemblages and experimental design have been described in detail by Jollands et al. (2014). Experimental conditions are given in Table 2. Three orientations, four fO2 conditions (three at 1300 °C), three temperatures and four Ti-source buffer assemblages (three at 1500 °C) were used, giving a total of over 100 different conditions in 11 experiments.

Table 2

Experimental conditions

Experiment T t t log fO2 log fO2
no. (°C) (days) (s) (bars) (ΔQFM)
TODE1 1300 28 ·0 2 ·4156 × 106 –0 ·7 +6 ·6
TODE4 1303 30 ·1 2 ·5992 × 106 –7 ·7 –0 ·4
TODE6 1300 16 ·8 1 ·4481 × 106 –12 ·1 –4 ·8
TODE9 1400 43 ·3 3 ·744 × 106 –0 ·7 +5 ·6
TODE13 1405 20 ·9 1 ·8023 × 106 –3 ·1 +3 ·2
TODE5 1400 18 ·8 1 ·626 × 106 –8 ·0 –1 ·7
TODE7 1403 13 ·7 1 ·182 × 106 –12 ·0 –5 ·7
TODE12 1502 4 ·9 4 ·248 × 105 –0 ·7 +4 ·7
TODE11 1500 3 ·6 3 ·132 × 105 –2 ·7 +2 ·7
TODE14 1500 2 ·3 2 ·0064 × 105 –7 ·0 –1 ·6
TODE8 1499 1 ·9 1 ·656 × 105 –11 ·0 –5 ·6
TODE3(ol) 1301 30 ·1 2 ·5992 × 105 –8 ·8 –1 ·5
Experiment T t t log fO2 log fO2
no. (°C) (days) (s) (bars) (ΔQFM)
TODE1 1300 28 ·0 2 ·4156 × 106 –0 ·7 +6 ·6
TODE4 1303 30 ·1 2 ·5992 × 106 –7 ·7 –0 ·4
TODE6 1300 16 ·8 1 ·4481 × 106 –12 ·1 –4 ·8
TODE9 1400 43 ·3 3 ·744 × 106 –0 ·7 +5 ·6
TODE13 1405 20 ·9 1 ·8023 × 106 –3 ·1 +3 ·2
TODE5 1400 18 ·8 1 ·626 × 106 –8 ·0 –1 ·7
TODE7 1403 13 ·7 1 ·182 × 106 –12 ·0 –5 ·7
TODE12 1502 4 ·9 4 ·248 × 105 –0 ·7 +4 ·7
TODE11 1500 3 ·6 3 ·132 × 105 –2 ·7 +2 ·7
TODE14 1500 2 ·3 2 ·0064 × 105 –7 ·0 –1 ·6
TODE8 1499 1 ·9 1 ·656 × 105 –11 ·0 –5 ·6
TODE3(ol) 1301 30 ·1 2 ·5992 × 105 –8 ·8 –1 ·5

Values of the quartz–fayalite–magnetite (QFM) equilibrium are from O’Neill (1987). TODE3 investigated San Carlos olivine; all others investigated pure forsterite.

#### Time series

To verify that the duration of the diffusion anneal does not appreciably affect the measured diffusivity, two experimental conditions were run for various times. In both cases, the time series were conducted to observe diffusion along the c-axis at ∼1400 °C from the fo–kar–prEn buffer (high aSiO2). The time series experiments were complementary to experiments TODE13 (1405 °C, QFM + 3·2) and TODE5 (1400 °C, QFM – 1·7). For each time series, three crystals were prepared with the standard crystal-powder technique, placed onto a Pt tray and lowered into the pre-heated gas mixing furnace. After some time, the Pt tray was lifted out of the furnace, one crystal was removed and the other two were returned. This was then repeated twice to give three experiments with identical conditions but different durations.

The experiment at QFM + 3·2 was run for 71, 141·3 and 267·7 h (∼3, 6 and 11 days) and at QFM – 1·7 for 59·5, 102·5 and 190·5 h (∼2·5, 4 and 8 days).

#### Diffusion in San Carlos olivine

Preliminary experiments in San Carlos olivine were carried out using randomly oriented crystals, cut then polished using the same method as for the pure forsterite. Four buffer assemblages were sintered, each containing an olivine phase of approximately Fo90 (i.e. San Carlos olivine). These were, in order of increasing aSiO2, olivine–ferropericlase–ulvöspinel (ol–fpr–ulv), olivine–ulvöspinel–ilmenite (ol–ulv–ilm), olivine–ilmenite–armalcolite (ol–ilm–arm) and olivine–armalcolite–orthopyroxene (ol–arm–opx). These were sintered at QFM – 1·5 at 1300 °C for 24 h, then pulverized and pasted onto the crystal surface, as in the forsterite experiments. The diffusion anneal was run for 30 days at the same conditions as for the sinter.

#### Hydroxylating diffusion profiles

A forsterite crystal showing a Ti diffusion profile with the shape of a hockey stick (TODE5; fo–kar–prEn) was equilibrated in hydrous conditions at 850 °C to hydroxylate individual defects, thus allowing analysis by Fourier transform infrared (FTIR) spectroscopy. The experiment was based on the concept of ‘two-stage high-temperature annealing experiments’ of Bai & Kohlstedt (1993) using a simpler experimental design that also allows large single crystals to be preserved without cracking.

The crystal was packed inside the corresponding Ti source buffer with ∼15 mg water inside a 6·3 mm outside diameter ‘cold-seal’ Ag capsule. The capsule design has been described by Hack & Mavrogenes (2006), who modified the design of Woodland & O’Neill (1997). The setup is advantageous in that the thermocouple is directly inside the lid, thus reducing temperature gradients between the point of measurement and the temperature at the charge. The capsule was placed into an MgO–graphite–NaCl assembly, surrounded by Teflon foil, and placed into a 15·9 mm pressure vessel in an end-loaded Boyd-type piston cylinder. The experimental design is depicted in

( are available for downloading at htttp://www.petrology.oxfordjournals.org). Temperature was controlled using a type B thermocouple held inside a mullite sleeve. The temperature was increased at 50 °C per minute to allow the large mass of silver to equilibrate while ramping up. Internal (piston) pressure was ramped up to run conditions over the same time scale as the temperature ramp-up; this is to prevent either over-pressurization or leakage of liquid water from the capsule. No correction was made for the effect of pressure on thermocouple e.m.f.

The charge was held at 850 °C and 1·0 GPa for 48 h, then quenched by turning off the power. The capsule was then opened using a small hacksaw and the partially sintered material was removed from the capsule; this was then ground to expose the crystal, which was doubly polished to around 500 µm thick using 6 µm diamond paste for FTIR analysis.

### Analytical methods

#### LA-ICP-MS measurement of trace element diffusion

Analyses of elemental concentration profiles were conducted by LA-ICP-MS using a Lambda Physik Compex 110 Excimer 193 nm laser coupled to an Agilent 7700 series ICP-MS system at the Australian National University. The beam was set to a 6 μm × 100 μm rectangular slit, and scanned from low to high Ti concentrations across the forsterite crystals. The analytical setup has been described by Jollands et al. (2014).

29Si was counted (0·01 s dwell time per sweep) to act as an internal standard for quantification. 61Ni, 59Co and 55Mn were counted for 0·1 s each per sweep, and 47Ti (± 49Ti; used only in initial analyses) and 48Ti were counted for 0·2 s and 0·05 s per sweep, respectively. In addition, the likely contaminants aluminium and iron were analysed; 27Al and 57Fe were both counted for 0·1 s per sweep. 57Fe was counted for only 0·01 s per sweep in the San Carlos olivine experiments.

The total dwell time summed to a sweep time of ∼0·8 s, and coupled with a motorized stage movement of 1 μm s–1, yielded just over one measurement per micrometre. As the laser slit has a width of around 6 μm, the data from the ICP-MS system can be regarded as an effective moving average, with each point sampled repeatedly as it moves beneath the laser beam. An example of a raw data trace is presented in Fig. 2, which shows counts against distance in micrometres (equivalent to time in seconds, given the stage movement of 1 µm s–1). For each crystal, about five laser traverses were made to ensure diffusion distance did not change laterally across the crystal; that is, to validate the assumption of one-dimensional diffusion. It is important to avoid contamination by adjacent scans owing to sputtering. With such scans omitted, reproducibility between scans is excellent (Fig. 3). The crystal–powder interface was located manually on laser sweeps using Si counts and Ti spikes, as there is a concomitant sharp rise in Ti content and drop in Si counts at the interface.

Fig. 2

(a, b) Raw data trace extracted directly from ICP-MS software (Agilent MassHunter). The scan was conducted from crystal core to rim (diffusion interface). Strong diffusion profiles of the divalent cations along with contaminant Fe are clear, along with a double-step Ti diffusion profile and diffusion-out of Al, inversely correlated with Ti. Mg and Si remain constant throughout the crystal region. The crystal–buffer interface is marked, as is the point at which the laser was turned on (shutter opened). (a) and (b) are from the same transect, split for clarity.

Fig. 2

(a, b) Raw data trace extracted directly from ICP-MS software (Agilent MassHunter). The scan was conducted from crystal core to rim (diffusion interface). Strong diffusion profiles of the divalent cations along with contaminant Fe are clear, along with a double-step Ti diffusion profile and diffusion-out of Al, inversely correlated with Ti. Mg and Si remain constant throughout the crystal region. The crystal–buffer interface is marked, as is the point at which the laser was turned on (shutter opened). (a) and (b) are from the same transect, split for clarity.

Fig. 3

Reproducibility of multiple LA-ICP-MS transects from the same experiment. The Ti concentrations are intentionally offset for illustrative purposes.

Fig. 3

Reproducibility of multiple LA-ICP-MS transects from the same experiment. The Ti concentrations are intentionally offset for illustrative purposes.

The primary standard used was NIST SRM 610 glass, which was analysed in the same session as the samples at least every five scans. NIST SRM 612 was used as a secondary standard. Raw laser data were then processed using Iolite freeware (Paton et al., 2011), which allows continuous visualization of background and standard values over the course of the entire analytical session. The values used for NIST SRM 610 and 612 glass were GeoREM preferred values.

Interference from 48Ca in the NIST610 glass standard on 48Ti counts yields anomalously low 48Ti values on the unknown. An empirical correction for this has been made by taking over 100 000 values of the ratios 47Ti/48Ti and 49Ti/48Ti from various analysis sessions, and taking an average of the interquartile range. This gives a correction factor of 48Ti(μg g–1) × 1·65 = true Ti (μg g–1); this should hold true whenever analyzing calcium-free samples using NIST610 as a primary standard.

#### FTIR analysis of hydrogenated diffusion profile

Unpolarized infrared spectra were obtained with a Bruker Tensor 27 spectrometer attached to a Bruker Hyperion microscope with a liquid nitrogen-cooled MCT detector at the Australian National University. The sample was placed over a hole in an aluminium slide and the microscope was optically focused on the centre of the sample in the vertical axis. The beam was narrowed with a 50 µm × 50 µm aperture, and measurements were taken every 25 µm to give overlapping analyses; analysis positions were controlled using a mapping stage attached to the microscope. Further details have been given by Jollands et al. (2016).

The spectra were corrected for background then for adsorbed water and CO2 bands. The baseline was then subtracted using a concave rubberband method with three iterations and 64 baseline points (all using Bruker OPUS software).

The intensity of silica overtone bands did not vary substantially across the scan, so no thickness correction was made. Furthermore, because orientation and thickness are (approximately) constant across the scan, the change in absorbance is directly related to a change in concentration, such that the relative change in OH associated with each point defect is very precisely determined.

As this experiment was conducted only to determine the speciation of Ti in diffusion experiments, quantification of the concentration of OH associated with different defects is not necessary. Indeed, quantifying total OH from FTIR absorbance is problematic because of considerable uncertainties between different absorption coefficients, which can give uncertainties of over 100% on derived water contents [discussed by Jollands et al. (2016)]. However, ignoring these uncertainties, and using the Bell et al. (2003) calibration for all point defects, we estimate that the water content (as µg g–1 H2O) at the crystal edge is 30–50 µg g–1.

## RESULTS

### Titanium

#### Treatment of diffusion profiles

A principal advantage of using scanning LA-ICP-MS to measure relatively long (hundreds of micrometres) diffusion profiles is that the shape of the profile can be very precisely determined (Fig. 4). The solution of Fick’s law for concentration-independent one-dimensional diffusion predicts a concentration–distance relationship that follows an error function shaped curve (Crank, 1975). However, no such curves were observed for our Ti concentration profiles; only experiments at the lowest fO2 produced profile shapes that approach error-function curves (

). All other diffusion profiles were either ‘hockey stick’ (Fig. 4b), with a constant decrease between interface and background concentrations, or were ‘double steps’ (Fig. 4a), with an initial concentration decrease (from the interface), a plateau, and a second decrease to background values, or were intermediate between these end-members (Fig. 4c), with the plateau expressed as a shallow linear concentration decrease rather than a flat line.
Fig. 4

Examples of processed diffusion profiles. (a) ‘Double-step’ Ti diffusion profile, characteristic of Ti diffusion at high fO2. (b) Hockey-stick diffusion profiles, shown by Ti at low fO2. Anisotropy is also shown, with diffusion profiles from the same experiment along the c-, b- and a-axes. (c) Intermediate shape Ti diffusion profile, where the flat plateau is replaced by a linear decrease. (d) Mn diffusion profile showing excellent agreement with the error function [equation (1); continuous line]. (e) Mn diffusion profile where the interface concentration has been depleted during the experiment. Two fits are shown. In red the theoretical curve [equation (1)] is fitted only to the diffusion-in section. In black, equation (3.26) of Crank (1975) is fitted to the whole curve. The two fits agree within 0·1 orders of magnitude. (f) High Fe contamination at high fO2 leads to Fe having a linear diffusion profile (concentration dependent) and the divalent cations mirroring the profile at lower concentrations. Such Mn, Ni and Co data are not considered when fitting the overall D–T–fO2aSiO2 relationships.

Fig. 4

Examples of processed diffusion profiles. (a) ‘Double-step’ Ti diffusion profile, characteristic of Ti diffusion at high fO2. (b) Hockey-stick diffusion profiles, shown by Ti at low fO2. Anisotropy is also shown, with diffusion profiles from the same experiment along the c-, b- and a-axes. (c) Intermediate shape Ti diffusion profile, where the flat plateau is replaced by a linear decrease. (d) Mn diffusion profile showing excellent agreement with the error function [equation (1); continuous line]. (e) Mn diffusion profile where the interface concentration has been depleted during the experiment. Two fits are shown. In red the theoretical curve [equation (1)] is fitted only to the diffusion-in section. In black, equation (3.26) of Crank (1975) is fitted to the whole curve. The two fits agree within 0·1 orders of magnitude. (f) High Fe contamination at high fO2 leads to Fe having a linear diffusion profile (concentration dependent) and the divalent cations mirroring the profile at lower concentrations. Such Mn, Ni and Co data are not considered when fitting the overall D–T–fO2aSiO2 relationships.

As we cannot fit error-function curves to our diffusion profiles, we cannot calculate the diffusion coefficients (D) according to standard solutions to Fick’s law [see, for example, equation (1) below]. A solution to Fick’s law that accounts for our profile shapes is presented in an accompanying paper (Petrischcheva et al., in preparation).

For the purposes of this study, a semi-quantitative ‘mobility parameter’, the maximum diffusive penetration distance into the crystal, is used in lieu of a diffusion coefficient (D). This is determined visually as the point along the diffusion profile where the Ti concentration first rises above the background crystal value, nominally by at least 1 μg g–1. This approach is permissible only because the Ti concentration does not asymptotically approach the background value at the end of the diffusion profile (as it would if it followed the error function). This penetration value is converted to metres, squared and divided by time (s) to give dimensions of m2 s–1, and will be referred to as the mobility parameter M. As it is an approximation to the diffusion coefficient, its magnitude is similar to those of diffusion coefficients, and it will be treated similarly and referred to using logarithms. The results from the time series (Fig. 5) show that the mobility parameter is nearly independent of time. The conversion of M to D is discussed below where Ni, Co and Mn diffusion values are compared with that of Ti. Values of M, where available for each Ti diffusion condition, are given in Table 3. The quoted errors are standard deviations of generally about five measurements—the errors are extremely small owing to excellent reproducibility of transects across the crystal.

Fig. 5

Diffusion profiles from time series. (a1) Time series conducted at 1400 °C, c-axis, fo–kar–prEn buffer, QFM – 1·7 (59·5, 102·5, 190·5, 451·7 h). The longest profile is from experiment TODE5. (b1) 1400 °C, c-axis, fo–kar–prEn buffer, QFM + 3·2 (71, 141·3, 267·7, 500·6 h). Longest profile is experiment TODE13. (a2, b2) show that time-normalized distance–diffusion can be considered effectively time-independent. The duration of the anneal changes the diffusion distance but not the (apparent) interface concentrations, although the sharp inflexion at the interface makes this difficult to quantify with scanning LA-ICP-MS. Colours represent experimental duration (lightest = longest, darkest = shortest), as shown in (a.1).

Fig. 5

Diffusion profiles from time series. (a1) Time series conducted at 1400 °C, c-axis, fo–kar–prEn buffer, QFM – 1·7 (59·5, 102·5, 190·5, 451·7 h). The longest profile is from experiment TODE5. (b1) 1400 °C, c-axis, fo–kar–prEn buffer, QFM + 3·2 (71, 141·3, 267·7, 500·6 h). Longest profile is experiment TODE13. (a2, b2) show that time-normalized distance–diffusion can be considered effectively time-independent. The duration of the anneal changes the diffusion distance but not the (apparent) interface concentrations, although the sharp inflexion at the interface makes this difficult to quantify with scanning LA-ICP-MS. Colours represent experimental duration (lightest = longest, darkest = shortest), as shown in (a.1).

Table 3

Ti, Mn, Ni and Co diffusion coefficients determined at different T, fO2, buffering assemblage and orientation conditions

log fO2 fo–kar–prEn   fo–gei–kar   fo–qan–gei   fo–per–qan
(ΔQFM) c-axis b-axis a-axis c-axis b-axis a-axis c-axis b-axis a-axis c-axis b-axis a-axis
Ti diffusion [as penetration; log M(m2 s–1)]
1300 °C
+6 ·62 –14 ·9(0 ·10,6) –15 ·4(0 ·14,6) –16 ·0(0 ·03,3) –16 ·0(0 ·18,5) –16 ·2(0 ·35,5) –16 ·6(n.a.,1) –16 ·7(0 ·43,4) –17 ·5(0 ·11,2) –17 ·2(0 ·52,5) –17 ·3(0 ·28,6) –16 ·5(0 ·57,6) –17 ·5(0 ·30,3)
–0 ·40 –13 ·4(0 ·03,3) –14 ·2(0 ·00,4) –14 ·8(0 ·03,4) –14 ·2(0 ·06,4) –15 ·1(0 ·00,3) –15 ·8(0 ·05,4) –15 ·6(0 ·12,4) –16 ·2(0 ·28,4) –17 ·3(0 ·14,4) –15 ·8(0 ·04,4) –16 ·7(0 ·31,4) –17 ·1(0 ·11,4)
–4 ·75 –12 ·7(0 ·01,4) –13 ·5(0 ·02,5) –14 ·3(0 ·03,6) –13 ·6(0 ·03,5) –14 ·5(0 ·03,3) –15 ·1(0 ·11,5) –14 ·2(0 ·04,5) –15 ·1(0 ·07,3) –15 ·6(0 ·14,4) –14 ·6(0 ·04,5) –15 ·6(0 ·09,5) –16 ·0(0 ·11,5)
1400 °C
+5 ·62 –13 ·6(0 ·05,5) –14 ·0(0 ·04,5) –14 ·7(0 ·03,4) –14 ·5(0 ·04,5) –15 ·3(n.a.,1) –16 ·0(0 ·03,2) –15 ·3(0 ·05,5) –15 ·5(0 ·07,3) –16 ·4(0 ·14,3) –15 ·9(0 ·07,5) –16 ·5(0 ·08,3) –16 ·6(0 ·05,3)
+3 ·18 –13 ·6(0 ·01,5) –14 ·3(0 ·01,5) –14 ·8(0 ·03,5) –14 ·7(0 ·02,5) –15 ·3(0 ·03,4) –15 ·9(0 ·03,3) –15 ·4(0 ·03,5) –16 ·0(0 ·12,4) –16 ·7(0 ·21,3) –15 ·8(0 ·05,5) –16 ·4(0 ·08,5) –17 ·0(0 ·14,2)
–1 ·66 –12 ·6(0 ·02,5) –13 ·3(0 ·01,5) n .d. –13 ·4(0 ·01,5) –14 ·1(0 ·03,5) –14 ·9(0 ·04,5) –14 ·3(0 ·03,5) –15 ·0(0 ·04,3) –15 ·6(0 ·03,2) –14 ·6(0 ·05,5) –14 ·4(0 ·53,5) n .d.
–5 ·72 –12 ·0(0 ·03,3) –12 ·8(0 ·00,4) –13 ·5(0 ·02,5) –12 ·3(0 ·03,4) –13 ·1(0 ·03,5) –13 ·7(0 ·07,4) –12 ·7(0 ·04,4) –13 ·5(0 ·05,5) –14 ·1(0 ·08,4) –13 ·1(0 ·14,4) –13 ·9(0 ·18,5) –14 ·5(0 ·11,5)
1500 °C
+4 ·71 n .s. n .s. n .s. –14 ·0(0 ·07,5) n .d. –15 ·2(0 ·08,3) –14 ·5(0 ·05,4) –15 ·1(0 ·13,3) –15 ·7(0 ·16,5) –15 (0 ·13,3) –15 ·5(0 ·2,4) –15 ·9(0 ·29,3)
+2 ·66 n .s. n .s. n .s. –13 ·8(0 ·01,4) –14 ·3(0 ·02,4) –15 ·1(0 ·05,3) –14 ·4(0 ·02,2) –15 ·0(0 ·06,2) –15 ·5(0 ·00,3) –14 ·7(0 ·18,4) –15 ·4(0 ·19,4) –15 ·8(0 ·18,3)
–1 ·59 n .s. n .s. n .s. –12 ·7(0 ·02,5) –13 ·5(0 ·01,5) –14 ·1(0 ·03,5) –13 ·1(0 ·02,5) –13 ·8(0 ·05,6) –14 ·5(0 ·05,3) –13 ·5(0 ·04,6) –14 ·4(0 ·01,5) –15 ·0(0 ·13,4)
–5 ·61 n .s. n .s. n .s. –12 ·1(0 ·02,4) –12 ·9(0 ·03,4) –13 ·4(0 ·02,4) –12 ·1(0 ·02,4) –12 ·9(0 ·03,4) –13 ·5(0 ·05,5) –12 ·6(0 ·05,4) –13 ·4(0 ·01,4) –13 ·9(0 ·03,4)
Mn diffusion [as log D (m2 s–1) fitted to equation (1)]
1300 °C
+6 ·62 –15 ·8(0 ·08,3) –16 ·3(0 ·02,3) –16 ·5(0 ·26,3) –16 ·6(0 ·03,3) –17 ·1(0 ·09,3) –17 ·8(0 ·36,2) –16 ·7(0 ·06,3) n .d. –17 ·8(0 ·08,3) –16 ·7(0 ·09,3) –17 ·7(0 ·43,2) –17 ·6(0 ·21,3)
–0 ·40 –14 ·9(0 ·01,3) –15 ·3(0 ·01,3) –15 ·9(0 ·03,3) –15 ·5(0 ·02,3) –16 ·2(0 ·03,3) –16 ·8(0 ·07,3) –16 ·6(0 ·05,3) –17 ·1(0 ·06,3) –17 ·6(0 ·12,3) –16 ·6(0 ·09,3) –17 ·2(0 ·06,3) –17 ·8(0 ·17,2)
–4 ·75 –14 ·5(0 ·02,2) –14 ·9(0 ·04,3) –15 ·6(0 ·07,3) –15 ·3(0 ·06,2) –16 ·0(0 ·12,3) –16 ·4(0 ·09,3) –15 ·7(0 ·06,3) –16 ·2(0 ·17,3) –16 ·8(0 ·17,3) –16 ·1(0 ·06,3) –16 ·8(0 ·10,3) –17 ·4(0 ·23,3)
1400 °C
+5 ·62 i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c.
+3 ·18 –14 ·6(0 ·01,3) –15 ·2(0 ·04,3) –15 ·7(0 ·08,3) –15 ·5(0 ·03,3) –16 ·1(0 ·11,3) –16 ·6(0 ·06,3) –15 ·8(0 ·02,3) –16 ·3(0 ·04,3) –16 ·7(0 ·04,3) –15 ·9(0 ·00,3) –16 ·0(0 ·02,3) –16 ·7(0 ·10,2)
–1 ·66 –14 ·1(0 ·05,3) –14 ·5(0 ·02,2) –14 ·9(0 ·00,1) –14 ·7(0 ·05,3) –15 ·4(0 ·03,3) –16 ·2(0 ·02,3) –15 ·4(0 ·04,3) –16 ·3(0 ·06,3) –16 ·8(0 ·06,3) –15 ·7(0 ·05,3) –16 ·3(0 ·24,3) b .d.l.
–5 ·72 –13 ·6(0 ·03,3) b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. –14 ·1(0 ·13,2) b .d.l. b .d.l. b .d.l. b .d.l. b .d.l.
1500 °C
+4 ·71 n .s. n .s. n .s. –14 ·9(0 ·07,3) –15 ·2(0 ·12,3) –15 ·8(0 ·13,3) –15 ·1(0 ·08,3) –15 ·7(0 ·13,3) –16 ·1(0 ·06,3) –15 ·2(0 ·03,3) –15 ·5(0 ·17,3) –16 ·1(0 ·1,3)
+2 ·66 n .s. n .s. n .s. –14 ·7(0 ·01,3) –15 ·2(0 ·07,3) –16 ·0(0 ·10,3) –15 ·1(0 ·04,3) –15 ·8(0 ·06,3) –16 ·2(0 ·05,3) –15 ·3(0 ·04,3) –15 ·9(0 ·12,3) –16 ·3(0 ·11,3)
–1 ·59 n .s. n .s. n .s. –14 ·3(0 ·01,3) –14 ·7(0 ·04,3) –15 ·2(0 ·02,3) –14 ·4(0 ·02,3) –14 ·9(0 ·08,3) –15 ·2(0 ·29,3) –14 ·7(0 ·01,3) –15 ·4(0 ·07,3) –16 ·0(0 ·03,3)
–5 ·61 n .s. n .s. n .s. –13 ·6(0 ·02,3) –14 ·1(0 ·02,3) –14 ·7(0 ·05,3) –13 ·6(0 ·05,3) b .d.l. b .d.l. –14 ·1(0 ·06,3) b .d.l. b .d.l.
log fO2 fo–kar–prEn   fo–gei–kar   fo–qan–gei   fo–per–qan
(ΔQFM) c-axis b-axis a-axis c-axis b-axis a-axis c-axis b-axis a-axis c-axis b-axis a-axis
Ti diffusion [as penetration; log M(m2 s–1)]
1300 °C
+6 ·62 –14 ·9(0 ·10,6) –15 ·4(0 ·14,6) –16 ·0(0 ·03,3) –16 ·0(0 ·18,5) –16 ·2(0 ·35,5) –16 ·6(n.a.,1) –16 ·7(0 ·43,4) –17 ·5(0 ·11,2) –17 ·2(0 ·52,5) –17 ·3(0 ·28,6) –16 ·5(0 ·57,6) –17 ·5(0 ·30,3)
–0 ·40 –13 ·4(0 ·03,3) –14 ·2(0 ·00,4) –14 ·8(0 ·03,4) –14 ·2(0 ·06,4) –15 ·1(0 ·00,3) –15 ·8(0 ·05,4) –15 ·6(0 ·12,4) –16 ·2(0 ·28,4) –17 ·3(0 ·14,4) –15 ·8(0 ·04,4) –16 ·7(0 ·31,4) –17 ·1(0 ·11,4)
–4 ·75 –12 ·7(0 ·01,4) –13 ·5(0 ·02,5) –14 ·3(0 ·03,6) –13 ·6(0 ·03,5) –14 ·5(0 ·03,3) –15 ·1(0 ·11,5) –14 ·2(0 ·04,5) –15 ·1(0 ·07,3) –15 ·6(0 ·14,4) –14 ·6(0 ·04,5) –15 ·6(0 ·09,5) –16 ·0(0 ·11,5)
1400 °C
+5 ·62 –13 ·6(0 ·05,5) –14 ·0(0 ·04,5) –14 ·7(0 ·03,4) –14 ·5(0 ·04,5) –15 ·3(n.a.,1) –16 ·0(0 ·03,2) –15 ·3(0 ·05,5) –15 ·5(0 ·07,3) –16 ·4(0 ·14,3) –15 ·9(0 ·07,5) –16 ·5(0 ·08,3) –16 ·6(0 ·05,3)
+3 ·18 –13 ·6(0 ·01,5) –14 ·3(0 ·01,5) –14 ·8(0 ·03,5) –14 ·7(0 ·02,5) –15 ·3(0 ·03,4) –15 ·9(0 ·03,3) –15 ·4(0 ·03,5) –16 ·0(0 ·12,4) –16 ·7(0 ·21,3) –15 ·8(0 ·05,5) –16 ·4(0 ·08,5) –17 ·0(0 ·14,2)
–1 ·66 –12 ·6(0 ·02,5) –13 ·3(0 ·01,5) n .d. –13 ·4(0 ·01,5) –14 ·1(0 ·03,5) –14 ·9(0 ·04,5) –14 ·3(0 ·03,5) –15 ·0(0 ·04,3) –15 ·6(0 ·03,2) –14 ·6(0 ·05,5) –14 ·4(0 ·53,5) n .d.
–5 ·72 –12 ·0(0 ·03,3) –12 ·8(0 ·00,4) –13 ·5(0 ·02,5) –12 ·3(0 ·03,4) –13 ·1(0 ·03,5) –13 ·7(0 ·07,4) –12 ·7(0 ·04,4) –13 ·5(0 ·05,5) –14 ·1(0 ·08,4) –13 ·1(0 ·14,4) –13 ·9(0 ·18,5) –14 ·5(0 ·11,5)
1500 °C
+4 ·71 n .s. n .s. n .s. –14 ·0(0 ·07,5) n .d. –15 ·2(0 ·08,3) –14 ·5(0 ·05,4) –15 ·1(0 ·13,3) –15 ·7(0 ·16,5) –15 (0 ·13,3) –15 ·5(0 ·2,4) –15 ·9(0 ·29,3)
+2 ·66 n .s. n .s. n .s. –13 ·8(0 ·01,4) –14 ·3(0 ·02,4) –15 ·1(0 ·05,3) –14 ·4(0 ·02,2) –15 ·0(0 ·06,2) –15 ·5(0 ·00,3) –14 ·7(0 ·18,4) –15 ·4(0 ·19,4) –15 ·8(0 ·18,3)
–1 ·59 n .s. n .s. n .s. –12 ·7(0 ·02,5) –13 ·5(0 ·01,5) –14 ·1(0 ·03,5) –13 ·1(0 ·02,5) –13 ·8(0 ·05,6) –14 ·5(0 ·05,3) –13 ·5(0 ·04,6) –14 ·4(0 ·01,5) –15 ·0(0 ·13,4)
–5 ·61 n .s. n .s. n .s. –12 ·1(0 ·02,4) –12 ·9(0 ·03,4) –13 ·4(0 ·02,4) –12 ·1(0 ·02,4) –12 ·9(0 ·03,4) –13 ·5(0 ·05,5) –12 ·6(0 ·05,4) –13 ·4(0 ·01,4) –13 ·9(0 ·03,4)
Mn diffusion [as log D (m2 s–1) fitted to equation (1)]
1300 °C
+6 ·62 –15 ·8(0 ·08,3) –16 ·3(0 ·02,3) –16 ·5(0 ·26,3) –16 ·6(0 ·03,3) –17 ·1(0 ·09,3) –17 ·8(0 ·36,2) –16 ·7(0 ·06,3) n .d. –17 ·8(0 ·08,3) –16 ·7(0 ·09,3) –17 ·7(0 ·43,2) –17 ·6(0 ·21,3)
–0 ·40 –14 ·9(0 ·01,3) –15 ·3(0 ·01,3) –15 ·9(0 ·03,3) –15 ·5(0 ·02,3) –16 ·2(0 ·03,3) –16 ·8(0 ·07,3) –16 ·6(0 ·05,3) –17 ·1(0 ·06,3) –17 ·6(0 ·12,3) –16 ·6(0 ·09,3) –17 ·2(0 ·06,3) –17 ·8(0 ·17,2)
–4 ·75 –14 ·5(0 ·02,2) –14 ·9(0 ·04,3) –15 ·6(0 ·07,3) –15 ·3(0 ·06,2) –16 ·0(0 ·12,3) –16 ·4(0 ·09,3) –15 ·7(0 ·06,3) –16 ·2(0 ·17,3) –16 ·8(0 ·17,3) –16 ·1(0 ·06,3) –16 ·8(0 ·10,3) –17 ·4(0 ·23,3)
1400 °C
+5 ·62 i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c.
+3 ·18 –14 ·6(0 ·01,3) –15 ·2(0 ·04,3) –15 ·7(0 ·08,3) –15 ·5(0 ·03,3) –16 ·1(0 ·11,3) –16 ·6(0 ·06,3) –15 ·8(0 ·02,3) –16 ·3(0 ·04,3) –16 ·7(0 ·04,3) –15 ·9(0 ·00,3) –16 ·0(0 ·02,3) –16 ·7(0 ·10,2)
–1 ·66 –14 ·1(0 ·05,3) –14 ·5(0 ·02,2) –14 ·9(0 ·00,1) –14 ·7(0 ·05,3) –15 ·4(0 ·03,3) –16 ·2(0 ·02,3) –15 ·4(0 ·04,3) –16 ·3(0 ·06,3) –16 ·8(0 ·06,3) –15 ·7(0 ·05,3) –16 ·3(0 ·24,3) b .d.l.
–5 ·72 –13 ·6(0 ·03,3) b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. –14 ·1(0 ·13,2) b .d.l. b .d.l. b .d.l. b .d.l. b .d.l.
1500 °C
+4 ·71 n .s. n .s. n .s. –14 ·9(0 ·07,3) –15 ·2(0 ·12,3) –15 ·8(0 ·13,3) –15 ·1(0 ·08,3) –15 ·7(0 ·13,3) –16 ·1(0 ·06,3) –15 ·2(0 ·03,3) –15 ·5(0 ·17,3) –16 ·1(0 ·1,3)
+2 ·66 n .s. n .s. n .s. –14 ·7(0 ·01,3) –15 ·2(0 ·07,3) –16 ·0(0 ·10,3) –15 ·1(0 ·04,3) –15 ·8(0 ·06,3) –16 ·2(0 ·05,3) –15 ·3(0 ·04,3) –15 ·9(0 ·12,3) –16 ·3(0 ·11,3)
–1 ·59 n .s. n .s. n .s. –14 ·3(0 ·01,3) –14 ·7(0 ·04,3) –15 ·2(0 ·02,3) –14 ·4(0 ·02,3) –14 ·9(0 ·08,3) –15 ·2(0 ·29,3) –14 ·7(0 ·01,3) –15 ·4(0 ·07,3) –16 ·0(0 ·03,3)
–5 ·61 n .s. n .s. n .s. –13 ·6(0 ·02,3) –14 ·1(0 ·02,3) –14 ·7(0 ·05,3) –13 ·6(0 ·05,3) b .d.l. b .d.l. –14 ·1(0 ·06,3) b .d.l. b .d.l.
log fO2 fo–prEn–kar   fo–kar–gei   fo–gei–qan   fo–qan–per
(ΔQFM) c-axis b-axis a-axis c-axis b-axis a-axis c-axis b-axis a-axis c-axis b-axis a-axis
Ni diffusion [as log D (m2 s–1) fitted to equation (1)]
1300 °C
+6 ·62 –15 ·9(0 ·12,3) –17 ·2(0 ·04,3) –17 ·3(0 ·27,3) –16 ·9(0 ·04,3) –17 ·5(0 ·17,3) –18 ·1(0 ·49,2) –17 ·1(0 ·10,3) –17 ·6(0 ·12,3) –17 ·8(0 ·18,3) –17 ·0(0 ·18,3) –17 ·6(0 ·46,3) –17 ·5(0 ·26,3)
–0 ·40 –14 ·8(0 ·03,3) –16 ·0(0 ·02,3) –16 ·5(0 ·16,3) –15 ·3(0 ·07,3) –16 ·6(0 ·10,3) –16 ·8(0 ·04,3) –15 ·8(0 ·02,3) –16 ·9(0 ·22,2) –16 ·3(0 ·02,3) –16 ·2(0 ·12,3) –17 ·1(0 ·05,3) –16 ·3(0 ·16,2)
–4 ·75 b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l.
1400 °C
+5 ·62 i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c.
+3 ·18 –14 ·6(0 ·02,3) –15 ·6(0 ·04,3) –16 ·1(0 ·02,3) –15 ·5(0 ·03,3) –16 ·2(0 ·12,3) –16 ·7(0 ·13,3) –15 ·9(0 ·03,3) –16 ·6(0 ·16,3) –17 ·2(0 ·04,3) –15 ·9(0 ·03,3) –16 ·3(0 ·09,3) –17 ·2(0 ·04,2)
–1 ·66 –13 ·9(0 ·05,3) –15 ·0(0 ·01,2) n.d. –14 ·5(0 ·02,3) –15 ·5(0 ·05,3) –15 ·9(0 ·08,3) –15 ·3(0 ·12,2) –16 ·0(0 ·06,3) –15 ·2(0 ·06,2) –15 ·2(0 ·21,3) –16 ·0(0 ·20,3) b .d.l.
–5 ·72 b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l.
1500 °C
+4 ·71 n .s. n .s. n .s. –15 ·0(0 ·02,3) –15 ·4(0 ·10,3) –16 ·2(0 ·41,3) –15 ·1(0 ·16,3) –16 ·0(0 ·31,3) –16 ·5(0 ·08,3) –15 ·2(0 ·06,3) –16 ·1(0 ·37,3) –16 ·5(0 ·1,2)
+2 ·66 n .s. n .s. n .s. –14 ·8(0 ·04,3) –15 ·5(0 ·08,3) –16 ·2(0 ·04,3) –15 ·2(0 ·02,3) –15 ·8(0 ·02,3) –16 ·2(0 ·09,3) –15 ·4(0 ·08,3) –16 ·1(0 ·23,3) –16 ·0(0 ·39,3)
–1 ·59 n .s. n .s. n .s. –14 ·1(0 ·21,3) –14 ·5(0 ·10,3) –15 ·0(0 ·13,3) b .d.l. b .d.l. –15 ·1(0 ·20,3) b .d.l. b .d.l. –14 ·9(0 ·06,3)
–5 ·61 n .s. n .s. n .s. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l.
Co diffusion [as log D(m2 s–1) fitted to equation (1)]
1300 °C
+6 ·62 –15 ·8(0 ·07,3) –16 ·5(0 ·04,3) –16 ·8(0 ·27,3) –16 ·6(0 ·02,3) –17 ·2(0 ·1,3) i .c. –16 ·6(0 ·03,3) –17 ·4(0 ·13,3) –17 ·6(0 ·16,3) –16 ·6(0 ·10,3) –17 ·4(0 ·34,3) –17 ·5(0 ·27,3)
–0 ·40 –14 ·7(0 ·02,3) –15 ·5(0 ·02,3) –16 ·1(0 ·02,3) –15 ·3(0 ·08,3) –16 ·1(0 ·02,3) –16 ·7(0 ·03,3) –16 ·4(0 ·07,3) –17 ·1(0 ·08,3) –17 ·6(0 ·16,3) –16 ·4(0 ·11,3) –17 ·1(0 ·09,3) –17 ·7(0 ·12,2)
–4 ·75 –14 ·3(0 ·03,2) –15 ·0(0 ·04,3) –15 ·8(0 ·07,3) –15 ·1(0 ·00,2) –15 ·9(0 ·15,3) –16 ·5(0 ·1,3) –15 ·7(0 ·12,3) –16 ·4(0 ·14,3) –16 ·7(0 ·16,3) –16 ·0(0 ·05,3) –17 ·2(0 ·62,3) –17 ·5(0 ·37,3)
1400 °C
+5 ·62 i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c.
+3 ·18 –14 ·7(0 ·01,3) –15 ·3(0 ·05,3) –15 ·8(0 ·07,3) –15 ·3(0 ·04,3) –16 ·1(0 ·12,3) –16 ·8(0 ·08,3) –15 ·7(0 ·01,3)   –15 ·8(0 ·00,3) –16 ·1(0 ·02,3) –16 ·9(0 ·08,2)
–1 ·66 –13 ·8(0 ·06,3) –14 ·6(0 ·03,2)  –14 ·5(0 ·05,3) –15 ·3(0 ·04,3) –16 ·0(0 ·02,3) –15 ·4(0 ·11,3) –16 ·4(0 ·05,3) –16 ·8(0 ·06,3) –15 ·6(0 ·09,3) –16 ·1(0 ·15,3) b .d.l.
–5 ·72 –13 ·4(0 ·07,3) b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. –15 ·4(0 ·04,3) –16 ·1(0 ·09,3) –16 ·4(0 ·32,3) b .d.l. b .d.l. b .d.l.
1500 °C
+4 ·71 n .s. n .s. n .s. –14 ·8(0 ·03,3) –15 ·3(0 ·11,3) –16 ·0(0 ·14,3) b .s.r. b .s.r. b .s.r. –15 ·2(0 ·03,3) –15 ·6(0 ·18,3) –16 ·3(0 ·11,3)
+2 ·66 n .s. n .s. n .s. –14 ·8(0 ·02,3) –15 ·2(0 ·07,3) –16 ·1(0 ·09,3) –15 ·0(0 ·08,3) –15 ·0(0 ·07,3) –16 ·3(0 ·06,3) –15 ·2(0 ·04,3) –15 ·9(0 ·11,3) –16 ·4(0 ·11,3)
–1 ·59 n .s. n .s. n .s. –14 ·1(0 ·03,3) –14 ·8(0 ·05,3) –15 ·3(0 ·08,3) –15 ·2(0 ·07,3) –15 ·8(0 ·07,3) –16 ·2(0 ·07,3) –14 ·6(0 ·08,3) –15 ·4(0 ·12,3) –16 ·1(0 ·06,3)
–5 ·61 n .s. n .s. n .s. –13 ·4(0 ·04,3) –14 ·1(0 ·1,3) –14 ·6(0 ·08,3) –14 ·3(0 ·05,3) –15 ·7(0 ·13,3) –15 ·3(0 ·26,3) b .d.l. b .d.l. b .d.l.
log fO2 fo–prEn–kar   fo–kar–gei   fo–gei–qan   fo–qan–per
(ΔQFM) c-axis b-axis a-axis c-axis b-axis a-axis c-axis b-axis a-axis c-axis b-axis a-axis
Ni diffusion [as log D (m2 s–1) fitted to equation (1)]
1300 °C
+6 ·62 –15 ·9(0 ·12,3) –17 ·2(0 ·04,3) –17 ·3(0 ·27,3) –16 ·9(0 ·04,3) –17 ·5(0 ·17,3) –18 ·1(0 ·49,2) –17 ·1(0 ·10,3) –17 ·6(0 ·12,3) –17 ·8(0 ·18,3) –17 ·0(0 ·18,3) –17 ·6(0 ·46,3) –17 ·5(0 ·26,3)
–0 ·40 –14 ·8(0 ·03,3) –16 ·0(0 ·02,3) –16 ·5(0 ·16,3) –15 ·3(0 ·07,3) –16 ·6(0 ·10,3) –16 ·8(0 ·04,3) –15 ·8(0 ·02,3) –16 ·9(0 ·22,2) –16 ·3(0 ·02,3) –16 ·2(0 ·12,3) –17 ·1(0 ·05,3) –16 ·3(0 ·16,2)
–4 ·75 b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l.
1400 °C
+5 ·62 i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c.
+3 ·18 –14 ·6(0 ·02,3) –15 ·6(0 ·04,3) –16 ·1(0 ·02,3) –15 ·5(0 ·03,3) –16 ·2(0 ·12,3) –16 ·7(0 ·13,3) –15 ·9(0 ·03,3) –16 ·6(0 ·16,3) –17 ·2(0 ·04,3) –15 ·9(0 ·03,3) –16 ·3(0 ·09,3) –17 ·2(0 ·04,2)
–1 ·66 –13 ·9(0 ·05,3) –15 ·0(0 ·01,2) n.d. –14 ·5(0 ·02,3) –15 ·5(0 ·05,3) –15 ·9(0 ·08,3) –15 ·3(0 ·12,2) –16 ·0(0 ·06,3) –15 ·2(0 ·06,2) –15 ·2(0 ·21,3) –16 ·0(0 ·20,3) b .d.l.
–5 ·72 b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l.
1500 °C
+4 ·71 n .s. n .s. n .s. –15 ·0(0 ·02,3) –15 ·4(0 ·10,3) –16 ·2(0 ·41,3) –15 ·1(0 ·16,3) –16 ·0(0 ·31,3) –16 ·5(0 ·08,3) –15 ·2(0 ·06,3) –16 ·1(0 ·37,3) –16 ·5(0 ·1,2)
+2 ·66 n .s. n .s. n .s. –14 ·8(0 ·04,3) –15 ·5(0 ·08,3) –16 ·2(0 ·04,3) –15 ·2(0 ·02,3) –15 ·8(0 ·02,3) –16 ·2(0 ·09,3) –15 ·4(0 ·08,3) –16 ·1(0 ·23,3) –16 ·0(0 ·39,3)
–1 ·59 n .s. n .s. n .s. –14 ·1(0 ·21,3) –14 ·5(0 ·10,3) –15 ·0(0 ·13,3) b .d.l. b .d.l. –15 ·1(0 ·20,3) b .d.l. b .d.l. –14 ·9(0 ·06,3)
–5 ·61 n .s. n .s. n .s. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. b .d.l.
Co diffusion [as log D(m2 s–1) fitted to equation (1)]
1300 °C
+6 ·62 –15 ·8(0 ·07,3) –16 ·5(0 ·04,3) –16 ·8(0 ·27,3) –16 ·6(0 ·02,3) –17 ·2(0 ·1,3) i .c. –16 ·6(0 ·03,3) –17 ·4(0 ·13,3) –17 ·6(0 ·16,3) –16 ·6(0 ·10,3) –17 ·4(0 ·34,3) –17 ·5(0 ·27,3)
–0 ·40 –14 ·7(0 ·02,3) –15 ·5(0 ·02,3) –16 ·1(0 ·02,3) –15 ·3(0 ·08,3) –16 ·1(0 ·02,3) –16 ·7(0 ·03,3) –16 ·4(0 ·07,3) –17 ·1(0 ·08,3) –17 ·6(0 ·16,3) –16 ·4(0 ·11,3) –17 ·1(0 ·09,3) –17 ·7(0 ·12,2)
–4 ·75 –14 ·3(0 ·03,2) –15 ·0(0 ·04,3) –15 ·8(0 ·07,3) –15 ·1(0 ·00,2) –15 ·9(0 ·15,3) –16 ·5(0 ·1,3) –15 ·7(0 ·12,3) –16 ·4(0 ·14,3) –16 ·7(0 ·16,3) –16 ·0(0 ·05,3) –17 ·2(0 ·62,3) –17 ·5(0 ·37,3)
1400 °C
+5 ·62 i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c. i .c.
+3 ·18 –14 ·7(0 ·01,3) –15 ·3(0 ·05,3) –15 ·8(0 ·07,3) –15 ·3(0 ·04,3) –16 ·1(0 ·12,3) –16 ·8(0 ·08,3) –15 ·7(0 ·01,3)   –15 ·8(0 ·00,3) –16 ·1(0 ·02,3) –16 ·9(0 ·08,2)
–1 ·66 –13 ·8(0 ·06,3) –14 ·6(0 ·03,2)  –14 ·5(0 ·05,3) –15 ·3(0 ·04,3) –16 ·0(0 ·02,3) –15 ·4(0 ·11,3) –16 ·4(0 ·05,3) –16 ·8(0 ·06,3) –15 ·6(0 ·09,3) –16 ·1(0 ·15,3) b .d.l.
–5 ·72 –13 ·4(0 ·07,3) b .d.l. b .d.l. b .d.l. b .d.l. b .d.l. –15 ·4(0 ·04,3) –16 ·1(0 ·09,3) –16 ·4(0 ·32,3) b .d.l. b .d.l. b .d.l.
1500 °C
+4 ·71 n .s. n .s. n .s. –14 ·8(0 ·03,3) –15 ·3(0 ·11,3) –16 ·0(0 ·14,3) b .s.r. b .s.r. b .s.r. –15 ·2(0 ·03,3) –15 ·6(0 ·18,3) –16 ·3(0 ·11,3)
+2 ·66 n .s. n .s. n .s. –14 ·8(0 ·02,3) –15 ·2(0 ·07,3) –16 ·1(0 ·09,3) –15 ·0(0 ·08,3) –15 ·0(0 ·07,3) –16 ·3(0 ·06,3) –15 ·2(0 ·04,3) –15 ·9(0 ·11,3) –16 ·4(0 ·11,3)
–1 ·59 n .s. n .s. n .s. –14 ·1(0 ·03,3) –14 ·8(0 ·05,3) –15 ·3(0 ·08,3) –15 ·2(0 ·07,3) –15 ·8(0 ·07,3) –16 ·2(0 ·07,3) –14 ·6(0 ·08,3) –15 ·4(0 ·12,3) –16 ·1(0 ·06,3)
–5 ·61 n .s. n .s. n .s. –13 ·4(0 ·04,3) –14 ·1(0 ·1,3) –14 ·6(0 ·08,3) –14 ·3(0 ·05,3) –15 ·7(0 ·13,3) –15 ·3(0 ·26,3) b .d.l. b .d.l. b .d.l.

The numbers outside and inside the parentheses, respectively, are the mean diffusion coefficient and standard deviation from repeated measurements and number of repeats. i .c., iron contamination, which occurred in the 1400 °C air experiment. Ti diffusion values are still quoted for this experiment but may be accelerated by Fe. n .s., not stable, referring to the fo–prEn–kar buffer being partially melted at 1500 °C. b .d.l., below detection limit, generally relating to low interface concentrations of Ni, Co and Mn at low fO2 and high T. b .s.r., below spatial resolution of the laser beam—profiles shorter than around 20 μm cannot realistically be measured. n.d., not determined for miscellaneous reasons. n.a., not applicable; standard deviations are not quoted where only one measurement was made.

#### Temperature

Diffusion is faster at higher temperatures; a 100 °C temperature increase leads to 1–1·5 orders of magnitude faster diffusion (e.g. Fig. 6, Table 3). When all other variables are fixed, the variation of log M with inverse temperature is nearly linear, as expected for an Arrhenius process. Temperature is only one of the variables affecting diffusion in this case; the orientation must be taken into account, along with aSiO2 and fO2, which themselves vary as a function of temperature.

Fig. 6

(a) Relationship between inverse temperature and diffusive penetration for Ti, along c-axis, at similar fO2 conditions. (b) Logarithm of aSiO2 of the four buffering assemblages over the studied temperatures, calculated using thermodynamic data from Hermann et al. (2005) and Holland & Powell (2011). The convergence in Ti diffusivity at high temperatures between the two intermediate buffers reflects the aSiO2 convergence at the same conditions.

Fig. 6

(a) Relationship between inverse temperature and diffusive penetration for Ti, along c-axis, at similar fO2 conditions. (b) Logarithm of aSiO2 of the four buffering assemblages over the studied temperatures, calculated using thermodynamic data from Hermann et al. (2005) and Holland & Powell (2011). The convergence in Ti diffusivity at high temperatures between the two intermediate buffers reflects the aSiO2 convergence at the same conditions.

#### Orientation effects

Diffusion is anisotropic (Figs 4c and 7), with diffusion along c > b > a (i.e. fastest parallel to the c-axis). The change is around one log M unit difference as a function of orientation, with regular, systematic variations (Fig. 7). There does not appear to be a resolvable effect of fO2 on diffusive anisotropy and orientation exerts no obvious control over profile shape, although the double step is well resolvable only in long (generally > 100 µm) profiles (see Fig. 7a, inset, where the a-axis profile shows only a slight step).

Fig. 7

Anisotropy of Ti diffusion as a function of crystallographic axis. (a) Example Arrhenius plot showing Ti diffusive penetration as a function of temperature and crystallographic axis. Inset shows the diffusion profiles of the 1400 °C experiments. (b) All Ti diffusion data showing the difference in diffusive penetration from the c- and a- or b-axes. As the experimental data were collected over a range of fO2, silica activity and temperature conditions, the coefficients vary systematically and together over five orders of magnitude.

Fig. 7

Anisotropy of Ti diffusion as a function of crystallographic axis. (a) Example Arrhenius plot showing Ti diffusive penetration as a function of temperature and crystallographic axis. Inset shows the diffusion profiles of the 1400 °C experiments. (b) All Ti diffusion data showing the difference in diffusive penetration from the c- and a- or b-axes. As the experimental data were collected over a range of fO2, silica activity and temperature conditions, the coefficients vary systematically and together over five orders of magnitude.

#### Buffer assemblage

The chemical activity of the buffer assemblage has a significant effect on the Ti diffusivity, profile shape and interface concentrations in olivine. To illustrate this, a compilation of profiles along the c-axis at 1400 °C is shown in Fig. 8 with one diffusion profile for each fO2 and buffering assemblage. A more complete set of c-axis diffusion data in presented in

.
Fig. 8

Compilation of Ti diffusion profiles measured at 1400 °C along the c-axis. The figures represent different fO2 conditions from log fO2 = –0·68 (a) to log fO2 = –12 (d). In each figure, the colours represent different aSiO2 conditions; orange, fo–per–qan (lowest aSiO2); pink, fo–qan–gei; purple, fo–gei–kar; blue, fo–kar–prEn (highest aSiO2). The strong positive influence of aSiO2 on the diffusion profile length should be noted. The profile shape changes systematically from stepped at high fO2 to near-linear at low fO2, with aSiO2 also influencing profile shapes at moderate fO2. The Ti concentrations on the steps are approximately constant at 15–20 μg g–1 regardless of fO2 or aSiO2.

Fig. 8

Compilation of Ti diffusion profiles measured at 1400 °C along the c-axis. The figures represent different fO2 conditions from log fO2 = –0·68 (a) to log fO2 = –12 (d). In each figure, the colours represent different aSiO2 conditions; orange, fo–per–qan (lowest aSiO2); pink, fo–qan–gei; purple, fo–gei–kar; blue, fo–kar–prEn (highest aSiO2). The strong positive influence of aSiO2 on the diffusion profile length should be noted. The profile shape changes systematically from stepped at high fO2 to near-linear at low fO2, with aSiO2 also influencing profile shapes at moderate fO2. The Ti concentrations on the steps are approximately constant at 15–20 μg g–1 regardless of fO2 or aSiO2.

Ti diffusion is faster and apparent interface concentrations are higher when the external assemblage has a higher activity of silica (Figs 6 and 8). In the high-fO2 experiments showing the double-step shape, diffusive penetration is over an order of magnitude greater when the external buffer assemblage contains protoenstatite (fo–kar–prEn) compared with when the buffer contains periclase (fo–per–qan). The difference in penetration distance between these assemblages is slightly lower in the more reducing experiments. At 1300 and 1400 °C, the diffusion profiles from the low-aSiO2 buffers fo–per–qan and fo–qan–gei are of very similar lengths, overlapping in some cases, but they diverge in the 1500 °C experiments (Fig. 6). This relates to the change in aSiO2 for each buffer as a function of temperature (Fig. 6). Silica activity was calculated using a combination of free energy data from Hermann et al. (2005) for qandilite and karooite and from Holland & Powell (2011) for all other phases (assuming SiO2 is present as beta quartz and Mg2Si2O6 is protoenstatite). This method of calculating aSiO2 has been described by Zhukova et al. (2014). In one experiment (TODE5), aSiO2 affects the profile shape; the highest silica activity experiment shows a hockey-stick profile, whereas low silica activity experiments show the double-step shape (Fig. 8c).

#### Oxygen fugacity

The rate of Ti diffusion increases considerably as fO2 is decreased: we observe nearly three orders of magnitude difference in M over ∼11 log units fO2 at 1300 and 1400 °C. At high oxygen fugacity, the profiles tend towards the ‘double-step’ shape (Fig. 8a), but at very reducing conditions, they become hockey-stick shaped (Fig. 8d). The change of log M as a function of log fO2 is sigmoidal, with smaller changes at high- and low-fO2 conditions and greatest change in the central region (around QFM). This will be further discussed below.

#### Interface concentrations

The concentrations at the crystal–buffer interface should be the equilibrium concentrations imposed by the buffer paragenesis as discussed by Hermann et al. (2005). However, the concentrations found from extrapolation of our measured concentration profiles (Table 4) are well below those expected from the study of Hermann et al. (2005). Furthermore, the expected dependence of interface concentrations on the activity of silica is only partly resolved. Comparing only periclase- and protoenstatite-buffered experiments, the concentration of Ti at the crystal edge increases as the activity of silica increases. There is, however, considerable uncertainty, with overlap between concentrations in the low-aSiO2 conditions (Fig. 9a, Table 3). Titanium concentrations increase as a function of decreasing oxygen fugacity but show no clear relationship with temperature. The interface concentrations are in the tens to hundreds of μg g–1 range, about two orders of magnitude lower than the equilibrium concentrations measured by Hermann et al. (2005) (Fig. 9b). In the case of the hockey-stick shaped profiles, or the classical error function profiles, the concentration can be projected back to the crystal edge (with relatively high precision) using a curve fit. For the double-step shape diffusion profiles, with a sharp rise in Ti concentration near the crystal edge, precise determination of the interface concentration using data from scanning LA-ICP-MS is difficult, but order-of-magnitude estimates should nevertheless be possible. However, the scale of our LA-ICP-MS scanning method is too coarse to pick up diffusion length scales of a few microns, and it may be that the apparent discrepancy between the interface concentrations obtained by extrapolating our observable profiles and the Ti equilibrium concentrations of Hermann et al. (2005) is due to very slow diffusion of Ti on tetrahedral sites, which we will discuss further below.

Table 4

Interface concentrations of Ti, all axes, at all fO2 and aSiO2 conditions, determined from concentration–distance plots

Buffer ln aSiO2 log fO2

–0 ·7 ∼ –3 ∼ –7 ∼ –12
1300 °C
fo–per–qan –4 ·4 35  (28)  36  (5) 246  (89)
fo–qan–gei –3 ·9 33  (11)  50  (8) 63  (33)
fo–gei–kar –2 38  (21)  40  (8) 83  (30)
fo–kar–prEn –0 ·4 36  (13)  29  (3) 388  (34)
1400 °C
fo–per–qan –4 ·6 15  (10) 23  (14) 61  (20) 94  (35)
fo–qan–gei –3 ·4 17  (6) 15  (4) 57  (7) 143  (23)
fo–gei–kar –2 24  (9) 21  (6) 77  (3) 307  (44)
fo–kar–prEn –0 ·4 57  (14) 49  (14) 221  (53) 350  (46)
1500 °C
fo–per–qan –3 ·9 16  (9) 29  (16) 58  (25) 60  (14)
fo–qan–gei –3 ·2 22  (6) 27  (7) 31  (7) 203  (43)
fo–gei–kar –2 ·2 27  (6) 36  (12) 24  (4) 294  (91)
Buffer ln aSiO2 log fO2

–0 ·7 ∼ –3 ∼ –7 ∼ –12
1300 °C
fo–per–qan –4 ·4 35  (28)  36  (5) 246  (89)
fo–qan–gei –3 ·9 33  (11)  50  (8) 63  (33)
fo–gei–kar –2 38  (21)  40  (8) 83  (30)
fo–kar–prEn –0 ·4 36  (13)  29  (3) 388  (34)
1400 °C
fo–per–qan –4 ·6 15  (10) 23  (14) 61  (20) 94  (35)
fo–qan–gei –3 ·4 17  (6) 15  (4) 57  (7) 143  (23)
fo–gei–kar –2 24  (9) 21  (6) 77  (3) 307  (44)
fo–kar–prEn –0 ·4 57  (14) 49  (14) 221  (53) 350  (46)
1500 °C
fo–per–qan –3 ·9 16  (9) 29  (16) 58  (25) 60  (14)
fo–qan–gei –3 ·2 22  (6) 27  (7) 31  (7) 203  (43)
fo–gei–kar –2 ·2 27  (6) 36  (12) 24  (4) 294  (91)

Mean and 1σ are given; these are calculated after rejecting anomalous values generally > 100% away from median.

Fig. 9

Interface concentrations of Ti as a function of temperature. (a) Data from this study at around QFM – 1·5, for four aSiO2 buffers. The lack of temperature dependence is interpreted as a result of unconstrained Al contamination at the interface, allowing Ti–Al coupled substitution. (b) Data from this study (qandilite–geikielite buffer) compared with Ti solubility data (qandilite-buffered) from Hermann et al. (2005) and interface concentrations from a qandilite-buffered diffusion study determined by Rutherford backscattering spectroscopy (RBS) (Cherniak & Liang, 2014). Interface concentrations from this study (Ti on M-site and Ti–Al coupled substitution) are far below the maximum solubility (Ti on T-site) where diffusion is too slow to be seen by LA-ICP-MS. The diffusion rate of Ti on the T-sites should give profiles comfortably within the range of RBS, but the interface concentrations from Cherniak & Liang (2014) do not change over >500 °C and do not agree with those from the solubility study. We suggest that these reported interface concentrations may be analytical artefacts.

Fig. 9

Interface concentrations of Ti as a function of temperature. (a) Data from this study at around QFM – 1·5, for four aSiO2 buffers. The lack of temperature dependence is interpreted as a result of unconstrained Al contamination at the interface, allowing Ti–Al coupled substitution. (b) Data from this study (qandilite–geikielite buffer) compared with Ti solubility data (qandilite-buffered) from Hermann et al. (2005) and interface concentrations from a qandilite-buffered diffusion study determined by Rutherford backscattering spectroscopy (RBS) (Cherniak & Liang, 2014). Interface concentrations from this study (Ti on M-site and Ti–Al coupled substitution) are far below the maximum solubility (Ti on T-site) where diffusion is too slow to be seen by LA-ICP-MS. The diffusion rate of Ti on the T-sites should give profiles comfortably within the range of RBS, but the interface concentrations from Cherniak & Liang (2014) do not change over >500 °C and do not agree with those from the solubility study. We suggest that these reported interface concentrations may be analytical artefacts.

#### San Carlos olivine experiments

As experiments using San Carlos olivine were conducted only to observe profile shapes, the experiments were conducted at a single temperature (1300 °C) and fO2 (QFM – 1·5) condition, without consideration given to crystal orientation. Nevertheless, it is clear from the results that diffusion varies as a function of aSiO2, with diffusion faster under orthopyroxene-buffered conditions (high aSiO2) compared with ferropericlase-buffered assemblages (Fig. 10). The diffusion profile shapes are similar to those observed in pure forsterite; at high aSiO2 the profiles are intermediate to stepped, and at low aSiO2 the profiles appear error function shaped, but are too short to constrain the shape precisely. The amount of Ti on the step is higher than in the pure forsterite; around 100 μg g–1 at the inflection down to background values. Apparent Ti interface concentrations vary between around 50 μg g–1 at low aSiO2 to around 150–200 μg g–1 at high aSiO2. The diffusion rate is similar to that observed in pure forsterite, but not quantifiable given the preliminary nature of the San Carlos olivine experiments.

Fig. 10

Ti diffusion profiles from experiments on randomly oriented San Carlos olivine at 1300 °C, QFM – 1·5. The fastest diffusion and highest apparent interface concentrations correspond to the highest aSiO2 buffer ol–arm–opx (equivalent to fo–kar–prEn in Fe-free experiments).

Fig. 10

Ti diffusion profiles from experiments on randomly oriented San Carlos olivine at 1300 °C, QFM – 1·5. The fastest diffusion and highest apparent interface concentrations correspond to the highest aSiO2 buffer ol–arm–opx (equivalent to fo–kar–prEn in Fe-free experiments).

#### Hydroxylation experiment

The intensity of all peaks in the structurally bound OH region of the IR spectra decreases from rim to core of the analysed crystal (Fig. 11). The Si–O overtones on the FTIR spectra are comprised of that representing [001] (the direction of Ti flux) along with some contribution from [010] and [100]. In other words, the crystal was slightly rotated around the [001] axis during or before the experiment, such that half of the absorbance is from the E//[001] contribution and the other half is a combination of E//[010] and E//[100]; this is fortuitous as all possible peaks are present in the measured spectra. The peaks at 3525 and 3572 cm–1 were attributed to the Ti-clinohumite (MgTi4+H2O4) point defect (Berry et al., 2005, 2007b; Hermann et al., 2007; Padrón-Navarta et al., 2014), the strong peak at 3350 cm–1 is interpreted as H+ charge balancing Ti3+ (Ti3+H+SiO4; Berry et al., 2007a), and the low broad peak at 3160 cm–1 is consistent with hydroxylated oxygen associated with an M1-site vacancy (MgH2SiO4; e.g. Lemaire et al., 2004). The relative change in absorbance is directly related to the change in concentration of the hydroxylated point defect away from the interface, given that the thickness and orientation of the sample are the same across the transect. This then gives an indication of the original concentration of the relevant anhydrous point defect, but it should be noted that not all anhydrous Ti3+ point defects are hydroxylated at 850 °C, 1·5 GPa, but probably all or most of the Ti4+ defects are (see Jollands et al., 2016). The Ti-clinohumite point defect profile shows a step similar to the Ti profiles from experiments at higher fO2, and the M-site vacancy profile has a similar hockey-stick shape to the Ti3+ defect. By comparing the integrated areas underneath these peaks with the LA-ICP-MS trace from the same crystal (Fig. 11), it is clear that the hydroxylated Ti3+ profile closely resembles the total Ti profile—both have the hockey-stick profile shape.

Fig. 11

FTIR data from hydroxylation of a diffusion profile (TODE5, 1400 °C, QFM – 1·7, fo–kar–prEn buffer). This profile was selected for hydroxylation as the first example of a linear profile as fO2 is decreased. The spectra are composed of a Ti-clinohumite point defect (Ti4+) (doublet), a strong Ti3+ defect and a low, broad Mg vacancy defect. The linear LA-ICP-MS profile can be deconvolved into two Ti profiles; Ti3+ is linear and Ti4+ is stepped. Further description of the hydrous defects, including potential formation reactions, is given in the text.

Fig. 11

FTIR data from hydroxylation of a diffusion profile (TODE5, 1400 °C, QFM – 1·7, fo–kar–prEn buffer). This profile was selected for hydroxylation as the first example of a linear profile as fO2 is decreased. The spectra are composed of a Ti-clinohumite point defect (Ti4+) (doublet), a strong Ti3+ defect and a low, broad Mg vacancy defect. The linear LA-ICP-MS profile can be deconvolved into two Ti profiles; Ti3+ is linear and Ti4+ is stepped. Further description of the hydrous defects, including potential formation reactions, is given in the text.

### Mn, Ni, Co diffusion

#### Treatment of diffusion profiles

In many cases, the diffusion of Mn, Co and Ni can be fitted to a simple error function [equation (1)], assuming diffusion is one-dimensional through a semi-infinite medium from an infinite source reservoir (Fig. 4d) [rearranged from equation (3.13) of Crank (1975)]:

(1)
$C(x)=Ci+(C0−Ci).erfc(x4Dt).$
Here C(x) is the concentration of the diffusant at position x (in metres), C0 is the interface concentration, Ci is the background concentration, t is the time in seconds and D is the diffusion coefficient in m2 s–1. However, in some experiments, the concentration profile shows a hump, decreasing towards the interface region (Fig. 4e), suggesting that the source buffer is not behaving as an infinite reservoir. This problem was also encountered by Chakraborty et al. (1994) and Zhukova et al. (2014). Such profiles can be generated if the source concentration at the interface is steadily depleted during the time scale of the experiment. Crank (1975) gave an equation for a linear decrease with time [his equation (3.26)]. Zhukova et al. (2014) showed by numerical simulation that the error in D caused by fitting such profiles using equation (1) was less than 0·2 log units, similar to the typical standard deviation obtained from replicate measurements (see Zhukova et al., 2014, fig. 6). With this in mind, and also because we do not know if the rate of depletion of the source concentration was linear (an exponential decrease towards zero concentration seems physically more reasonable), we follow Chakraborty et al. (1994) and Zhukova et al. (2014) in fitting all profiles to equation (1) for the innermost part of the profile that is unaffected by this complication. An example of a depleting source profile fitted to both equations is given in Fig. 4e, showing agreement within 0·1 orders of magnitude.

In an experiment where Fe was a major contaminant (concentration greater than that of Mn, Ni and Co) and the fO2 is high (TODE9, 1400 °C, air), near hockey-stick profiles are observed for Fe, and these are mimicked by the other cations at lower concentrations (Fig. 4f). In the most reducing, highest temperature experiments where native metal is present in the buffer, Ni and Co are below detection limit, indicating that the solubilities of these cations in forsterite under these conditions are too low to be detected by our analytical method (∼1 µg g–1 limit of detection for Mn and Co; ∼10 µg g–1 for Ni, given the low abundance of 61Ni).

#### Effect of buffer assemblage

As with Ti, diffusion of Co, Ni and Mn varies with aSiO2, with diffusivities at protoenstatite-buffered conditions (high aSiO2) around 1 to 1.5 orders of magnitude faster than at the lowest aSiO2 (highest aMgO) (

), in agreement with Zhukova et al. (2014). The buffer assemblage does not exert any clear control on profile shape; the profiles consistently fit error function or depleted error function curves.

#### Effect of temperature, orientation and fO2

A series of Arrhenius plots for Mn, Ni and Co diffusion, where attempts have been made to fix all variables except temperature and one other, are presented in

. These show that diffusion coefficients systematically increase with temperature following linear Arrhenius relationships. Mn, Ni and Co diffuse faster as fO2 is decreased. The variation is around one order of magnitude over the studied range. Diffusion is fastest along the c-axis, and slowest along the a-axis with around one order of magnitude difference between the two. Taken together, the total variation of Ni, Co or Mn diffusivity at any temperature is over 4–5 orders of magnitude. Experiments conducted along the a-axis buffered by periclase at high fO2 show diffusion 4–5 log D units slower than experiments along the c-axis, buffered by enstatite at low fO2. The concentrations of Mn, Ni and Co are unbuffered in these experiments, and given that at low fO2 the interface concentrations decrease over the course of the anneal, only the interface concentrations from the high-fO2 experiments (air and pure CO2) are given in Table 5
Table 5

Interface concentrations of Mn, Ni and Co from c-axis experiments in either pure CO2 or air  (high fO2)

Buffer aSiO2 ln aSiO2 Mn Ni Co
1300 °C, log fO2 –0 ·68
fo–per–qan 0 ·01 –4 ·4 105  (25) 163  (29) 175  (23)
fo–qan–gei 0 ·02 –3 ·9 85  (13) 233  (32) 175  (25)
fo–gei–kar 0 ·13 –2 ·0 208  (33) 607  (105) 358  (53)
fo–kar–prEn 0 ·67 –0 ·4 390  (75) 893  (49) 745  (48)
1400 °C, log fO2 –3 ·07
fo–per–qan 0 ·01 –4 ·6 250  (13) 255  (67) 267  (15)
fo–qan–gei 0 ·03 –3 ·4 205  (5) 557  (32) 457  (167)
fo–gei–kar 0 ·13 –2 ·0 413  (21) 1060  (40) 577  (31)
fo–kar–prEn 0 ·65 –0 ·4 390  (10) 847  (50) 628  (10)
1500 °C, log fO2 –0 ·68
fo–per–qan 0 ·02 –3 ·9 152  (10) 217  (40) 213  (13)
fo–qan–gei 0 ·04 –3 ·2 132  (10) 358  (58) 203  (21)
fo–gei–kar 0 ·11 –2 ·2 192  (24) 520  (61) 212  (28)
1500 °C, log fO2 –2 ·74
fo–per–qan 0 ·02 –3 ·9 323  (15) 252  (64) 322  (10)
fo–qan–gei 0 ·04 –3 ·2 237  (6) 397  (21) 380  (18)
fo–gei–kar 0 ·11 –2 ·2 480  (10) 668  (28) 643  (6)
Buffer aSiO2 ln aSiO2 Mn Ni Co
1300 °C, log fO2 –0 ·68
fo–per–qan 0 ·01 –4 ·4 105  (25) 163  (29) 175  (23)
fo–qan–gei 0 ·02 –3 ·9 85  (13) 233  (32) 175  (25)
fo–gei–kar 0 ·13 –2 ·0 208  (33) 607  (105) 358  (53)
fo–kar–prEn 0 ·67 –0 ·4 390  (75) 893  (49) 745  (48)
1400 °C, log fO2 –3 ·07
fo–per–qan 0 ·01 –4 ·6 250  (13) 255  (67) 267  (15)
fo–qan–gei 0 ·03 –3 ·4 205  (5) 557  (32) 457  (167)
fo–gei–kar 0 ·13 –2 ·0 413  (21) 1060  (40) 577  (31)
fo–kar–prEn 0 ·65 –0 ·4 390  (10) 847  (50) 628  (10)
1500 °C, log fO2 –0 ·68
fo–per–qan 0 ·02 –3 ·9 152  (10) 217  (40) 213  (13)
fo–qan–gei 0 ·04 –3 ·2 132  (10) 358  (58) 203  (21)
fo–gei–kar 0 ·11 –2 ·2 192  (24) 520  (61) 212  (28)
1500 °C, log fO2 –2 ·74
fo–per–qan 0 ·02 –3 ·9 323  (15) 252  (64) 322  (10)
fo–qan–gei 0 ·04 –3 ·2 237  (6) 397  (21) 380  (18)
fo–gei–kar 0 ·11 –2 ·2 480  (10) 668  (28) 643  (6)

Concentrations are determined by reading from concentration–distance plots, errors are 1σ from repeated transects. The 1400 °C air experiment was heavily contaminated by Fe so is omitted.

.

### Relationships between diffusion of Ti and other elements

The relationship between the diffusion of Ti and that of the divalent cations is variable. In the high-fO2 experiments with stepped profile shapes where Ti diffusion is relatively slow, the divalent cations diffuse relatively rapidly. In contrast, in the lower fO2 experiments where Ti diffusion profiles are hockey-stick shaped, fast Ti diffusion relates to fast Ni, Co and Mn diffusion (Fig. 12).

Fig. 12

Ti diffusion vs Mn, Ni and Co diffusion. Where Ti diffusion is slow (Ti4+), Mn, Ni and Co are faster. As Ti diffusion becomes faster (Ti3+), the rates of diffusion overlap. The log D = log M line is indicated to show where log M and log D coincide in terms of profile length—this is determined from an arbitrary diffusive penetration from an error function shaped profile. The penetration is the point at which the diffusant concentration is 10% above background values on a profile with 500 μg g–1 at the interface and 1 μg g–1 in the background.

Fig. 12

Ti diffusion vs Mn, Ni and Co diffusion. Where Ti diffusion is slow (Ti4+), Mn, Ni and Co are faster. As Ti diffusion becomes faster (Ti3+), the rates of diffusion overlap. The log D = log M line is indicated to show where log M and log D coincide in terms of profile length—this is determined from an arbitrary diffusive penetration from an error function shaped profile. The penetration is the point at which the diffusant concentration is 10% above background values on a profile with 500 μg g–1 at the interface and 1 μg g–1 in the background.

Aluminium, the major pre-existing contaminant in the synthetic forsterite used in this study (Zhukova et al., 2014), has a complicated relationship with Ti. Over the main length of the Ti diffusion profiles, Al profiles inversely mirror the Ti profiles, as the initial Al diffuses out (e.g. Fig. 2). Near the interface, however, Ti and Al appear to increase together. As well as in forsterite, this is seen in experiments in San Carlos olivine, which has 5–10 times more Al in the crystal background (Fig. 13).

Fig. 13

Ti–Al diffusion in both pure forsterite (c-axis) and natural San Carlos olivine (random orientation). As Ti diffuses in, Al diffuses out, suggesting that the Ti is displacing the Al from stable tetrahedral substitution into the faster diffusing octahedral sites. In the near interface region, Ti and Al concentrations increase together, suggesting slow, coupled diffusion with Al on the tetrahedral site charge balancing Ti on the octahedral site. This behaviour is not seen in all experiments; the amount of interface Al contamination is unconstrained.

Fig. 13

Ti–Al diffusion in both pure forsterite (c-axis) and natural San Carlos olivine (random orientation). As Ti diffuses in, Al diffuses out, suggesting that the Ti is displacing the Al from stable tetrahedral substitution into the faster diffusing octahedral sites. In the near interface region, Ti and Al concentrations increase together, suggesting slow, coupled diffusion with Al on the tetrahedral site charge balancing Ti on the octahedral site. This behaviour is not seen in all experiments; the amount of interface Al contamination is unconstrained.

## DISCUSSION

### Titanium

#### Charge balancing titanium incorporation in forsterite and the effect of chemical activity

The pure forsterite structure is a distorted hexagonally close-packed (hcp) array of oxygen anions, with half of the octahedral sites (‘M-sites’) occupied by magnesium. Two octahedral sites can be distinguished; M1 and M2. One-eighth of the tetrahedral sites (‘T-sites’) are occupied by silicon. Ti incorporation in pure, dry (hydrogen-free) forsterite with negligible impurities (a nominally aluminium- and iron-free system) probably occurs by three substitution mechanisms. The size of Ti4+ allows substitution for either magnesium or silicon in the forsterite lattice, whereas Ti3+ is likely on only the M-sites owing to both size constraints and the excess octahedral site preference energy of the Ti3+ cation with its [Ar]3d1 electron configuration.

The equilibrium substitution, first alluded to by Hartman (1969) then observed in synthesis experiments (Hermann et al., 2005; Berry et al., 2007b; Rodina et al., 2011), sees Ti4+ exchanging for Si4+ on the tetrahedral site. This is the most stable configuration in anhydrous settings; no charge balance is necessary and despite a large difference in ionic radii [0·26 Å for Si4+ versus 0·42 Å for Ti4+ in tetrahedral coordination, from Shannon (1976)], a limited extent of solid solution is possible. Ti4+ can also substitute for Mg2+ on the octahedral sites; however, in the absence of impurities, this substitution must be charge balanced by a vacancy on another M-site, forming a Ti4+[vac]SiO4 complex ([vac] represents a vacant Mg site). M-site Ti3+ substitution is possible as $Ti4/33+$[vac]2/3SiO4. The smaller ionic radius of Ti4+ means it should preferentially sit on the M1-site in olivine, whereas Ti3+ should be able to sit on either M1 or M2, with a slight preference for M1 (Papike et al., 2005).

Where Ti substitutes directly for Si, the highest concentrations are expected when qandilite is present in the buffering assemblage (Hermann et al., 2005) [reaction (2)]. This corresponds to low aSiO2 (i.e. fo–per–qan and fo–qan–gei buffers, Fig. 1), encouraging substitution on the T-site:

(2)
$Mg2Ti4+O4=Mg2Ti4+O4. qan ol$

In contrast, substitution of Ti (as Ti4+ or Ti3+) for Mg should be more favourable where aSiO2 is high and aMgO is low; this encourages M-site substitution. Reaction (3) describes Ti4+ substituting for Mg in the presence of protoenstatite and karooite (fo–kar–prEn):

(3)
$2MgTi24+O5+5Mg2Si2O6=4Ti4+[vac]SiO4+6Mg2SiO4. kar prEn ol ol$
Synthesis experiments show that equilibrium Ti substitution in forsterite is dominated by the MgTi4+O4 substitution (Hermann et al., 2005); however, such experiments do not have the resolution to address whether a small fraction of Ti may also be on the octahedral sites.

The results from this study show very much lower interface concentrations than those found by Hermann et al. (2005); moreover, the highest apparent Ti interface concentrations occur when the external buffer assemblage contains protoenstatite, and the lowest where periclase is present (Fig. 9), suggesting that Ti is on the M-site, charge balanced by an M-site vacancy. The maximum concentration of Ti4+ on the M-site is likely to be only a few tens of parts per million, if it is analogous to Zr4+ or Hf4+ (Jollands et al., 2014). Also, the diffusion measurements show conclusively that the fastest diffusion occurs when aSiO2 is high (Fig. 14), implying that Ti is exploiting M-site vacancies and diffusing as either $Ti4/33+$[vac]2/3SiO4 or Ti4+[vac]SiO4.

Fig. 14

The relationship between aSiO2 and diffusion from experiments with similar fO2. Error bars are 1σ from repeat measurements. (a) Ti diffusion (as diffusive penetration; log M), showing strong positive relationship between diffusion and activity of silica. (b) Equivalent plot for Mn diffusion as log D [fitted to equation (1)]. Similar plots for Ni and Co and different conditions can be found in

.

Fig. 14

The relationship between aSiO2 and diffusion from experiments with similar fO2. Error bars are 1σ from repeat measurements. (a) Ti diffusion (as diffusive penetration; log M), showing strong positive relationship between diffusion and activity of silica. (b) Equivalent plot for Mn diffusion as log D [fitted to equation (1)]. Similar plots for Ni and Co and different conditions can be found in

.

It appears that the Ti interface concentration data inferred in this study from extrapolation of the measured diffusion profiles do not represent true equilibrium (i.e. predominantly Ti4+ on the T-site). It is possible that the Ti contents at the interface of the crystals are those expected from the synthesis studies, but occur over very short length scales that cannot be seen using our analytical procedure [a profile using secondary ion mass spectrometry (SIMS) depth profiling or nanoSIMS step profiling could possibly resolve this]. The diffusion of Ti on the tetrahedral site must be balanced by a counter flux of Si in the opposite direction, with Si generally agreed to be the slowest diffusing major element in olivines (e.g. Bejina et al., 1999; Costa & Chakraborty, 2008; Dohmen et al., 2010; Fei et al., 2012). Using the available data for Si self-diffusion in olivine, we would expect that over the duration of the experiments yielding the longest diffusion profiles of Ti on the M-sites, Ti existing on the Si sites would have diffused some 0·5–2 µm, far below the resolution of scanning LA-ICP-MS. Conversely, the concentrations of Ti that we do see at the near-interface ends of the diffusion profiles, which are of the order of 102 µg g–1 Ti, may be the equilibrium amount of octahedrally coordinated Ti4+ only. This represents only a per cent or so of the tetrahedrally coordinated Ti4+ found by Hermann et al. (2005), and would be fully invisible to that study, or the subsequent spectroscopic measurements of Berry et al. (2007b).

The most likely contaminant-balanced substitution involves Al3+. Alumina is a ubiquitous component of 1 atm furnace assemblies, and is also present in the forsterite and San Carlos starting materials. Al3+ can substitute for either Si4+ or Mg2+, and when substituting for Si4+ allows highly charged cations to substitute octahedrally without the necessity for vacancies. Ti3+ and Ti4+ in Al-bearing forsterite could potentially form the (MgTi3+)AlO4 and (Mg3/2$Ti1/24+$)AlO4 complexes, respectively. Given that the concentration of Al often rises steeply towards the interface in concert with Ti (Fig. 13), we assume that the interface Ti concentrations derived in this study are predominantly the result of a Ti–Al coupled substitution. As these near-interface Al concentrations occur over length scales that are too short for full characterization using our analytical methods, and are unconstrained by any buffering, it is not possible to obtain quantitative information regarding Ti defects from interface concentration measurements.

#### The effect of oxygen fugacity on Ti diffusion

The Ti diffusion rate, along with Mn, Ni and Co diffusion, increases as fO2 is reduced. Oxygen fugacity changes can affect point defect types and concentrations, and hence diffusion rates, in a variety of ways. In the experiments reported here, Ti, O, Fe (contaminant), Ni, Co and Mn are all redox-sensitive to some degree over the T–fO2 range studied.

The redox changes of Ni and Co are from M2+ to M0 (O’Neill & Pownceby, 1993a). As the experimental conditions become more reducing, Co2+ and Ni2+ are effectively removed from the buffering assemblage by reduction to the metal, leaving no detectable Ni or Co in the forsterite of the lowest fO2 experiments. The presence or absence of Co2+ and Ni2+ should have no effect on total point defect chemistry. Berry et al. (2007a) observed an almost negligible proportion of Mn3+ in forsterite under far more oxidizing conditions than encountered in this study, so all Mn in olivine can be confidently expected to be divalent. Iron valence should undergo an almost complete change between Fe3+ and Fe2+ over the experimental range of fO2, with trivalent Fe in the octahedral sites inducing point defect formation (M-site vacancies) for charge balance. This effect is greatest at high fO2, so cannot explain the increase in Ti diffusion rate at more reducing conditions. Vacancies may also increase in the oxygen sub-lattice as fO2 decreases, but these should induce no change in vacancy concentration on either the M- or T-sites. Oxygen vacancies may decrease defect migration energies associated with either the M- or T-sites, but this would lower the activation energy at lower fO2; no evidence of this is seen.

The enhancement of both Ti interface concentrations and diffusion as fO2 is decreased may therefore be a function of a valence state change from Ti4+ to Ti3+. This can be described by the general reaction, for the fo–kar–prEn buffer:

(4)
$34Ti4/33+[vac]2/3SiO4+14Mg2Si2O6+14O2 ol opx =Ti4+[vac]SiO4+14Mg2SiO4. ol ol$

The equilibrium constant of (4) is then

(5)
$K=(XTi4+[vac]SiO4ol.γTi4+[vac]SiO4ol)(XMg2SiO4ol.γMg2SiO4ol)1/4(XTi4/33+[vac]2/3SiO4ol.γTi4/33+[vac]2/3SiO4ol)3/4(XMg2Si2O6ol.γMg2Si2O6ol)1/4fO21/4$
where X is mole fraction and γ is activity coefficient. In the present study the pure phases forsterite and enstatite have an activity of one (a = Xγ = 1), such that they can be removed from the equilibrium expression. Assuming Henry’s law behaviour, the expression can then be simplified and rearranged:
(6)
where K* is a modified equilibrium constant indicating that the standard state of the Ti end-members is taken at infinite dilution. Assuming that ΣTi = Ti3+ + Ti4+, and following Berry and O'Neill (2004), (6) is arranged to
(7)
$Ti3+ΣTi=11+10(log⁡K*+14log⁡fO2).$
This curve defines a sigmoid in log fO2–Ti3+/ΣTi space. The ¼log fO2 term in (7) [related to the ¼O2 term in (4)] fixes the slope of the sigmoid and log K* defines the curve position. This expression is defined for the equilibrium condition (i.e. the relative amounts of Ti3+ and Ti4+ at the interface), but can also be instructive for explaining the diffusive behaviour.

The Ti mobility (log M) is plotted against log fO2 in Fig. 15 along with curves with the slope (but not position) defined by (7). As equation (7) gives the proportion of Ti3+ over total Ti concentration at equilibrium and the data represent a kinetic process, the fit of the data to this relationship is qualitative. Nevertheless, the similarity suggests that a valence change from Ti4+ to Ti3+ may be responsible for the change in diffusion rates. In addition, the position of the curves suggests that the transition from predominantly Ti4+ to Ti3+ is a function of silica activity; Ti3+ becomes the major valence at higher fO2 where enstatite is present compared with where periclase is present.

Fig. 15

Ti diffusion as a function of fO2 at fixed orientation and temperature. Data from the four buffer assemblages are shown, along with the theoretical curve describing a one-electron transfer reaction fitted to the data. Dashed line shows the Ti3+/ΣTi curve determined by Mallmann & O’Neill (2009) [M&O’N (2009)], arbitrarily placed on the y-axis for reference. The data from this study suggest that Ti3+ may be present in natural samples at considerably higher fO2 than expected from the solubility studies.

Fig. 15

Ti diffusion as a function of fO2 at fixed orientation and temperature. Data from the four buffer assemblages are shown, along with the theoretical curve describing a one-electron transfer reaction fitted to the data. Dashed line shows the Ti3+/ΣTi curve determined by Mallmann & O’Neill (2009) [M&O’N (2009)], arbitrarily placed on the y-axis for reference. The data from this study suggest that Ti3+ may be present in natural samples at considerably higher fO2 than expected from the solubility studies.

Trivalent Ti is rarely reported in terrestrial minerals, with Ti3+ minerals (e.g. tistarite) existing only in very reducing extraterrestrial parageneses. The transition in olivine (Mallmann & O’Neill, 2009) is also expected to occur at more reducing conditions than experienced in these experiments. The Ti4+-bearing phases in the buffer assemblages are stable throughout the experiments, with the exception of the fo–qan–gei assemblage in the most reducing experiments. XRD of buffer powders showed that this changed to an assemblage of fo–gei–kar over the course of the experiment; the reason for this change is unclear.

The Ti3+ content is expected to have the equilibrium value at the crystal interface of ∼200–300 μg g–1 Ti3+ (Fig. 8) against ∼0·2–1 wt % Ti4+ on the T-sites (e.g. Hermann et al., 2005), giving Ti3+/ΣTi of <0·01, below the sensitivity of either X-ray absorption near edge structure (XANES) or Mössbauer spectroscopy. This highlights a useful by-product of diffusion measurements: the potential to observe the less favourable species that would normally be unmeasurable in conventional equilibrium studies.

#### Hydroxylation processes

In an attempt to detect contributions of Ti3+ and Ti4+ to a diffusion profile, we have chosen a sample from the center of the sigmoid curve for the hydroxylation. The obtained FTIR spectra show that a hockey-stick shaped diffusion profile can be resolved into two separate Ti profiles (Fig. 11); one owing to Ti4+, revealed as the Ti-clinohumite point defect substitution (Ti4+ on the M-site, 2H+ on the T-site), and the other owing to Ti3+, revealed as the trivalent substitution, with Ti3+ charge balanced by a single hydrogen on the M-site (Ti3+H+SiO4). The Ti4+ profile shows a step akin to that observed in the high-fO2 Ti diffusion experiments. The Ti3+ profile is linear and is similar in shape to the LA-ICP-MS Ti profile from this experiment.

For these defects to form inside the crystal, hydrogen must diffuse in. H+ diffusion must be charge balanced by diffusion-in of a species with net negative charge, the most likely carriers being M-site vacancies.

The mechanisms for hydroxylation have been treated in detail elsewhere (Jollands et al., 2016), but, briefly, the reaction to produce the Ti3+H+SiO4 defect involves loss of M-site vacancies from the crystal:

(8)
Formation of the Ti-clinohumite point defect (MgTiH2O4) requires Ti4+ to move from tetrahedral to octahedral coordination:
(9)
$Mg2Ti4+O4+12Mg2Si2O6+H2O=MgTi4+H2O4+Mg2SiO4. ol opx ol ol$

It is unlikely that Ti changes valence at all, rather it simply forms the most stable hydroxylated defect in its current valence state. This is not the case for all elements; Ti exhibits this behaviour because the anhydrous defects have different site preferences and it is not easy to translate from one to the other without rearranging silicon cations on the slowly diffusing tetrahedral sublattice to some extent. Similar behaviour is not seen in the Cr2+–Cr3+ system, where both cations are octahedrally coordinated in both dry and wet systems (e.g. Jollands et al., 2015).

Taken together, the results from the hydroxylation experiment suggest that the hockey-stick shaped Ti profiles at moderately low fO2 (which, we emphasize, were obtained in anhydrous diffusion anneals) predominantly comprise Ti3+ on the M-site, with Ti4+ at low concentrations on the T-site. When hydroxylated, the Ti4+ moves to the M-site, and the Ti3+ replaces its charge-balancing vacancy portion with a single proton. The Ti-clinohumite point defect profile shows the same characteristic double step as observed in high-fO2 experiments; thus we assume the stepped profiles represent Ti on the tetrahedral site. The linear profiles at low fO2 are likely to reflect trivalent Ti3+ on the M-sites, although, as in this experiment, they may show subordinate amounts of Ti4+ on the T-site that are likely to be missed by conventional trace element measurement techniques.

#### Diffusion profile shapes and potential diffusion mechanisms

The Ti concentration–distance profiles observed in this study cannot be fitted to a solution to simple concentration-independent diffusion; that is, an error function (Crank, 1975). The profiles from high-fO2 experiments show a double step, and those from low fO2 show a hockey-stick shape. In addition, at moderate fO2, the aSiO2 of the experiment also exerts control on the profile shape, and at a given temperature log M is positively correlated with the tendency towards hockey-stick shaped profiles.

Similarly shaped profiles have been reported previously in the geological literature: Li in olivine (Dohmen et al., 2010) has been shown to have stepped profile shapes. The hockey-stick shape observed at low fO2, thought to relate to Ti3+, has also been observed for diffusion of (trivalent) chromium in olivine at high fO2 (Jollands et al., in preparation) and the diffusion of various trivalent cations in periclase (Van Orman et al., 2009; Crispin & Van Orman, 2010).

The diffusion of Ti3+ is conceptually simple as it has only one potential substitution mechanism in the pure system: the vacancy mechanism $Ti4/33+$[vac]2/3SiO4. Diffusion of Ti3+ therefore involves a simple exchange of two Ti3+ and a vacancy for three Mg2+ ions. As the cations move with their own vacancy, the diffusing Ti effectively creates its own pathway. Higher Ti concentrations lead to higher vacancy concentrations, which in turn lead to faster Ti diffusion. The diffusion coefficient is therefore non-constant and will decrease into the crystal, which when modeled gives the observed linear diffusion profile shapes (e.g. Van Orman et al., 2009; Crispin & Van Orman, 2010).

The diffusion of Ti4+ is conceptually more complex. If Ti4+ were only able to substitute on the M-site, charge balanced by an M-site vacancy, the profile shapes should be either error functions [as seen for Zr4+ and Hf4+ in forsterite by Jollands et al. (2014)] or hockey-stick shapes, depending on the binding energy between the cation and its vacancy (e.g. Crispin & Van Orman, 2010). Neither of these end-members, or intermediate shapes, are observed.

Interpretation of XANES data (Berry et al., 2007b) suggests that, in the anhydrous setting, Ti4+ is predominantly located on the tetrahedral site as Mg2Ti4+O4. However, the dependence of diffusion rates on chemical potentials suggests that Ti4+ diffuses on the M-sites as Ti4+[vac]SiO4: mobility is higher where aSiO2 is high. We propose that the resolution of this apparent paradox lies with the known ability of Ti4+ to substitute for either Mg or Si in the forsterite structure. This explanation may also account for the extremely low interface concentrations of Ti4+ found here, compared with those at equilibrium reported by Hermann et al. (2005). The potential for two different site locations in the lattice, with different diffusivities, was invoked by Dohmen et al. (2010) to explain their stepped Li diffusion profile shapes, and can be modified to explain Ti4+ diffusion in this case.

It can be assumed that the diffusion of Ti located on the tetrahedral site in forsterite will be at a similar rate to Si self-diffusion in the lattice; Si vacancies or cations must jump each time Ti does to allow it to move. Likewise, the diffusion of Ti4+ that sits on the octahedral site will occur about the same rate as that for the octahedral site cation, Mg2+, for the same reason. Currently available data for Si and Mg self-diffusion in olivine generally agree that Mg diffuses significantly faster than Si, although the magnitude of the difference is still debated [see Brady & Cherniak (2010) for a compilation of data]. In any case, the rate differences are great enough that Si in forsterite can be considered immobile on the time scales of Mg self-diffusion.

Thus, the diffusion of Ti4+ can be conceptualized as follows. The majority of Ti in the first few hundred to thousand nanometres will dissolve into the Si site as predicted by Hermann et al. (2005), but this cannot be seen by the analytical techniques used in this study owing to the short diffusion profile lengths. A small amount of Ti4+ substitutes for Mg, charge balanced by a vacancy, and as such, is considerably more mobile than T-site Ti4+. As this less stable Ti4+ defect moves through the crystal it may come into contact with a vacant T-site, or a T-site occupied by a cation forming a defect with lower stability than Mg2Ti4+O4. At this point, the Ti4+ will jump into the tetrahedral site, leaving behind M-site vacancies. For example, let us consider a situation where Ti4+ on the octahedral site comes into contact with Al3+ on the tetrahedral site, charge balanced by Al3+ on the M-site:

(10)
$2Ti4+[vac]SiO4+2(MgAl)AlO4+Mg2SiO4 ol ol ol =2Mg2Ti4+O4+3Al4/3[vac]2/3SiO4. ol ol$

The Ti4+ displaces the Al3+ into the M-site, effectively swapping mobilities. Ti4+ that was previously more mobile has become far less so; it is ‘trapped’ on the T-site, and Al3+ that was bound on the T-site is now free to move on the M-site. Reaction (10) is convenient as it also explains the strong negative correlation between Al and Ti diffusion (Fig. 13); Al appears to diffuse out as Ti diffuses in (excluding the near-interface region). The Ti ‘trapping’ reaction does not necessarily need to be described by reaction (10), any interaction between Ti4+ on the M-site and a vacant T-site should give the same result. However, the suggestion that Ti–Al interaction gives the observed profile shapes is corroborated by the experiments on San Carlos olivine; these also show stepped profiles with about five times higher ‘plateau’ concentrations in crystals with 5–10 times more aluminium, and the same Al-out–Ti-in relationships as the forsterite experiments (Fig. 13).

The concentration of Ti in the plateau of the double-step shape (always 15–20 µg g–1 in forsterite) is therefore the concentration of pre-existing tetrahedral site defects that can be accessed by Ti; the Ti content reaches a new, stable baseline in the crystal as these vacancies become saturated. The front of the diffusion profile, where the concentration of Ti drops from the plateau to the background value, effectively represents a wave of Ti moving through the crystal, diffusing only on the M-sites but then dropping onto the T-sites where it has much greater stability and lower diffusivity. Further details of the diffusion and trapping processes are given elsewhere (Petrischcheva et al., in preparation), including the mechanism by which the diffusion profiles evolve with time.

The final mechanism, observed in experiments at high and low fO2, is coupled Ti–Al diffusion. Al3+ sits relatively comfortably on the T-site (ionic radius = 0·39 Å) and as such is able to balance highly charged cations on the M-sites. Diffusion of (MgTi3+)AlO4 and (Mg3/2$Ti1/24+$)AlO4 is expected to be intermediate between the rates of M-site and T-site diffusion (e.g. Spandler & O’Neill, 2010). In many profiles Al and Ti rise sharply together towards the interface (Fig. 13), suggesting that they are at this point linked. Therefore, the observed interface concentrations are unconstrained; they represent the degree of experimental Al contamination, which varies from sample to sample and even between transects on single crystals.

#### Comparison with other studies

Spandler et al. (2007) and Spandler & O’Neill (2010) investigated the diffusion of Ti between an approximately basaltic melt and natural olivine [mid-ocean ridge basalt (MORB) phenocrysts] (Spandler et al., 2007) and San Carlos olivine crucibles (Spandler & O’Neill, 2010) and found Ti showing concentration-independent diffusion with around the same rate as for other divalent cations and for Mg–Fe interdiffusion (e.g. Dohmen et al., 2007) over length scales of hundreds of micrometres (LA-ICP-MS and electron microprobe measurements). In contrast, Cherniak & Liang (2014) found Ti diffusion in olivine occurring significantly slower (but still following error function shaped profiles), around the rate of Si self-diffusion, although this was measured over tens to hundreds of nanometres using Rutherford backscattering spectroscopy (RBS). In neither study were any double-step or linear concentration–distance profiles observed.

The studies of Spandler et al. (2007) and Spandler & O’Neill (2010) are probably the most relevant in terms of providing geologically useful trace element diffusion coefficients as they most closely resemble a natural system, and include the buffering of all chemical activities by use of a silicate melt as the diffusant source or sink. On the other hand, these studies do not allow a thorough treatment of diffusion mechanisms, as the experiments were conducted only at 1300 °C and 1 atm, each at a single fO2.

The study of Cherniak & Liang (2014) used a powder source, as in this study, but consideration was not given to buffering as those researchers used only a two-phase assemblage of forsterite–qandilite. As a result, the chemical activity of the source is unconstrained, and it will migrate to either forsterite–qandilite–periclase or forsterite–qandilite–geikielite during the experiment according to uncontrolled experimental variables such as impurity contents and volatility-related transport. Admittedly, this may be only a minor problem in the SiO2–MgO–TiO2 system: Hermann et al. (2005) showed that when substituting for Si in the tetrahedral site, the concentration of Ti in forsterite is the same regardless of which side of the qandilite–forsterite tie-line the experiment is on. That Ti substitutes for Si in the Cherniak & Liang (2014) experiments is suggested by their determined diffusion coefficients that are closer to Si self-diffusion coefficients than M-site diffusion coefficients. However, an aspect of concern in their data regards the interface concentrations of Ti. Over the 500 °C range of temperature they do not change significantly, whereas solubility relations would predict that equilibrium interface concentrations should decrease systematically with temperature (Hermann et al., 2005) (Fig. 9). Another concern is the detection limit of Ti by RBS, which was not stated in the paper but realistically is several hundred µg g–1. Given that diffusion profiles follow a trend of concentration decay into the crystal, this means that only Ti at the edge of the crystals should be detectable by RBS, and, given the maximum permissible Ti concentrations in forsterite (Hermann et al., 2005), only at the higher temperatures used by Cherniak & Liang (2014) (Fig. 9).

The relatively slow Ti diffusion found by Cherniak & Liang (2014) seems to be inconsistent (by several orders of magnitude) with the relatively fast Ti diffusivity found in this study, and by Spandler et al. (2007) and Spandler & O’Neill (2010). The vast discrepancies between these studies are not simply explained. Burgess & Cooper (2013) suggested that extended planar defects associated with Ti near the surface of olivines may explain the relatively fast diffusion rates reported by Spandler et al. (2007) and Spandler & O’Neill (2010), but this explanation overlooks the fact that the Spandler et al. (2007) results were obtained from diffusion of Ti out of the crystal starting with natural concentrations. Rather, such defects on olivine surfaces are much more likely to influence the experiments of Cherniak & Liang (2014), which involved high TiO2 activities and measurement of tens of nanometre length-scale profiles. The most complex systems, investigated by Spandler et al. (2007) and Spandler & O’Neill (2010), show very simple diffusion profiles with Ti presumably moving on M-sites, although the charge-balance mechanisms are unclear in their case; Na+ and Li+ may be involved, for example. That our preliminary experiments on San Carlos olivine also showed stepped profiles suggests that the error-function shapes observed by Spandler et al. (2007) and Spandler & O’Neill (2010) are not a result of Fe3+ in natural olivine.

### Mn, Co, and Ni diffusion

#### Diffusion as a function of T, aSiO2 and fO2

Diffusion of the divalent cations in olivine is influenced by oxygen fugacity, crystal orientation, temperature and chemical activity, so a single expression to describe diffusion rate must account for all of these variables. To this end, following a modification of the methods of Zhukova et al. (2014), all divalent diffusion data have been fitted to the general equation

(11)
$log⁡D=log⁡D0+nlog⁡aSiO2+mlog⁡fO2−[Ealn⁡(10)RT]$
where n and m are the dependences of diffusion on silica activity and oxygen fugacity, respectively, and Ea is the activation energy. This method assumes no change in the dependence of each parameter on each other parameter (i.e. the dependence of diffusivity on aSiO2 does not change as a function of temperature, for example). The values of n and m are therefore only considered to be semi-quantitative.

The data were fitted to this equation using non-linear least-squares regression for each diffusing element and each orientation. After regression, the values of n and m were set to the nearest logical fraction and the process was repeated. The n and m values for Co and Mn converged to around the same value (2/3 and –1/8, respectively), whereas the Ni data were better fitted to m = –1/7. The discrepancy between the m values is small enough that the diffusion mechanisms can be assumed to be the same for all cations; Ni data can also be fitted to a –1/8 dependence within error. The 2/3 dependence of diffusion on aSiO2 is the same as predicted by Zhukova et al. (2014) using only forsterite–enstatite and forsterite–periclase buffering assemblages.

Values of log D0 and Ea (along with m and n) are given in Table 6. The predicted log D values from the equation are plotted against measured values in Fig. 16, including 1:1 lines, showing that the values in Table 6 describe well the aSiO2, fO2, orientation and temperature dependence of diffusion. Although this equation describes diffusion under the conditions investigated, it does not account for compositional variations in olivine [e.g. Mg/(Mg + Fe2+)] that are expected to alter both the diffusion rate and fO2 dependence (e.g. Chakraborty et al., 1994; Chakraborty, 1997), or the pressure effect; these cations should diffuse more slowly at higher pressures (Holzapfel et al., 2007).

Table 6

Parameters from equation (11) derived by least-squares regression of all data

log D0 (principal axes)

n m Ea
a b c   (kJ mol–1
Ni –1 ·96 –1 ·56 –0 ·71 2/3 –1/8 459 ·5
Co –2 ·65 –2 ·05 –1 ·35 2/3 –1/8 439 ·5
Mn –1 ·54 –0 ·99 –0 ·20 2/3 –1/8 472 ·7
log D0 (principal axes)

n m Ea
a b c   (kJ mol–1
Ni –1 ·96 –1 ·56 –0 ·71 2/3 –1/8 459 ·5
Co –2 ·65 –2 ·05 –1 ·35 2/3 –1/8 439 ·5
Mn –1 ·54 –0 ·99 –0 ·20 2/3 –1/8 472 ·7
Fig. 16

Predicted diffusion coefficients from equation (11) at the experimental fO2 and aSiO2 conditions compared with measured coefficients for Mn (a), Co (b) and Ni (c) including 1:1 lines. These fits suggest that Mn, Ni and Co diffusion (and hence M-site vacancy concentrations) are all subjected to a 2/3 dependence on aSiO2 and a –1/8 to –1/7 dependence on fO2.

Fig. 16

Predicted diffusion coefficients from equation (11) at the experimental fO2 and aSiO2 conditions compared with measured coefficients for Mn (a), Co (b) and Ni (c) including 1:1 lines. These fits suggest that Mn, Ni and Co diffusion (and hence M-site vacancy concentrations) are all subjected to a 2/3 dependence on aSiO2 and a –1/8 to –1/7 dependence on fO2.

#### Point defect chemistry

The predominant intrinsic point defect in pure forsterite is expected to be the Mg–Frenkel vacancy–interstitial complex (Pluschkell & Engell, 1968; Walker et al., 2009) whereby a magnesium ion leaves an octahedral site and relocates to a nearby interstitial vacancy. The concentration of this defect, however, should not be affected by either chemical activities or oxygen fugacity; the concentration should be increased only through raising temperature. We must assume therefore that the concentration of Mg vacancies (and with it the rate of M-site diffusion) is being increased by another mechanism in parallel with the Frenkel reaction.

Zhukova et al. (2014) invoked an anti-Schottky mechanism [favoured by the point defect models of Pluschkell & Engell (1968) and Smyth & Stocker (1975)] to explain their observed 2/3 power dependence of Ni and Co diffusion on aSiO2; by dissolving silica into the forsterite lattice Mg vacancies are created by effectively building new Mg-deficient formula units (in Kröger–Vink notation):

(12)
$2MgMg×+2SiO2=Mg2SiO4+Sii••••+2VMg//.$
However, the anti-Schottky reaction (12) liberates no free oxygen, so no fO2 dependence should be observed. Potentially, the diffusion of Ti is affecting that of Ni, Co and Mn; the –1/8 dependence of diffusion on fO2 may be a function of changing Ti valence state. Although Ti4+ is expected to diffuse on the M-site, charge balanced by a vacancy, it becomes trapped on the tetrahedral site. At this point the vacancy is expected to diffuse out of the crystal, possibly accompanying aluminium. Ti3+, on the other hand, retains its vacancies; it cannot substitute into the tetrahedral site given its large ionic radius. The concentration of vacancies should therefore be proportional to Ti3+ concentrations, which vary as a function of fO2. Equation (7) defines the relationship between Ti3+/ΣTi and fO2 as a sigmoidal curve, with an approximately –1/7 slope over the inflexion. Potentially, rather than any change in bulk point defect concentration as a function of fO2, the concentration of Ti3+ is changing, which is in turn affecting the concentration of M-site vacancies, and hence the diffusion of the divalent cations. It would be expected then that the Mn, Ni and Co diffusion profiles should be dependent on the concentration of Ti3+ [i.e. should not fit equation (1), which specifies concentration independence] but this is difficult to evaluate given that the cations often show depleting interface concentrations. The potential for diffusive interference is also suggested in Fig. 12; where Ti diffuses slowly (as mostly Ti4+), Ni, Co and Mn diffuse considerably faster than Ti, whereas when Ti diffusion is faster (as Ti3+) the divalent cations have approximately the same diffusivity; they diffuse faster and their rate of diffusion is increased by presence of Ti3+. This has implications for diffusion in natural systems, as the mobility of divalent cations may be increased by the simultaneous diffusion of a trivalent cation (e.g. Fe3+, Cr3+ and Al3) with vacancies for charge balance. Therefore, it may not be valid to use frozen diffusion profiles of divalent cations for time scale determination without consideration of the effect of diffusive interference from other cations. In this system, as little as 200–400 μg g–1 of Ti3+ is enough to change diffusion coefficients or divalent cations by an order of magnitude.

#### Comparison with other studies

There has been considerable previous research on the diffusion of Ni, Co and Mn in olivine (Fig. 17) experimentally as a function of pressure (Holzapfel et al., 2007), oxygen fugacity (Jurewicz & Watson, 1988a), orientation (Spandler & O’Neill, 2010), chemical potentials (Zhukova et al., 2014), temperature (Ito et al., 1999) and olivine composition (Morioka, 1980, 1981). It has been established that diffusion is faster at higher temperature, lower pressure, higher silica activity, and higher oxygen fugacity (in natural olivine), and is anisotropic; Dc > Db ∼ Da. The results from this study generally agree with literature data in terms of the temperature, orientation and silica activity (pressure was not investigated). The exception is at low fO2, where we suggest that Ti3+ diffusion has interfered with the M-site vacancy concentration and hence divalent cation diffusion rate.

Fig. 17

Comparison of diffusion data collected in this and other studies where more than one temperature was investigated. Ti, Mn, Ni and Co data from this study represent c-axis diffusion, high aSiO2 at log fO2 of –0·7 (labelled ‘low fO2’) and –11 (labelled ‘high fO2’). ‘High’ aSiO2 refers to (proto)enstatite-buffered, ‘low’ aSiO2 refers to periclase-buffered. It should be noted that we estimate equivalent length scales (according to the definition presented above) of diffusion where log D = log M – 1·5. Labels are as follows. I, Ito et al. (1999): 1 atm, c-axis, forsterite, unbuffered aSiO2, log fO2 = –0·7. C&L, Cherniak & Liang (2014): 1 atm, c- and b-axes, forsterite, low or unbuffered aSiO2, fO2 = Fe–FeO or Ni–NiO. Z, Zhukova et al. (2014): 1 atm, a-axis, forsterite, high aSiO2, log fO2 = –0·7. M1, Morioka (1980): 1 atm, c-axis, forsterite, unbuffered aSiO2, log fO2 = –0·7. M2, Morioka (1981): as M1. J&W, Jurewicz & Watson (1988b): 1 atm, c-axis, St Johns olivine, high aSiO2, log fO2 = –8. H, Holzapfel et al. (2007): 12 GPa, c-axis, forsterite and San Carlos olivine, low aSiO2, fO2 ∼ Ni–NiO. S&O, Spandler & O’Neill (2010): (Ti) 1 atm, c-axis, San Carlos olivine, high aSiO2 (basalt melt), log fO2 = –8.

Fig. 17

Comparison of diffusion data collected in this and other studies where more than one temperature was investigated. Ti, Mn, Ni and Co data from this study represent c-axis diffusion, high aSiO2 at log fO2 of –0·7 (labelled ‘low fO2’) and –11 (labelled ‘high fO2’). ‘High’ aSiO2 refers to (proto)enstatite-buffered, ‘low’ aSiO2 refers to periclase-buffered. It should be noted that we estimate equivalent length scales (according to the definition presented above) of diffusion where log D = log M – 1·5. Labels are as follows. I, Ito et al. (1999): 1 atm, c-axis, forsterite, unbuffered aSiO2, log fO2 = –0·7. C&L, Cherniak & Liang (2014): 1 atm, c- and b-axes, forsterite, low or unbuffered aSiO2, fO2 = Fe–FeO or Ni–NiO. Z, Zhukova et al. (2014): 1 atm, a-axis, forsterite, high aSiO2, log fO2 = –0·7. M1, Morioka (1980): 1 atm, c-axis, forsterite, unbuffered aSiO2, log fO2 = –0·7. M2, Morioka (1981): as M1. J&W, Jurewicz & Watson (1988b): 1 atm, c-axis, St Johns olivine, high aSiO2, log fO2 = –8. H, Holzapfel et al. (2007): 12 GPa, c-axis, forsterite and San Carlos olivine, low aSiO2, fO2 ∼ Ni–NiO. S&O, Spandler & O’Neill (2010): (Ti) 1 atm, c-axis, San Carlos olivine, high aSiO2 (basalt melt), log fO2 = –8.

Jurewicz & Watson (1988b) found diffusion of divalent cations in natural olivine to be faster at higher oxygen fugacity, whereas data from this study show that diffusion rates increase (by up to an order of magnitude) with decreasing oxygen fugacity (over c. 11 log units of fO2). This disagreement can be explained simply by considering the different defect populations of natural and synthetic olivines. Natural olivine contains predominantly divalent iron, but small amounts [of the low hundreds of µg g–1 level according to Dohmen & Chakraborty (2007) and Mallmann & O’Neill (2009)] are trivalent at formation conditions. By conducting diffusion experiments within the limited range of fO2 at which such natural olivines are stable, the Fe3+/ΣFe ratio is modified, and with it the pre-existing point defect concentration of the olivine, as the Fe2+ to Fe3+ electron transfer must be charge balanced by addition of M-site vacancies. In contrast, pure forsterite with small amounts of Ti as either Mg2TiO4 or $Ti4/33+$[vac]2/3SiO4 will have an extrinsic vacancy concentration that increases as fO2 decreases.

## IMPLICATIONS

1. Ti3+ shows concentration-dependent diffusion in forsterite. This may also apply to many other systems where the diffusing species require vacancies for charge balance.

2. Concentration–distance profiles showing a step that may have been otherwise interpreted as an overgrowth followed by diffusive exchange may result instead from complex diffusion mechanisms. A viable way to distinguish between element zonation by diffusion or by overgrowth is by analyzing multiple elements with known variations in diffusivity and diffusive anisotropy.

3. Elements that have more than one potential crystallographic site or valence, or require vacancies for charge balance, are likely to show diffusive behaviour that does not allow simple fits to the error function. If they do, explanations must be given as to why. Explanations could include alternative charge-balance mechanisms, concentrations lower than intrinsic vacancy populations or fully bound cation–vacancy pairs.

4. Failure to control chemical activities in diffusion experiments can lead to large discrepancies in diffusion coefficients. By simply moving from one side of a buffer assemblage tie-line to the other, the measured diffusivity can change by an order of magnitude. Notably, this includes the classic diffusion couple experiments (two-crystal olivine–olivine diffusion).

5. Diffusion of trivalent cations charge balanced by M-site vacancies may speed up the diffusion of passively diffusing divalent cations. The effect of Fe contamination is considerable; it changes the shape of divalent cation diffusion profiles. Conversely, the effect of Ti3+ is extremely subtle; the divalent cations still follow error function profiles but diffuse at slightly faster rates. Potential diffusive interference must be considered when extracting time scales from frozen diffusion profiles.

6. Ti3+ may be present in natural systems at considerably higher fO2 than previously determined using equilibrium data. The concentrations will be considerably lower than that of Ti4+, but even tens of µg g–1 level Ti3+ on the octahedral site could affect M-site vacancy concentrations and hence divalent cation diffusion rates.

## ACKNOWLEDGEMENTS

The author wishes to thank Charlotte Allen and Jung-Woo Park for assistance with LA-ICP-MS, Ulrike Troitsch for help with powder XRD, and Dave Clark, Dean Scott and Jeremy Wykes for general experimental assistance. Sumit Chakraborty, Ralf Dohmen, Thomas Müller, Frauke Petersen and Christian Bratfisch are thanked for their assistance and hospitality during the writing of much of this paper at the Ruhr Universität, Bochum, and Rainer Abart, Elena Petrischcheva and Faheem Najm are thanked for many useful discussions and hospitality in Vienna. Rick Ryerson and an anonymous reviewer are thanked for their useful comments and suggestions.

### FUNDING

This work was supported by the Australian Research Council (DP110103134 to J.H., H.O’N. and C.S.).

### SUPPLEMENTARY DATA

for this paper are available at Journal of Petrology online.

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