Defect chemistry

The CH2O exponent for DSi can be understood using defect chemistry. Formation of a Mg vacancy, VMg’’, requires less energy than a Si vacancy, VSi’’’’ [Brodholt and Refson, 2000]. The concentration of VMg’’ is therefore much higher than that of VSi’’’’; [VMg’’] >> [VSi’’’’] in

The DSi should be proportional to the density of silicon vacancies [Costa and Chakraborty, 2008; Kohlstedt, 2006]. In addition, Si⁴⁺ is surrounded by O^2- in tetrahedrons, and therefore, VO••

is also needed for Si⁴⁺ migration. We can expect a certain proportion of VSi’’’’ should be associated with VO•• due to the Coulomb force and the Si migration is dominated by VO•• -associated VSi’’’’. Hence, DSi should be also proportional to [VO••]. This idea is supported by the oxygen partial pressure (PO2) exponent for DSi in olivine reported by Houlier et al. [1990], DSi ∝ (PO2)^-0.19 ≈ (PO2)^-1/6, which suggests DSi ∝ [VO••] because [VO••] ∝ (PO2)^-1/6 [Stocker and Smyth, 1978]. Though Houlier et al.’s [1990] results were obtained in iron-bearing olivine, the proportional relationship between DSi and [VO••] should be also the case for iron-free forsterite.

As a result, we have:

118

which agrees with our experimental results, DSi ∝ (CH2O)^0.32±0.07.

Natural olivine contains about 10 % of the Fe2SiO4 component. The defect chemistry of iron-bearing olivine should be the same as that of forsterite if the iron ions are ferrous under reducing conditions. This means that the charge neutrality condition should be [(OH)O•] = 2[VMe’’] for the CH2O values as that in the present study. If the CH2O is much higher, a large proportion of metal vacancies is occupied by H⁺ and the charge neutrality condition becomes [(OH)O•] = [HMe’].

On the other hand, if a sufficiently large proportion of ferric iron exists, namely under oxidizing conditions, the defect chemistry would be different because of the positively excess-charged FeMe•. In this case, if CH2O is extremely low, the dominant charge neutrality condition in olivine is [FeMe•] = 2[VMe’’]. If CH2O is higher, it would be replaced by [FeMe•] = [HMe’]. If CH2O

is extremely high, the contribution of FeMe• becomes negligible in comparison with that of (OH)O• and the charge neutrality condition should be [(OH)O•] = [HMe’] as also found for reducing conditions.

Now let us consider which charge neutrality condition applies for the upper mantle conditions. In the upper mantle, the ratio of [FeMe•]/[MeMe×] is around 10^-5-10^-6 in anhydrous olivine [Karato, 2008]. Consequently, 10^-5 of [H⁺]/[MeMe×] (i.e., ~1 μg/g of H2O in olivine) is enough to satisfy the condition [(OH)O•] > [FeMe•]. Therefore, the dominant charge neutrality condition will be [(OH)O•] = 2[VMe’’] when CH2O > 1 wt. ppm. On the other hand, If all H⁺ ions enter VMe’’ to form HMe’ at CH2O = 1 wt. ppm, we have [HMe’] ≈ 1.6×10^-5. By using the relationships [VMe’’] ∝ (CH2O)^1/3 and [HMe’] ∝ (CH2O)^2/3 (Table 3.3), and the experimental result that [VMe] ≈ [VMe’’] + [HMe’] ≈ 4.2×10^-4 at CH2O ≈ 16 wt. ppm [Wang et al., 2004], CH2O >3700 wt. ppm is required to make the condition [HMe’] > 2[VMe’’] true. Therefore, [(OH)O•] = [HMe’]

dominates the charge neutrality condition only if CH2O is at least >3700 wt. ppm. Hence, [(OH)O•]

= 2[VMe’’] is the charge neutrality condition for olivine under upper mantle conditions where CH2O is at the level of several hundred wt. ppm [Hirschmann, 2006; Workman and Hart, 2005].

We also have to consider the case in which H⁺ is incorporated into the Si vacancies. If CH2O

is relatively low, the dominant Si vacancy should be VSi’’’’. With increasing CH2O, H⁺ is trapped

119 in the Si vacancies and the dominant Si vacancy should change from VSi’’’’ to HSi’’’, (2H)Si’’, (3H)Si’, and finally (4H)Si×.

Let us consider the CH2O exponent for DSi when the dominant Si vacancy is VSi’’’’. When the charge neutrality conditions are [FeMe•] = 2[VMe’’], as well as [(OH)O•] = [HMe’], [VSi’’’’] and [VO••] are independent with CH2O. Therefore, DSi ∝ [VSi’’’’] × [VO••] ∝ (CH2O)⁰. When [FeMe•] = 2[HMe’], we have DSi ∝ (CH2O)^-0.5. This relationship indicates that increasing CH2O makes olivine even harder under oxidizing conditions. For the charge neutrality condition of [(OH)O•] = 2[VMe’’], we have DSi ∝ (CH2O)^1/3, which is the case in this study. Thus, the CH2O exponent of 1/3 is the largest value that can be obtained if the dominant Si vacancy is VSi’’’’ (Table 3.3).

Next, let us consider the case that H⁺ ions are trapped in the Si vacancies. When the dominant Si vacancy is HSi’’’, the CH2O exponent for DSi is 1/2 or 2/3. In the case of (2H)Si’’, the CH2O exponent will be 1. If the dominant Si vacancy is (3H)Si’ or (4H)Si×, the CH2O exponent will be even higher, i.e. up to 2.5 (Table 3.3). Thus, the CH2O exponent will be higher than 1/3 if H⁺ ions are incorporated into Si vacancies.

We can expect a certain proportion of VSi’’’’ are associated with VO•• due to the Coulomb force, and Si diffusion is dominated by VO•• associated VSi’’’’. Since VSi’’’’ has four charges, and HSi’’’, (2H)Si’’, (3H)Si’, or (4H)Si× has three or less charges, the probability of association of VSi’’’’ and Vo^•• should be much higher than that of hydrated VSi’’’’ and VO••. Therefore, HSi’’’, (2H)Si’’, (3H)Si’, or (4H)Si× could dominate Si diffusion only if CH2O is extremely high and all of Si vacancies are hydrated.

We note that the incorporation of H⁺ ions into Si vacancies is unlikely in the upper mantle, at least in the oceanic mantle. Our experimental results demonstrate that the species of Si vacancies that dominates Si self-diffusion should not change with CH2O from 1 to 800 wt. ppm.

However, our experimental results do not show increase of CH2O exponent up to 800 wt. ppm.

Higher CH2O conditions are unlikely in upper mantle judging from petrological studies (i.e., ~70-160 wt. ppm of water in depleted mantle [Workman and Hart, 2005], and four to five times higher in enriched mantle [Hirschmann, 2006]). Therefore, the CH2O exponent of 1/3 should be the maximum for the realistic mantle.

120

Table 3.3. Dependencies of defect species and DSi on CH2O under four charge neutrality conditions, expressed as the exponent r in the relationship [A]∝(CH2O)^r. VSi denotes the silicon vacancy species, namely, VSi’’’’, (H)Si’’’, (2H)Si’’, (3H)Si’, or (4H)Si×. Defect species exponent data in the table are from Kohlstedt [2006]. Note that the H⁺ should not be trapped by Si vacancies under the charge neutrality condition of [FeMe•] = 2[VMe’’] because this condition is the case where CH2O is extremely low.

Charge neutrality condition [VMe’’] [HMe’] [VO••] Si defect species [VSi] DSi

exponent for DSi should be identical to that for creep rates. However, deformation studies [Hirth and Kohlstedt, 2003; Jung and Karato, 2001; Karato et al., 1986; Mei and Kohlstedt, 2000a; b]

on olivine aggregates claimed a much larger CH2O exponent, 1.2±0.4(Fig. 3.5). In this paper, we have concluded that the CH2O exponent value r = 1.2±0.4[Hirth and Kohlstedt, 2003], obtained from deformation experiments, is an overestimate.