Fei et al. (2013, 2014) systematically measured silicon and oxygen self-diffusion coefficients (DSi and DO, respectively) in iron-free forsterite as functions of water contents, showing that DSi ∝ CH2O0.32±0.07, DO ∝ CH2O0.05±0.06. These water-content exponents are much smaller than that expected on the basis of the assumption that self-diffusion coefficient of a chemical species is simply proportional to its defect density. In this study, we propose a new defect chemistry model to explain the above relationships.
The DSi should be proportional to the density of silicon defects, namely, DSi ∝ [VSi’’’’]. On the other hand, Si4+ is tightly surrounded by four-coordinated O2-, and therefore, migration of Si4+ should be enhanced if a surrounding O2- is missing. We can expect that a certain proportion of VSi’’’’ is associated with VO•• due to their excess charges with the opposite signs. As a result, Si migration is probably dominated by VO••-associated VSi’’’’. DSi is thus also proportional to [VO••] and we have,
DSi ∝ [VSi’’’’] × [VO••].
Under the charge neutrality condition of [(OH)O•] = 2[VMg’’] in hydrous forsterite, we have,
[VSi’’’’] ∝ (fH2O)2/3
[VO••] ∝ (fH2O)-1/3.
DSi ∝ [VSi’’’’] × [VO••] ∝ (fH2O)1/3,
which agrees well with the experimental results: DSi∝ (CH2O)0.32±0.07 (Fei et al., 2013).
Fig. 1. Hopping of Si from one octahedral to its neighbor site. If an oxygen ion is missing
from the octahedral, the hopping of Si becomes much easier as shown in thick arrow.
In hydrous olivine/forsterite, hydrogen exists as hydroxyl, (OH)-. Oxygen ions could diffuse either by hopping of O2- without H+ or by hopping of O in (OH)-. Because H+-associated O has a lower Coulomb force due to the excess charge by H+, the hopping probability of (OH)- should be higher than that of O2-. Thus, the O diffusion should be dominated by O2- of (OH)-. As a result, we have,
DO∝ [VO••] × ([O2-]hopping + [(OH)-]hopping) ≈[VO••] × [(OH)-]hopping.
Under the charge neutrality condition of [(OH)O•] = 2[VMg’’], we have
[VO••] ∝ (fH2O)-1/3
[(OH)O•]un-associated ∝ (fH2O)1/3.
DO ∝ [VO••] ×[(OH)O•]un-associated ∝ (fH2O)0.
This model suggests that DO is independent from CH2O, which agrees well with the experimental results, DO ∝ CH2O0.05±0.06 (Fei et al., 2014).
Fig. 2. Hopping of (OH)- in the crystal structure of forsterite in one unit cell looking along  direction. (OH)- is less charged than O2-, namely, weaker bonding with the surrounding cations, and therefore the hopping probability of (OH)- should be higher.