Mechanisms of diffusion

where COH (x) is the hydroxyl concentration at a distance x from the first sample edge, X is the width of the sample,

t is time,

erfc is the complementary error function = 1-erf, C0 is the observed maximum hydroxyl content,

D~i

is the chemical diffusivity for the mobile species parallel to i = [100], [010]

or [001] under the experimental conditions.

This relationship is valid if the profile from each side does not overlap significantly.

A full treatment for diffusion into a finite slab is given in Carslaw and Jaeger (1959, p. 96) or by Schmalzried (1981, p. 84).

3.3. Mechanisms of diffusion

In an ideal crystalline solid, chemical reactions and mass transfer can occur only because these solids contain imperfections in the structure that permit mobility of

species. Points defects are the major imperfections involved in hydrogen diffusion in the system H₂O - (Mg, Fe)₂SiO₄.

Diffusion and point defects

The nature of point defect in complex minerals is not very well understood;

the basic concepts come from studies of simple crystals such as NaCl. Two kinds of point defects can occur in crystalline material: intrinsic and extrinsic point defects.

The extrinsic point defects result from impurity ions or variations of oxidation state.

One of the simplest intrinsic defects is a vacant cation site charge balanced by a vacant anion site to maintain the electrical neutrality. Such a combination is called Schottky defect. In contrast, the association of an interstitial cation (or anion), which is balancing a cation (or anion) vacancy is called Frenkel defect. Schottky and Frenckel defects do not affect stoichiometry.

The diffusion processes may involve other types of point defects. Figure 3.1.

presents the elementary diffusion mechanisms in a 2-dimensional lattice. Mechanisms (1) and (2) are direct ion exchanges. Their activation energies are, in both cases, very high. The most probable mechanisms for diffusion involve vacancies (3) or interstitial defects (4, 5) in the crystal structure.

In this study, hydrogen diffusion is assumed to involve polarons (i.e., a polaron is a hole or an electron-deficient site formed in the valence band, when an electron moves from the valence band into the conduction band; Serway et al., 1997).

The polaron is a charge carrier and appears as a positive charge h⁺ or h^•. Here, the polaron is localized on iron atoms occupying octahedrally coordinated metal cation sites, metal vacancies, and/or silicon vacancies to charge balance the protons.

Figure 3. 1. Various atomic mechanisms of diffusion (redrawn from Putnis, 1992).

(1) and (2) are exchange mechanisms without involving vacancies (3) is a vacancy migration mechanism

(4) and (5) are interstitial migration mechanisms.

Activation energy

As diffusion is thermally activated process, it can be described by an Arrhenius law:

(

)

D

D~_i = ~_io exp − _i /R Eq. 17

where D~_i

is the chemical diffusion coefficient parallel to i =[100], [010] or [001],

~o

Di is the pre-exponential term, Qi is the activation energy for diffusion, T is the temperature in Kelvin and R is the gas constant.

Each point defect mechanism has its own activation energy. In certain cases, it may be possible, by comparing experimental values and theoretic models, to identify the point defect mechanism that controls diffusion.

3.4. Ionic diffusion in olivine

Over the past decade, a number of studies have focused on the diffusion of ionic species in silicates (Béjina and Jaoul, 1997), especially olivine (Houlier et al., 1988, 1990; Dohmen et al., 2002). Several approaches have been used. Experimental studies have generally involved two different diffusion processes: ionic self-diffusion and interdiffusion (typically Fe-Mg interdiffusion in olivine), as well as computer simulation (Walker et al., 2003). The diffusion coefficient of a given species is strongly dependent of the chemical composition of the solid. Notably, cation vacancy concentrations vary with trivalent cation content, which is a function of oxygen fugacity. The results of experimental studies to date are not perfectly consistent with each other due to the anisotropy of diffusion. Within anisotropic solids, ionic diffusion measurements must be performed as functions of the crystallographic orientation of the solid. For example oxygen diffusion in diopside is anisotropic, with the [010] axis the slowest direction of diffusion (Ingrin et al., 2001).

Previous research on ionic diffusion in olivine (Houlier et al., 1988, 1990;

Béjina and Jaoul, 1997; Dohmen et al., 2002) has shown that, at 1400°C, the silicon ion is the slowest species to diffuse in olivine (~10^-20m²/s) followed by oxygen (10

-18m²/s), divalent metal cations (Mg, Fe, ~10^-15 m²/s), and metal vacancies (10^-11 m²/s).

In brief, the self-diffusion of atomic species in olivine is characterized as follows.

Oxygen: Experimental results (Gérard and Jaoul, 1989, Ryerson, 1989) suggested an interstitial mechanism for oxygen diffusion with a very weak anisotropy such as D[010]>D[100]≈D[001]. Recent computer simulations (Walker et al., 2003) suggested that an increase in oxygen fugacity would change the interstitial diffusion to a vacancy

mechanism and that diffusion is isotropic. Silicon: According to numerous experimental studies (Anderson et al., 1989; Houlier et al., 1990; Bejina and Jaoul, 1997; Dohmen et al., 2002), silicon is the slowest species to diffuse in olivine or forsterite. The anisotropy of diffusion is a function of the oxygen fugacity with D_[100]>D_[010]>D_[001] buffered with MgSiO₃ and D_[010]>D_[100]>D_[001] buffered with MgO (Anderson et al., 1989). Oxygen and silicon self-diffusivities are both lower in forsterite than in iron-bearing olivine (Houlier et al, 1988). Magnesium: Chakraborty et al. (1994) have demonstrated a dependency of Mg diffusivity on pressure and oxygen fugacity. The Mg diffusion coefficient decreases with increasing pressure and with decreasing oxygen fugacity. The anisotropy of diffusion is very different from the simulation for oxygen or silicon self-diffusion with D[001]>D[100]>D[010]. Iron:

From Nakamura and Schmalzried (1983), it is known that the dominant defects involve changes in Fe redox state and that the Fe point defect population is proportional to f O21/n

where n varies with pressure and . The anisotropy of diffusion is similar to Mg diffusion but at low f O

SiO2

a

2, the [100] axis was reported to be the fastest direction of diffusion (Jurewicz and Watson, 1988). This is again attributed to a change in the diffusion mechanisms. Hydrogen diffusion mechanisms in olivine are detailed in the next subsection.