Diffusion in a multi-component system (1) Diffusion without interaction (2) Diffusion with electrostatic (chemical) interaction

Which D? 1.Which species (Mg,, Si or O)? 2. Diffusion through grains or diffusion along grain-boundaries? (m=2 or 3)

diffusion in olivine (volume diffusion)

dislocation (slip direction, slip plane: slip system)

Stress-strain field and energy of a dislocation Stress field Energy of a dislocation (J/m)

Dislocation creep The Orowan equation Non-linear rheology (strain-rate~ , n~3) Anisotropic rheology: depends on the slip system v=v  for low stress (m~2) n

dislocation density vs. stress

Stress dependence of strain-rate

Grain-size dependence of strength : grain-size reduction can result in significant weakening

An example of “deformation mechanism map”

Slip systems: deformation by dislocation motion is anisotropic

Plastic anisotropy (olivine) Durham et al. (1977) Bai et al. (1991) Rate of deformation depends on the orientation of crystal (slip system).

von Mises condition

Slip systems and deformation The strength of a polycrystalline material is controlled largely by the strength of the hardest (strongest) slip system. The deformation microstructure (lattice preferred orientation) is large controlled by the softest slip system.

The principle of lattice preferred orientation

Slip Systems and LPO Seismic anisotropy is likely due to lattice preferred orientation (LPO). Deformation of a crystal occurs by crystallographic slip on certain planes along certain directions (slip systems). During deformation, a crystal rotates to direction in which microscopic shear coincides with imposed macroscopic shear to form LPO. Therefore, if the dominant slip system changes, LPO will change (fabric transition), then the nature of seismic anisotropy will change. olivine

Deformation along the [001] orientation is more enhanced by water than deformation along the [100] orientation.

Type A: “dry” low stress Type B: “wet” high stress Type C: “wet” low stress Water-induced fabric transitions in olivine Distribution of orientation of crystallographic axes is non-uniform after deformation (lattice preferred orientation). The pattern of orientation distribution changes with water content (and stress,----). Jung and Karato (2001)

A lattice preferred orientation diagram for olivine (at high temperatures) (Jung and Karato, 2001) Dominant LPO depends on the physical conditions of deformation. This diagram was constructed based on high-T data. What modifications could one need to apply this to lower-T?

Thermal activation under stress jump probability At low stress At high stress

The Peierls mechanism At high stresses, the activation enthalpy becomes stress dependent.-> highly non-linear creep H*: enthalpy of formation of a kink pair  p: Peierls stress  slip system dependent (anisotropic) Effective activation enthalpy decreases with stress. Highly non-linear rheology (important at high stress, low temperature)

Pressure effects Pressure effects are large. In a simple model, pressure either enhances or suppresses deformation.

Reliable quantitative rheological data from currently available apparatus are limited to P<0.5 GPa (15 km depth: Rheology of more than 95% of the mantle is unconstrained!).

30-100% for P 2 -P 1 <0.5 GPa 3-10% for P 2 -P 1 <15 GPa Although uncertainties in each measurements are larger at higher-P experiments, the pressure effects (V*) can be much better constrained by higher-P experiments.

Various methods of deformation experiments under high-pressures Multianvil apparatus stress-relaxation tests D-DIA Rotational Drickamer Apparatus (RDA) Very high-P Mostly at room T Unknown strain rate (results are not relevant to most regions of Earth’s interior.) DAC Stress changes with time in one experiment. Complications in interpretation Constant displacement rate deformation experiments Easy X-ray stress (strain) measurements Strain is limited. Pressure may be limited. Constant shear strain-rate deformation experiments Large strain possible High-pressure can be achieved. Stress (strain) is heterogeneous. (complications in stress measurements)

Effect of pressure at the presence of water (water-saturated conditions) Increased water fugacity enhances deformation at high P. Pressure suppresses mobility of defects (V* effect).  non-monotonic dependence on P (Karato, 1989) pressure, GPa log viscosity

How could water be dissolved in nominally anhydrous minerals? Water (hydrogen) is dissolved in nominally anhydrous minerals as “point defects” (impurities). [Similar to impurities in Si (Ge).] (Karato, 1989; Bai and Kohlstedt, 1993)

Pressure effects under “wet” conditions can be more complicated. Fugacity of water affects rheological properties. Fugacity of water increases significantly with pressure.

Solubility of water in olivine Given a plausible atomistic model, we can quantify the relation between solubility of water and thermodynamic conditions (pressure, temperature). Solubility of water in olivine (mineral) increases with pressure. Kohlstedt et al. (1996)

Pressure effects on creep strength of olivine (“dry” conditions) Strength increases monotonically with P under “dry” conditions. Pressure, GPa Strength, GPa

Pressure effects on creep strength of olivine (“wet” conditions) Variation in the strength of olivine under “wet” conditions is different from that under “dry” conditions. The strength changes with P in a non-monotonic way. High-P data show much higher strength than low-P data would predict. pressure, GPa strength, GPa

A two-parameter (r, V*) equation fits nicely to the data. pressure, GPa water fugacity, GPa nornalized strength

The effects of water to reduce the viscosity are very large. (C OH : water content) (Karato and Jung, 2003)

Stress measurement from X-ray diffraction d-spacing becomes orientation-dependent under nonhydrostatic stress. Strain (rate) can also be measured from X-ray imaging.