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Diffusion in a multi-component system. (1) Diffusion without interaction. (2) Diffusion with electrostatic (chemical) interaction. Which D? Which species (Mg,, Si or O)? 2. Diffusion through grains or diffusion along grain-boundaries? (m=2 or 3). diffusion in olivine (volume diffusion).

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Diffusion in a multi-component system

(1) Diffusion without interaction

(2) Diffusion with electrostatic (chemical) interaction


Which D?

Which species (Mg,, Si or O)?

2. Diffusion through grains or diffusion along

grain-boundaries? (m=2 or 3)



(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


v=v(s) for low stress

Non-linear rheology (strain-rate~s, n~3)

Anisotropic rheology: depends on the slip system



Grain-size dependence of strength:

grain-size reduction can result in significant weakening

plastic anisotropy olivine
Plastic anisotropy (olivine)
  • Rate of deformation depends on the orientation of crystal (slip system).

Durham et al. (1977)

Bai et al. (1991)

slip systems and deformation
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.
slip systems and lpo
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.



Deformation along the [001] orientation

is more enhanced by water than deformation

along the [100] orientation.


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,----).

Type A: “dry” low stress

Type B: “wet” high stress

Type C: “wet” low stress

Jung and Karato (2001)


A lattice preferred orientation diagram for

olivine (at high temperatures)

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?

(Jung and Karato, 2001)


Thermal activation under stress

jump probability

At low stress

At high stress

the peierls mechanism
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

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 P2-P1<0.5 GPa

3-10% for P2-P1<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

Rotational Drickamer

Apparatus (RDA)

Multianvil apparatus

stress-relaxation tests



Very high-P

Mostly at room T

Unknown strain rate

(results are not relevant to

most regions of Earth’s interior.)

Stress changes with time in

one experiment.

Complications in interpretation

Constant shear strain-rate

deformation experiments

Large strain possible

High-pressure can be achieved.

Stress (strain) is heterogeneous.

(complications in stress measurements)

Constant displacement rate

deformation experiments

Easy X-ray stress (strain)


Strain is limited.

Pressure may be limited.

effect of pressure at the presence of water water saturated conditions
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

log viscosity

pressure, GPa

(Karato, 1989)

how could water be dissolved in nominally anhydrous minerals
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
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
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
Pressure effects on creep strength of olivine (“dry” conditions)
  • Strength increases monotonically with P under “dry” conditions.

Strength, GPa

Pressure, GPa

pressure effects on creep strength of olivine wet conditions
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.

strength, GPa

pressure, GPa


nornalized strength

water fugacity, GPa

A two-parameter (r, V*) equation

fits nicely to the data.

nornalized strength

pressure, GPa


The effects of water to

reduce the viscosity

are very large.

(COH: water content)

(Karato and Jung, 2003)


Stress measurement from X-ray diffraction

d-spacing becomes


under nonhydrostatic stress.

Strain (rate) can also be

measured from X-ray imaging.