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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(1) Diffusion without interaction
(2) Diffusion with electrostatic (chemical) interaction
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)
of a dislocation
Energy of a dislocation (J/m)
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 reduction can result in significant weakening
Durham et al. (1977)
Bai et al. (1991)
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.
is more enhanced by water than deformation
along the  orientation.
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)
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)
At low stress
At high stress
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 are large.
In a simple model,
pressure either enhances or
available apparatus are limited to P<0.5 GPa (15 km depth:
Rheology of more than 95% of the mantle is unconstrained!).
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.
experiments under high-pressures
Mostly at room T
Unknown strain rate
(results are not relevant to
most regions of Earth’s interior.)
Stress changes with time in
Complications in interpretation
Constant shear strain-rate
Large strain possible
High-pressure can be achieved.
Stress (strain) is heterogeneous.
(complications in stress measurements)
Constant displacement rate
Easy X-ray stress (strain)
Strain is limited.
Pressure may be limited.
(Karato, 1989; Bai and Kohlstedt, 1993)
Kohlstedt et al. (1996)
water fugacity, GPa
A two-parameter (r, V*) equation
fits nicely to the data.
reduce the viscosity
are very large.
(COH: water content)
(Karato and Jung, 2003)
under nonhydrostatic stress.
Strain (rate) can also be
measured from X-ray imaging.