Transcript Document

Diffusion in a multi-component system
(1) Diffusion without interaction
˜  x BDA  x ADB
D
x A  xB  1
(2) Diffusion with electrostatic (chemical) interaction
˜    
D

DA

DB
Which D?
1. Which species (Mg,, Si or O)?
2. Diffusion through grains or diffusion along
grain-boundaries? (m=2 or 3)
A A A    ( Mg2SiO4 )
1
˜ 
D
2
 i
i
i

i Di
3

7
2
1
4


Mg
S
i
D
D
DO
i  Di   Di
Deff
V
L B
diffusion in olivine (volume diffusion)
dislocation
(slip direction, slip plane: slip system)
Stress-strain field and energy
of a dislocation
Stress field

b
2Kr
Energy of a dislocation (J/m)
E

b 2
R
log
4 K
bo
~ b 2
When a dislocation moves through a crystal with a dimension L, then an average
st rain of ~ bL will be created. Thus, when a dislocation moves a small dist ance L , then
an increment of strain will be
 ~
b L
L L
  bL .
(10-1)
Therefore
Ý
 

t
 b Lt  bL
d
dt
 bv  bL
d
dt
.
(10-2)
Because we take a limit of L  0 , the second term can in most cases be neglect ed, and
equat ion (10-2) is reduced to,
Ý
  bv .
(10-3)
Dislocation creep
The Orowan equation
Ý  bv


 
2  m
 b
v=v for low stress
(m~2)
v
2Dc j
bRT log Lb
Non-linear rheology (strain-rate~,n n~3)
Anisotropic rheology: depends on the slip system
dislocation density vs. stress

 
2  m
 b
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
Tabl e 5-1. Slip syst ems of t ypical mat erials
crystal st ruct ure
fcc metal
bcc metal
hcp metal
B1 (NaCl-t ype)
quartz
spinel
garnet
olivine
orthopyroxene
clinopyroxene
wadsleyit e
perovskite (cubic)
ilmenit e
Burgers vector (glide direction)
1
2  110 
1
2  111, <100>
1
1
3  1120 , <0001>, 3  1123 
<110>
1
1
3  1120 , <0001>, 3  1123 
1
2  110 
1
2  111
1
2  111, <100>
[100], [001]
[001]
glide plane
{111}, {110}, {100}
{110}, {112}, {123}
(0001), {1100}
{110}, {100}, {111}
(0001), {1010}, {101 1}
{110}, {100}
{110}
(010), (100), (001), {0kl}
(100)
(100)
[001]
1
3  1120 
(0001)
Plastic anisotropy (olivine)
• Rate of deformation
depends on the
orientation of crystal
(slip system).
Durham et al. (1977)
Bai et al. (1991)
von Mises condition
von Mises condition and the independent slip systems
Plastic deformation by dislocat ion mot ion occurs only wit h a limited geomet ry.
Therefore if only one slip system operates, t hen only one type of simple shear
deformation can occur for a given crystal. Deformation of a poly cryst alline material
cannot occur by d islocation glide in such a case. For a homogeneous deformation of a
polycrystalline material to occur by dislocation motion, a certain number of independent
slip syst ems must be present . Consider a polycryst alline mat erials made of crystals with
random orientation. If homogeneous deformat ion were to occur in such a material, a
crystal must be a ble to change its shape in an arbitrary fashion. Consequen tly, the
deformation by d islocat ion glide must be able t o creat e any arbitrary st rain. The total
st rain due to different slip systems can be writt en as,
ijT  ijk   12  k (n kj lik  nik l kj )
k
(5-65)
k
where ijk is st rain caused by the k-th slip system. Since the volume is conserved by
plastic deformation,
T
11
  T22   T33  0 .
(5-66)
Equat ions (2-65) and (2-66) give a set of five equations to determine five unknowns,  k .
Consequently, five independent slip systems, i.e., five independent sets of (n, l) are
needed to achieve homogeneous deformation. This condit ion is referred to as the 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.
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)
Type A: “dry” low stress
Type B: “wet” high stress
Type C: “wet” low stress
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
Jump probabilit y =
probability



exp  H RT   exp  H RT 
*
*

*  H *  A and H *  H *  A
H
At low stress
o
o
At low st ress,
and
exp

H *  
 RT
 exp
 exp  
 exp
 exp 
H *  
 RT
* 2A
H
 RT RT
.
At high
st ress,
At high
stress
exp

H *  
 RT
H *  
 RT
H *  
 RT
 exp exp ART
H o*
 RT
The Peierls mechanism
At high stresses, the activation enthalpy becomes
stress dependent.-> highly non-linear creep
Ý

  
 2
2
Bb
exp
* A
H
 RT
  2 expART 
*
H

 A of formation of a kink pair
P enthalpy
H*:
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!).
Ý  A exp

n
V *
V*


* PV *
E
 RT
RT

Ý


n
Ý

P1 P2 V * 
P2-P1
0.5 GPa
1%
1%
5%
0.1%
10%
P / P
 / 
Ý / 
Ý

T / T
d / d
V *
V*



PV * P
RT P

H * T
RT T

15 GPa
5%
10%
20%
3%
10%
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
DAC
Multianvil apparatus
stress-relaxation tests
Very high-P
Stress changes with time in
Mostly at room T
one experiment.
Unknown strain rate
Complications in interpretation
(results are not relevant to
most regions of Earth’s interior.)
D-DIA
Rotational Drickamer
Apparatus (RDA)
Constant shear strain-rate
deformation experiments
Large strain possible
Constant displacement rate
High-pressure can be achieved.
deformation experiments
Stress (strain) is heterogeneous.
Easy X-ray stress (strain)
(complications in stress measurements)
measurements
Strain is limited.
Pressure may be limited.
• Increased water
fugacity enhances
deformation at high P.
• Pressure suppresses
mobility of defects
(V* effect).
• non-monotonic
dependence on P
log viscosity
Effect of pressure at the presence of
water (water-saturated conditions)
pressure, GPa
(Karato, 1989)
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
Pressure effects on creep strength
of olivine (“wet” conditions)
fugacity effect
V* effect
pressure, GPa
• 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.
water fugacity, GPa
A two-parameter (r, V*) equation
fits nicely to the data.
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
orientation-dependent
under nonhydrostatic stress.
Strain (rate) can also be
measured from X-ray imaging.