Diffusion in a multi-component system

€

˜ D = x

B

D

A

+ x

A

D

B

€

x

A

+ x

B

= 1

(1) Diffusion without interaction

(2) Diffusion with electrostatic (chemical) interaction

€

˜ D = α +β( )α

D A +β

DB

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

€

Aα 1Aα 2

Aα 3− − − ( Mg2SiO4 )

€

˜ D =α i

i∑

α i

Dii∑

= 72

DMg+ 1

DSi+ 4

DO

€

Deffi = DV

i + πδL DB

i

diffusion in olivine (volume diffusion)

dislocation(slip direction, slip plane: slip system)

Stress-strain field and energyof a dislocation

€

σ = μb2πKr

€

E = μb2

4πK log Rbo

( ) ~ μb2

Stress field

Energy of a dislocation (J/m)

When a dislocation moves through a crystal with a dimension L, then an averagestrain of ~

€

b

L

will be created. Thus, when a dislocation moves a small distance Δ L , then

an increment of strain will be

€

Δ ε ~

b

L

Δ L

L

= ρ b Δ L . (10-1)

Therefore

€

˙ ε =

Δ ε

Δ t

= b ρ

Δ L

Δ t

+ b Δ L

d ρ

dt

= b ρ v + b Δ L

d ρ

dt

. (10-2)

Because we take a limit of

€

Δ L → 0 , the second term can in most cases be neglected, andequation (10-2) is reduced to,

€

˙ ε = b ρ v . (10-3)

Dislocation creep

The Orowan equation

€

˙ ε = bρv

€

v =2πDc jσΩ

bRT logLb

Non-linear rheology (strain-rate~ , n~3)Anisotropic rheology: depends on the slip system

v=v for low stress

€

ρ =αb−2 σμ( )

m(m~2)

n

dislocation density vs. stress

€

ρ =αb−2 σμ( )

m

Stress dependence of strain-rate

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

An example of “deformation mechanism map”

Table 5-1. Slip systems of typical materials

crystal structure Burgers vector (glide direction) glide planefcc metalbcc metalhcp metalB1 (NaCl-type)quartzspinelgarnetolivineorthopyroxeneclinopyroxenewadsleyiteperovskite (cubic)ilmenite

€

1

2

< 110 >

€

1

2

< 111 > , <100>

€

1

3

< 11 2 0 > , <0001>,

€

1

3

< 11 2 3 >

<110>

€

1

3

< 11 2 0 > , <0001>,

€

1

3

< 11 2 3 >

€

1

2

< 110 >

€

1

2

< 111 >

€

1

2

< 111 > , <100>[100], [001][001]

[001]

€

1

3

< 11 2 0 >

{111}, {110}, {100}{110}, {112}, {123}(0001),

€

{ 1 1 0 0 }

{110}, {100}, {111}(0001),

€

{ 10 1 0 } ,

€

{ 10 1 1 }

{110}, {100}{110}(010), (100), (001), {0kl}(100)(100)

(0001)

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 and the independent slip systemsPlastic deformation by dislocation motion occurs only with a limited geometry.

Therefore if only one slip system operates, then only one type of simple sheardeformation can occur for a given crystal. Deformation of a poly crystalline materialcannot occur by d islocation glide in such a case. For a homogeneous deformation of apolycrystalline material to occur by dislocation motion, a certain number of independentslip systems must be present. Consider a polycrystalline materials made of crystals withrandom orientation. If homogeneous deformation were to occur in such a material, acrystal must be a ble to change its shape in an arbitrary fashion. Consequently, thedeformation by d islocation glide must be able to create any arbitrary strain. The totalstrain due to different slip systems can be written as,

€

ε

ij

T

= ε

ij

k

k

∑ =

1

2

γ

k

( n

j

k

l

i

k

+ n

i

k

l

j

k

)

k

∑ (5-65)

where

€

ε

ij

k is strain caused by th e k-th slip system. Since the volume is conserved by

plastic deformation,

€

ε

11

T

+ ε

22

T

+ ε

33

T

= 0 . (5-66)

Equations (2-65) and (2-66) give a set of five equations to determine five unknowns,

€

γ

k .Consequently, five independent slip systems, i.e., five independent sets of (n, l) areneeded to achieve homogeneous deformation. This condition is referred to as the vonMises condition.

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] orientationis more enhanced by water than deformation along the [100] orientation.

Type A: “dry” low stressType B: “wet” high stressType C: “wet” low stress

Water-induced fabric transitions in olivine

Distribution of orientationof crystallographic axes is non-uniform after deformation (lattice preferred orientation).

The pattern of orientationdistribution changes withwater 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 =

€

exp −

H

* +

σ( )

RT

( )− exp −

H

* −

σ( )

RT

( )

At low stress,

€

H

* +

= H

o

*

− A σ and

€

H

* −

= H

o

*

+ A σ

and

€

exp −

H

* +

σ( )

RT

( )− exp −

H

* −

σ( )

RT

( )≈ exp −

H

*

RT

( )

2 A σ

RT

.

At high stress,

€

exp −

H

* +

σ( )

RT

( )− exp −

H

* −

σ( )

RT

( )≈ exp −

H

* +

σ( )

RT

( )= exp

−

H

o

*

RT

( )exp

A σ

RT

jump probability

At low stress

At high stress

The Peierls mechanismAt high stresses, the activation enthalpy becomesstress dependent.-> highly non-linear creep

H*: enthalpy of formation of a kink pairσp: Peierls stress

slip system dependent (anisotropic)Effective activation enthalpy decreases with stress.Highly non-linear rheology (important at high stress,

low temperature)

€

˙ ε = Bb−2 σμ( )

2exp − H *−Aσ

RT( )∝ σ 2 exp AσRT( )

€

σP = H *

A

Pressure effects

Pressure effects are large.

In a simple model,pressure either enhances or suppresses deformation.

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

€

˙ ε = Aσ n exp − E*+PV *

RT( )

€

δV*

V* = 30-100% for P2-P1<0.5 GPa3-10% for P2-P1<15 GPa

€

δV*

V* = RTP1−P2 V*

δ˙ ε ˙ ε + n δσ

σ + PV *

RTδPP + H *

RTδTT( )

P2-P1 0.5 GPa 15 GPa

€

δ P / P 1% 5%

€

δ σ / σ 1% 10%

€

δ ˙ ε / ˙ ε 5% 20%

€

δ T / T 0.1% 3%

€

δ d / d 10% 10%

Although uncertainties in each measurements are largerat higher-P experiments, the pressure effects (V*) can be muchbetter constrained by higher-P experiments.

Various methods of deformationexperiments under high-pressures

Multianvil apparatusstress-relaxation tests D-DIA

Rotational DrickamerApparatus (RDA)

Very high-PMostly 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 interpretationConstant displacement rate deformation experimentsEasy X-ray stress (strain)

measurementsStrain is limited.

Pressure may be limited.

Constant shear strain-rate deformation experiments

Large strain possibleHigh-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

vis

cosi

ty

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

Str

engt

h, G

Pa

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.

fugacity effect

V* effect

pressure, GPa

stre

ngt

h, G

Pa

A two-parameter (r, V*) equationfits nicely to the data.

pressure, GPa

water fugacity, GPa

nor

nal

ized

str

engt

hn

orn

a liz

ed s

tren

gth

The effects of water toreduce the viscosity are very large.

(COH: water content)(Karato and Jung, 2003)

Stress measurement from X-ray diffraction

d-spacing becomesorientation-dependent under nonhydrostatic stress.

Strain (rate) can also bemeasured from X-ray imaging.