Lectures on
Rheology of Earth Materials
Fundamentals and frontiers in the study of
deformation of minerals and rocks
(at Tohoku University)
Shun-ichiro Karato
Yale University
Department of Geology & Geophysics
New Haven, CT, U.S.A.
June 25-27, 2003
Lecture Outline
1. Why rheology?
2. General background
defects and plastic deformation
thermodynamics
3. Some fundamentals of creep
3-1. Diffusion and diffusion creep
3-2. Dislocations, slip systems, and dislocation
creep
3-3. Deformation mechanism maps
3-4. Effects of phase transformations
3-5. Effects of pressure, water
4. Physical processes controlling the grain size
4-1. Grain-growth
4-2. Dynamic recrystallization
4-3. Nucleation-growth
5. Some applications
5-1. Lithosphere-asthenosphere
5-2. Rheology subducting slabs
5-3. Some unresolved problems
要点 1。スケーリング則
レオロジーでは時間依存性のある性質を扱うので
実験結果を直接地球には応用できない。
実験室での結果を地球に応用するときに物理的
モデルにもとづいたスケーリング則を確立しておく
必要がある。
例:流動則、結晶粒径(相転移)
要点 2。
• レオロジー的性質は温度、水、相転移、結
晶粒径などをとおして地球の進化、ダイナ
ミクスと密接に関係している。(密度や弾性
的性質などと違う。)
Why rheology?
• Rheology controls mantle convection.
– Mixing of geochemical reservoirs
• Rheology has strong influence on the way
in which seismic wave propagation is
affected by mantle convection.
– anelasticity, anisotropy
What causes velocity heterogeneity:
thermal or chemical origin (or both) ?
Much of the velocity heterogeneity in
the upper mantle has chemical origin
(ocean vs. continent).
To what extent is the deep mantle
chemically heterogeneous?
In regions where there is chemical
heterogeneity, what is the cause of
heterogeneity (which elements or
which minerals)?
Ritsema et al. (1999)
Models to explain geophysical and geochemical
observations
(Tackley, 2000)
Upper mantle
Transition zone
Trampert and van Heijst (2002)
Pattern of anisotropy is very different between
the upper mantle and the transition zone.
What does it tell us?
What do we want to know?
• How does rheology control convection pattern?
– What is the rheological stratification in Earth?
– How can we infer flow pattern from seismological observations?
• How does rheology affect mixing?
– Where are geochemical reservoirs and how have they survived (or
not survived)?
• Why plate tectonics on Earth and not on Venus?
– What are the mechanisms of localization of deformation?
• How have continents survived for billions of years?
– Is continental lithosphere ”dry” or “wet”?
Frontiers in the study of deformation
of minerals and rocks
• Extreme conditions
– High-pressure (whole Earth dynamics)
– Large strain
(anisotropy, grain-size evolution, strain
partitioning)
– Small strain
(seismic tomography-anelasticity
post-glacial rebound)
• Complexities
– deformation  chemical reaction(s) [metamorphism, phase
transformations, partial melting, hydrogen redistribution---]
– deformation  microstructural evolution
[instability, localization: “plate” formation, deep earthquakes]
Some fundamentals of mineral
and rock rheology
•
•
•
•
T-P dependence of plastic deformation
Stress, grain-size dependence
Dependence on chemical environment
Effects of phase transformations
Elastic deformation
Plastic deformation: involves atomic jumps
thermally activated processes
probability
T-dependence  time-dependence
Temperature dependence of elastic constant
and viscosity: ductile strength (viscosity) is highly
sensitive to T
Pressure dependence of elastic constant
and viscosity: ductile strength (viscosity) is highly
sensitive to P
Basic physics of plastic deformation
• Defects and plastic deformation
– thermo-chemical equilibria
– law of mass action
– fugacity
• Thermally activated processes
– origin of t-T dependence
– Boltzmann statistics
Ý    K (T ,P,C , ,  ,t)  N(T ,P,C,  , ,t )

K: rate (velo city) of motion of defects
 : plastic strain as sociated with a unit motion o f defect
N: number density of defects
T: temperature
P: pressure
C: chemical env ironment
¦ Defect densit y
P oint defects (dependent on T, P, C but not on  )
Dislocat ions (dependent on  but not onT, P and C)
Grain-boundaries (generally independent of thermodynamic
parameters)
¦ Defect mobilit y
This term depends strongly on T, P and st ress (and chemical
environment) and cont rols the dependence of rate of
deformation
Point defects
G(n)  ngf  TSconf (n) .
(5-8)
The configurat ional ent ropy, Sconf (n) , is given by (see Chapter 2),
Sconf (n)  kB log ( N Nn!)!n!
(5-9)
where kB is the Boltzmann constant. T he equilibrium stat e of t he system is t he one t hat
minimizes the Gibbs free energy so that , the equilibrium number of defect , n, is
determined by the equat ion,
 G( n )
n
 0.
(5-10)
Inserting the relat ion (5-9) and u sing the StirlingÕs formula, log n! n  log n  n , for
large n, one gets,
gf
n
N
n
N

exp( k
BT
gf
1exp( k
 exp 
)
BT
Sf
R
gf
)
~ exp( k
BT
).
 exp
(5-11)
E f PV f
RT
   exp
n
N o
E f PV f
RT

Box 5-1. Kršge r-Vin k notati on of point de fects
In considering a chemical reaction involving point defects in an ionic crystal, it is
important to consider the conservation of lat tice sites and of elect rost atic charge. T he
Kršg er-Vink notat ion of point defects is a useful tool to clearly define a point defect and
indicate its effect ive charge relat ive to the perfect lattice. A Kršg er-Vink not ation
contains three types of informat ion: X YZ . X indicates t he at omic species that sits on the
lat tice site Y, and Z represents t he effective charge (  is effective + charge,  is effective
2- charge and  is neutral). T hus for example, FeM ,FeM in olivine (Mg,Fe)2SiO4 indicate
a ferric iron (Fe3+) at M-site, a ferrous iron (Fe2+) and VM indicates a vacancy at M-site
with an effective charge of 2- (relative t o the perfect latt ice) respectively.
Effects of oxygen fugacity on defect concentration
1
2 O2
 2FeM  OO  VM  2FeM
[Fe M ] 2 [ V M ]
f O1/ 2
 K31
2
.
(5-31)
(5-32)
.

[ FeM ]  2[VM ]
(5-34)
1
[ FeM ]  2[VM
 ]  fO6 .
2
(5-35)
The Kröger-Vink Diagram for olivine
Effects of water

"
H2 O  M M
 VM
 2H   MOsurface .
   2
(5-46)
"
f H 2 O  K 46 VM
 H  aMO .
(5-47)
"
H   VM
 H'M
(5-48)
H  VM"  K 48H'M 
(5-49)
 
(5-50)
H'M
1
2
 
"
f H O VM
2
1
2
 21
aMO .
Let us consider an olivine crystal t hat is originally in chemical equilibrium under
"
water-poor conditions. In such a c ase, the dominant point defects are VM
a nd FeM .
Using equation (5-43), one has,
  f H O f
H'M
1
2
2
1
12
O2
 23
aMO .
(5-51)
The concent rat ion of t his defect increases with water fugacit y but the concent ration of
"
does not, and therefore at a certain point , the concent ration of this defect will exceed
VM
"
that of VM
. When this happens, t he dominant defects will be FeM and H'M . The charge
neutrality condit ion in this case is therefore,
H'M  FeM .
(5-52)
Using this relation toget her with (5-50) and (5-29), one gets,
  
H'M
FeM
 f H O f
1
4
2
1
8
O2
 14
aMO .
(5-53)
A Kröger-Vink Diagram for a water bearing system
Diffusion creep
c
J  D  x
(9-1)
cd    cd 0   exp(
(9-38)
J a  Dd
By noting
 cd
x
cd
x
exp(
2Dd cd 0   n
L
RT
 LÝ  LV2Ý  J a
Ý

 LÝ
L

J a
L
)
(9-39)
where L is grain-size, equation (9-39) becomes,
J a  Dd cd 0 
Ja ~
RT
  Jd.
cd (  n ) cd (   n )
,
L

n

 n
RT

)exp(
L
 n
RT
)
2Dc  n
L RT
2Dc   n 
L2 RT

2D  n 
L2 RT
Diffusion creep
Ý

n

A
Lm
D
n

B
Lm
exp

* PV *
E
 RT

n=1, m=2-3
Viscosity (  Ý) is independent o f stress
and strongly dependen t on grain-size:
  Lm.
m=2 for diffusion inside of grains, m=3 for diffusion
along the grain-boundary
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(at Tohoku University) Shun-ichiro Karato Yale University