Some remarks on micro-physics
of LPO (plastic anisotropy)
some tutorials
Shun-ichiro Karato
Yale University
Department of Geology & Geophysics
12/14/2009
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Why LPO?
upper mantle
D” layer
transition zone
x=
( )
VSH 2
VSV
Visser et al. (2008)
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Trampert and van Heijst (2002)
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Panning and Romanowicz (2006)
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• Seismic anisotropy is controlled mostly by LPO.
• But the relationships between LPO and flow geometry
are poorly known for most part of Earth’s interior.
• LPO is determined by the dominant slip systems (LPO)
that are controlled by a combination of many
microscopic processes (physics of LPO is complex!).
 experimental approach: (i) systematic, well-defined lab
experiments + (ii) scaling analysis
 theoretical approach: (i) key parameters (diffusion,
dislocation properties) + (ii) integration of multi-scale
physics of deformation
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olivine
wadsleyite
(preliminary results)
[100]
[010]
[001]
~1800 K
~1700 K
~1500 K
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LPO-flow geometry relationship depends on (i) materials, and (ii)
physical/chemical conditions (fabric transitions).
ei
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(
) (
s
,
; fH 2O ( P,T ) = e j
Tm ( P ) m ( P,T )
T
s
,
; fH 2O ( P,T )
Tm ( P ) m ( P,T )
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T
)
5
Origin of plastic anisotropy (dislocation creep)
How do fabric transitions occur? What controls LPO?
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Dislocation climb in an anisotropic crystal
æ Jx ö
æ Dxx
ç J ÷ = -ç 0
ç y÷
ç
çè J ÷ø
çè 0
z
à
¶c
¶t
0
Dyy
0
= Dxx
æ ¶c ö
0 ö ¶x
ç ÷
÷
÷
0 ç ¶c
÷ ç ¶y ÷
Dzz ÷ø ç ¶c ÷
è ¶z ø
¶2 c
¶x 2
+ Dyy
¶2c
¶y 2
climb rate z direction:
+ Dzz
µ
(D
¶2c
¶z 2
xx
y direction:
µ ( Dxx
x direction:
µ
plastic anisotropy:
Dxx + Dyy
Dxx + Dzz
plastic anisotropy:
Dxx + Dyy cz
Dxx + Dzz cy
(D
yy
+ Dyy
+ Dzz
+ Dzz
)
)
)
etc. (for diffusion-controlled model)
etc. (for diffusion + jog model)
à strong anisotropy if D along one axis is fast (quartz)
à weak anisotropy if D on one plane is fast (ppv?)
à anisotropy in jog density is critical
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Si diffusion in olivine is nearly isotropic.
--> diffusion controlled model does not explain large
plastic anisotropy
“dry”
Houlier et al. (1981)
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from Costa and Chakraborty (2008)
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Fabric transitions for olivine
A
B
C
D
E
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Observations for olivine
• Dominant slip direction is b=[100] (or [001])
• consistent with the role of kink/jog
• Stress-induced fabric transitions
• inconsistent with the simple diffusion-controlled model
• Larger water weakening effect for b=[001]
• cannot be explained by the simple diffusion-controlled model
• larger weakening effects for dislocations with larger Peierls stress
(or longer Burgers vector) E* µ E × E
E* µ E
k
l
P
j
l
anisotropy is largely due to dislocation-related properties
(not by diffusion)
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A model of fabric transitions
ei
(
) (
s
,
; fH 2O ( P,T ) = e j
Tm ( P ) m ( P,T )
T
(
e = A× 1+ B×
f Hr 2O
)
s
,
; fH 2O ( P,T )
Tm ( P ) m ( P,T )
T
æ
*æ
× s × exp ç - HRT ç 1 è
è
n
)
qösö
( ) ÷ø ÷ø
s
sP
• Stress/temperature-induced fabric transition (low T, high
stress) [kink energy (Peierls stress)]
log
A1
A2
=
H1*
RT
æ
çè 1 -
( )
s
s P1
q1 ö s1
÷ø -
H 2*
RT
æ
çè 1 -
( )
s
sP2
q2 ö s2
÷ø
• Water/temperature-induced fabric transition (high T, low
stress, high water fugacity) [diffusion (point defects), jog energy]
log
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r
A1 1+ B1 f H12O
A2 1+ B f r2
2 H 2O
=
H 1* - H 2*
RT
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Testing the jog (+ diffusion) - controlled model
some speculations on post-perovskite phase
Eshelby-Foreman theory for dislocation
energy with anisotropic elasticity
Jog-controlled climb model is consistent
with olivine data.
 [100](010) or [100](001)
for post-perovskite?
diffusion creep?
• large dislocation energy
• fast diffusion (Karki-Khanduja (2007))
post-perovskite
a= 0.2456 nm
b= 0.8042 nm
c= 0.6093 nm
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Conclusions
 Plastic anisotropy is caused mostly by anisotropic
dislocation properties (not much by diffusion anisotropy).
 Plastic anisotropy depends on T, stress, water content etc.
 lab studies: well-defined experiments (high-T, low
stress) + scaling analysis
[Direct applications of lab results without scaling analyses can lead
to misleading conclusions.]
 modeling: test with well-known materials (e.g., olivine)
and then apply to not-yet-studied minerals
[jog-controlled model (high-T plasticity model) works OK for olivine, and
suggests [100](010) or [100](001) (or [001](100))is the easiest slip system
in post-perovskite. But deformation in ppv might occur by diffusion creep.]
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Plastic anisotropy of post-perovskite ?
a= 0.2456 nm
b= 0.8042 nm
c= 0.6093 nm
(1) very small a/b, a/c ratio
à anisotropic kink density
or anisotropic jog density
(need to include anisotropic Cij)
(2) highly anisotropic diffusion (?)
à small effect because
Dyy + Dzz
Dxx + Dzz
controls anisotropy
à kink/jog density anisotropy likely dominates
Ek* µ El × EP
kink: Peierls potential + line energy
E *j µ El
jog: line energy
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Micro-physics of LPO
(reminder of ABC of LPO)
• LPO depends on macroscopic deformation geometry and
microscopic deformation mechanisms.
• Deformation by dislocation creep produces LPO.
• LPO formed by dislocation creep depends activity of slip
systems.
• LPO is largely controlled by easiest (+ some other) slip
system(s).
• The relative easiness of slip systems is controlled by the
relative rate of deformation that is controlled by (i)
anisotropy of dislocation energy (kink, jog formation
energy), (ii) by anisotropy of diffusion.
– These factors will change with T, P, stress, water content etc.
– Results at conditions different from Earth’s interior (e.g., low
T) cannot be applied to Earth’s interior.
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How should we investigate LPO
relevant to Earth’s interior?
(micro-physics of LPO is complex)
• Experimental approach: time scales are vastly different
between lab and Earth (need extrapolation)
• what kind of experiments should we conduct ?
• How should we extrapolate these results ?
• Theoretical (modeling) approach: creep processes
are complex
• How should we infer the dominant slip system(s)?
• Diffusion coefficients ?
• Dislocation properties ?
• How should we integrate ?
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Strong LPO develops by deformation only
through certain mechanisms
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• Classic diffusion-controlled high-T creep
model: can it explain fabric transitions?
• Peierls stress: How does it explain plastic
anisotropy at high-T?
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