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2 - Deformation of mantle minerals
Les Houches 2018
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Seismic data mantle structure and flow
horizontal shear waves velocities anisotropy (direction and amplitude)
Yuan and Beghein, 2013
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A multiscale problem
Ando et al, Nature, 2007
B. Reynard
rock
atomic
regional
103 km < 10-10 m
crystal
planet
Deformation and transformation processes span 10-9 s to 108 Yr
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q Deformation 101
q Agents of deformation at crystal scale ?
q Consequences on physical properties « Viscosity »
Preferred orientations and seismic velocities
q Experimental tools for HP deformation
q Where we stand for mantle minerals
q Deformation of multi-phase aggregates
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Deformation 101
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generic macroscopic case
Société Francophone des Biomatériaux Dentaires (SFBD) elastic deformation: linear and reversible
stress proportional to deformation : material elasticity, with E Young’s modulus, σ = E ε
σ
ε
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<- flow stress depends on T, P, microstructure, f(O2)…
Société Francophone des Biomatériaux Dentaires (SFBD) plastic deformation: non linear, irreversible
permanent deformation
σ = F/S
ε = ΔL/L0
generic macroscopic case
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P/T effects on rheology
T increase
P increase
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rock = polycrystal
What controls the macroscopic behavior ...?
Ando et al, Nature, 2007
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rock = polycrystal l individual crystals l grain boundaries l microstructure
l grain size(s) l grain orientations l grain shapes
l contrasts in mineral properties…. Ando et al, Nature, 2007
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What are the agents of deformation at the crystal and polycrystal scale ? Resulting physical properties ?
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Crystals deform through the motion of lattice defects
direct deformation: • many chemical bonds to break • energetically not favorable • high stress
S.Merkel Les Houches 2018
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physics of worms l Let's consider a worm...
l This how they do...
crystals do the same...
Move all legs forward ? High energy cost.
S.Merkel Les Houches 2018
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Crystals deform through the motion of lattice defects
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Point defects:
vacancy interstitial substitution
Mobility : diffusion
Rheoman.eu / Multiscale modeling of the mantle rheology, 2018, eds. P. Cordier, A. Goryaeva.
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Dislocation
Dislocations
Movie ref.: Kasher and Roberston, Acta Mater. 2012
Creep, in-situ
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dislocations q dislocation = linear defect
“line” in the crystal structure
q moves under the effect of a shear stress
q moves parallel to crystallographic plane(s) (more or less)
q Displacement prop. to some interatomic distance (Burgers vector)
shear stress
slip plane
edge dislocation
line
REF
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slip plane and directions depend on the structure
rule of thumb q Slip direction: short translation in the lattice q Slip plane: dense packing plane
example: MgO
MgO crystal structure
½<110> slip
½<110> slip in {100}, {110} or {111}
Which dislocation ?
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Dislocation climb
à Diffusion-mediated (high T)
Rheoman.eu / Multiscale modeling of the mantle rheology, 2018, eds. P. Cordier, A. Goryaeva.
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…multiple mechanisms:
l intra-granular - ‘Atomic’ diffusion (red arrows) - dislocation glide (blue edges) - dislocation climb (mediated by
diffusion)
l grain boundaries - diffusion (red arrows) - disclinations (spirals) - dislocation assisted boundary
sliding (green arrows, +planar defects)
- diffusion assisted boundary sliding
picture S. Merkel
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Consequences on microstructure and on physical properties
‘viscosity’
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Resulting properties: constitutive laws, from deformation to stress
!ε = !ε(P,T ,d ,COH ,...) ??????
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Deformation to stress: constitutive laws
!ε = Aσ ne−QRT
!ε = Aσ nd − pCOHe−E+PV*RT
activation energy
stress exponent
activation energy E activation volume V*
grain size exponent
!ε = Ae−ERT
1− στ
⎛
⎝⎜
⎞
⎠⎟p⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
q
Low temperature
Under pressure
Peierls stress
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Deformation to stress: constitutive laws
!ε = Aσ nd − pCOHe−E+PV*RT
Olivine dislocation creep, Mei and Kohlstedt 2000
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Deformation to stress: constitutive laws
!ε = Aσ nd − pCOHe−E+PV*RT
Olivine dislocation and diffusion creep, Mei and Kohlstedt 2000
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association of plastic mechanisms
in series
in parallel
e.g. grain boundary sliding + dislocation creep
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Why n=3 or how to build a constitutive law
Strain produced by a density ρ of dislocations with b burgers vector, moving a distance x :
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Why n=3 or how to build a constitutive law
Strain produced by a density ρ of dislocations with b burgers vector, moving a distance x :
ε = ρbx
!ε = ρbvρ ≈σGb⎛
⎝⎜
⎞
⎠⎟
2
constant (steady state) àwith v velocity
!ε∝σ 3v ≈ vC ∝σ
Weertman 1999: if dislocation climb + glide operate in series, and slower process is climb
à
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Exponents vs. deformation mechanisms
l Diffusion dominated regime - p > 0 - n ≈ 1 - newtonian viscosity, function of d and T
l Dislocation creep dominated regime - p ≈ 0 - n ≈ 2-5 - non-linear viscosity, depends on σ and T
l Grain boundary sliding - p ≈ 2 - n ≈ 2
Viscosity
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association of plastic mechanisms
in series
in parallel
e.g. grain boundary sliding + dislocation creep
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deformation maps dominant deformation mechanism, depending on external conditions
parameters: stress temperature grain size strain rate pressure…
several mechanisms can be active at the same time
Frost and Ashby, Deformation-Mechanism Maps
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When looking at experimental flow laws in the literature, keep in mind
• conditions under which they have been measured (P, T, ε, d, composition)
• several mechanisms may act concurrently - microstructural understanding (characterisation) is a must have for use of a flow law, and a fortiori for extrapolation
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Further consequences of pressure : the famous “a-slip to c-slip switch” in olivine slip systems
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Further consequences of pressure : the famous “a-slip to c-slip switch” in olivine slip systems
a
b
c
b
a
c
J. Durinck, P. Cordier
shear along <a> in (010)
shear along <c> in (010)
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Consequences on microstructure and on physical properties preferred orientations and seismic velocities
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Olivine (Mg,Fe)SiO4
P wave velocity 0 GPa – 300 K
Anisotropy: single crystal
Stereographic projection:
S. Merkel Les Houches 2018
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Bridgmanite Single crystal anisotropy in both Vp and Vs about 8 % at 1000 km prof., 13 % at 2500 km.
D. Mainprice: Seismic anisotropy in the deep earth from a mineral and rock physics perspective, in Treatise on geophysics, vol.2, 2007 Images : Rheoman.eu / Multiscale modeling of the mantle rheology, 2018, eds. P. Cordier, A. Goryaeva.
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Bridgmanite single crystal at 88 GPa and 2000K
D. Mainprice: Seismic anisotropy in the deep earth from a mineral and rock physics perspective, in Treatise on geophysics, vol.2, 2007
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Post-perovskite Single crystal anisotropy in Vp 15% and Vs 22% at 120 GPa.
D. Mainprice: Seismic anisotropy in the deep earth from a mineral and rock physics perspective, in Treatise on geophysics, vol.2, 2007 Images : Rheoman.eu / Multiscale modeling of the mantle rheology, 2018, eds. P. Cordier, A. Goryaeva.
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D. Mainprice: Seismic anisotropy in the deep earth from a mineral and rock physics perspective, in Treatise on geophysics, vol.2, 2007
Post-perovskite single crystal
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No preferred orientations Random fabric Anisotropies of each crystal cancel each other
Anisotropy : polycrystal
Les Houches 2018 S. Merkel
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Plastic deformation Dislocation glide Grain rotations Non-random crystal orientations
olivine ex. of slip system
Anisotropy: plastic deformation
Les Houches 2018 S. Merkel
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Experimental tools for deformation
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high-pressure deformation tools
No stress probe available / large frictions above 2 GPa
Early 2000: new deformation apparatus needed, solid pressure medium + new measurement « device » needed….
Griggs, Heard, Paterson…
Deformation-Dia (2003)
Rotational Drickamer (RDA, 2001)
D T-cup (2010?)
DAC
geot
herm
Multi-anvil
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Diamond anvil cells
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images S. Merkel
q Highest pressure tool (to lower mantle, core conditions)
q Sample sizes: 100’s microns x 10’s microns
q Laser heat, resistive heat: large T range
q « uncontrolled » deformation
few 100 microns
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Large volume presses q Lower pressures q Larger samples ! larger grain sizes possible q Smaller T gradients q Control H2O, f(O2) q (easier) decoupling of
deformation from pressure (constant V) Durham et al, 2002
D-Dia
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Les Houches 2018
ESRF ID06-LVP
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Highest pressures and shear geometry
Nishiyama et al, 2008,
Kawazoe et al, 2009
Experimental cell for RDA Yamazaki and Karato, Rev Sci Instr (2001)
Rotational Drickamer (RDA): opposed anvils D-Dia double-stage
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Measuring stresses under high pressure ?
Coupling high pressure tools with synchrotron radiation: in-situ measures
Inc. X rays
ψ
Inc. X rays
ψq Transparent anvils (sintered
diamonds, cBN, …) q Transparent high pressure
cell (amorphous boron – epoxy, graphite furnace, hBN…)
Compres.us
Wang et al. 2003
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D-Dia typical cell assemblies
1.6 mm 2.0 mm
Boron + epoxy
1.2 mm
crushable alumina
densified alumina
BN
graphite
1.2 mm
Sample
thermocouple
buffer rod (Vp, Vs)
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radiography for strain in-situ
Time (s)
Str
ain
(% (
dε/dt=2.7 10-6 s-1
dε/dt=1.7 10-5 s-1
Metal sheets above and below P = 7 GPa T = 1673 K
pictures P. Raterron
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diffraction for stress in-situ
nλ = 2dsinθ
www.hyperphysics.phy-astr.gsu.edu
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diffraction for stress in-situ polycrystal diffraction of monochromatic x-rays
nλ = 2dsinθ
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diffraction for stress in-situ
figure S. Merkel
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diffraction for stress in-situ Unrolled pattern (« cake »)
Measures: • isotropic part σP • t = σ1-σ3
elastic theory, Sing et al, 1998, Uchida et al, 1996
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diffraction for stress in-situ
q lattice strains: for each diffraction plane, a « stress » measure
macroscopic stress … ? not straigthforward ! Some tools exist but still no satisfactory way to have « absolute » stress measurements.
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X-ray diffraction for microstructure
peak intensities variations and broadening q preferred orientations
« texture » q microstrains, crystallite size
preferred orientation -> slip systems -> + elastic constants = seismic properties of rocks
inverse pole figure for the maximum compression direction,
assuming cylindrical symmetry olivine at 5GPa, 1600K, strain
11%
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q Experimental tools exist at the relevant P and T. f(O2), H2O can be (somewhat) controlled.
q Natural strain rates *convection* down to 10-14, 10-16 s-1… : not attainable experimentally.
q Go to natural microstructures
q Models
q Experiments and models need to be validated by geophysical observations and/or geological samples
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Numerical modeling state of the art : example of MgO
Amodeo et al, 2018 Amodeo et al, 2016 Amodeo et al, 2014
Atomistic calculations Dislocation cores, lattice scale
Dislocation dynamics « DD » Dislocation interactions
crystal scale
Finite elements models polycristal
Great for bridging temporal scales Complex compositions and large unit cells: difficult ! Grain growth, nucleation, … currently not possible
Multiscale modeling of the mantle rheology, 2018, eds. P. Cordier, A. Goryaeva.
Les Houches 2018
Where we stand on mantle minerals (experimental) deformation
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(non exhaustive) experimental data on mantle minerals deformation
slip systems/deformation mechanisms
controlled deformation experiments
comments
olivine +++ +++ increasingly sophisticated constitutive laws and deformation maps
majorite + +
wadsleyite ++ +(+)
ringwoodite ++ +(+) LVP difficult
bridgmanite ++ (+) DAC most of the stab. field
post-perovskite
++ - DAC only
periclase +++ at ambient PT
++ Commonly investigated in material science
perovskites +++ at ambient PT
+ ? Commonly investigated in material science
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Wadsleyite deformation map
Farla et al, PCM2015
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Merkel and Cordier, 2016 in Deep earth: physics and chemistry of the lower mantle and core.
Slip systems in lower mantle minerals
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Ex. of seismic anisotropy of post-perovskite and D’’
DAC experiments
Wu et al, 2017
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Ex. of seismic anisotropy of post-perovskite and D’’
-> (001)[100] and [010] or {110} were active
DAC experiments Modeling slip systems
Wu et al, 2017
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Wu et al, 2017
Shear wave splitting larger than geophysical observations. Other deformation mechanisms which do not produce anisotropy ? - Diffusion ? - Planar defects ? - Second phase ?
Ex. seismic anisotropy of post-perovskite and D’’
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Multi-phase deformation
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load bearing frame interconnected weak layer Handy, 1994
q Which material is the weakest / strongest? q How do stress and strain partition in the phases? q What is the viscosity of the aggregate ?
Dealing with polymineralic deformation
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Bridgmanite/perovskite and MgO aggregates:
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Bridgmanite « strong »
+ periclase « weak »
Marquardt and Miyagi, 2015
q Is MgO weaker or stronger than bridgmanite ? q Which one controls the overall behavior ?
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elastic and plastic interactions between these phases and microstructural evolution ?
Experiments – Girard et al, 2015, 50% MgO – Kaercher et al, 2015, textures on two
phases analogs
Girard et al, Science 2015
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Girard et al, Science 2015
Bridgmanite and MgO deformed to large strains in Rotational Drickamer under lower mantle conditions
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Deformation experiments (D-DIA)
starting material
q Analog for the lower mantle (Mg,Fe)SiO3+(Mg,Fe)O q CaGeO3-pv + MgO two-phase polycrystal (70-30) q CaGeO3-Pv single-phase polycrystal, same P-T-deformation
paths q + unpub. run at strain > 25% 2-phases polycrystal, 1000K.
Les Houches 2018
800K$1000K$
600K$
1200K$
(0.01$
(0.008$
(0.006$
(0.004$
(0.002$
0$
0.002$
0.004$
0.006$
0.008$
0.01$
30$ 50$ 70$ 90$ 110$ 130$ 150$ 170$ 190$
Q(100)$Q(110)$Q(200)$Q(210)$Q(211)$Q(220)$
Textures$CaGePv$D0754$–$maximum$compressive$stress$inverse$pole$figures$$
Q(hkl)$
DiffracLon$#$
001$ 110$
111$
75
800K$1000K$
600K$
1200K$
(0.01$
(0.008$
(0.006$
(0.004$
(0.002$
0$
0.002$
0.004$
0.006$
0.008$
0.01$
30$ 50$ 70$ 90$ 110$ 130$ 150$ 170$ 190$
Q(100)$Q(110)$Q(200)$Q(210)$Q(211)$Q(220)$
Textures$CaGePv$D0754$–$maximum$compressive$stress$inverse$pole$figures$$
Q(hkl)$
DiffracLon$#$
001$ 110$
111$
Pv lattice strains in two-phase sample Wang et al, 2013
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MgO lattice strains in two-phase sample Wang et al, 2013
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9(2) GPa, 600 K
Bulk strain, %0 2 4 6 8 10 12
D0748, GePv single phase sampleD0754, GePv two-phased sampleGePv+MgO average in two phased sample
7(2) GPa, 800 K
Bulk strain, %0 2 4 6 8 10
5(1) GPa, 1000 K
Bulk strain, %0 2 4 6 8 10
Diff
eren
tial s
tres
s, G
Pa
0
1
2
3
4
5
GePv in two-phase
composite
Avg. GePv
+MgO
GePv single-phase
Wang, Hilairet et al. G3, 2013
q Two-phase sample stress average calculated with a Taylor approximation (volumetrically weighted average of phase stresses)
q Mechanical behavior still controlled by the strong phase (Pv)
……Larger strains and/or higher MgO content ?
Les Houches 2018
Les Houches 2018
