JOURNAL OF PETROLOGY
VOLUME 51
NUMBER 4
PAGES 921^945
2010
doi:10.1093/petrology/egq006
Rheology of Synthetic Omphacite Aggregates
at High Pressure and HighTemperature
RASOUL H. MOGHADAM, CLAUDIA A. TREPMANN,
BERNHARD STO«CKHERT AND JO«RG RENNER
INSTITUTE FOR GEOLOGY, COLLABORATIVE RESEARCH CENTRE 526, MINERALOGY AND GEOPHYSICS,
RUHR-UNIVERSITY BOCHUM, D44780 BOCHUM, GERMANY
RECEIVED JULY 18, 2009; ACCEPTED FEBRUARY 19, 2010
ADVANCE ACCESS PUBLICATION MARCH 30, 2010
Eclogite rheology, an essential control on the mechanics of subduction
zones, is probably governed by the mechanical properties of the volumetrically dominant constituent, omphacite. Triaxial compression
tests were performed in a solid-medium apparatus on synthetic
aggregates with five compositions along the solid-solution series jadeite^diopside. The bulk water content of the synthetic material is of
the order of 102 H/Si.The deformation experiments were performed
at a confining pressure of 2·5 GPa and temperatures between 900
and 11008C (i.e. conditions in the stability field of jadeite). Strain
rates ranged from 106 to 104 s1. For a given composition, strength
decreases with increasing temperature and decreasing strain rate.
The relation between strength and composition changes with increasing temperature. Whereas jadeite is the weaker endmember at all
explored conditions, intermediate compositions are stronger than
either endmember at 9008C and 10008C. Samples with calcic compositions deformed at differential stresses above 500 MPa have
abundant mechanical (100)[001] twins. In contrast, almost all
deformation steps on jadeite aggregates yield strengths below
500 MPa and the resulting microstructures are dominated by
well-developed subgrain boundaries and locally recrystallized
grains. A weak crystallographic preferred orientation is developed in
the deformed samples. Combining mechanical data and observed
microstructures we hypothesize that the rate-controlling deformation
mechanism differs for the two compositional endmembers.
Deformation of diopside aggregates is controlled by dislocation
glide, whereas the kinetics of dislocation climb controls the deformation of jadeite aggregates at the explored conditions. Reasonable
parameters are found when fitting corresponding, micromechanically
motivated flow laws to our data.The variation in strength with composition obeys scaling based on the dependence of melting temperature
and shear modulus on composition. Extrapolation of the constrained
flow laws to geological strain rates indicates considerable strength
*Corresponding author. E-mail: [email protected]
for omphacite aggregates at temperatures up to 10008C. Continuous
deformation of eclogites by dislocation creep therefore appears unlikely
in subduction zones. We suspect that exhumed eclogites exhibiting
microstructures indicative of crystal plastic deformation experienced
high-stress episodes related to seismic events.
KEY WORDS: omphacite; solid solution; deformation; dislocation;
glide; climb; flow law; twinning
I N T RO D U C T I O N
Eclogites are found as xenoliths in kimberlites and alkali
basalts, as layers and lenses in alpine-type peridotite
bodies, and as blocks in blueschist-facies terrains, and also
constitute large volumes of the exhumed crust in highpressure metamorphic terrains (Carswell, 1990). Recently,
the term ‘eclogite engine’ has been coined to emphasize
that eclogitization is the expected universal fate of the
crustal portion of aging lithosphere and thus constitutes a
key player in plate tectonics (Anderson, 2007). In particular, eclogites are expected to replace the basalts and gabbros of the oceanic crust during subduction. The kinetics
of the eclogite-forming reactions suggest a transition at
depths exceeding about 50 km (e.g. Hacker, 1996; Kirby
et al., 1996). At greater depths a continuous eclogitic layer
with a thickness of several kilometres is predicted at the
top of the downgoing slab. The deformation characteristics
of this layer are of fundamental importance for stress
transfer at the plate boundary and therefore the dynamics
of subduction zones.
The Author(s) 2010. Published by Oxford University Press.
This is an Open Access article distributed under the terms of the
Creative Commons Attribution Non-Commercial License (http://
creativecommons.org/licenses/by-nc/2.5), which permits unrestricted
non-commercial use, distribution, and reproduction in any medium,
provided the original work is properly cited.
JOURNAL OF PETROLOGY
VOLUME 51
NUMBER 4
APRIL 2010
Fig. 1. (a) Phase diagram for the jadeite^diopside system (after Carpenter, 1980). Whereas omphacite crystals of intermediate composition may
exhibit an ordered structure (space group P2/n) up to temperatures of about 7008C, all aggregates tested in the current experiments are in the
disordered state (space group C2/c) as evidenced by X-ray diffraction. (b) Stability field of omphacite (om) in the system CaO^MgO^Na2O^
Al2O3^SiO2 calculated (1) assuming that free SiO2 does not occur in the presence of the pyroxene phase and (2) employing the thermodynamic
dataset of Holland & Powell (1998) and the mixing model of Green et al. (2007) by distributing cations according to the internal equilibrium 2
om ! jd þ di. The upper continuous line represents the univariant equilibrium between the high-pressure phase jadeite (jd) and the phase assemblage nepheline þ quartz. The calcic endmember diopside (di) is stable at all conditions. Curves between the two endmember curves correspond to intermediate compositions in steps of 25% in mol fraction. The field labeled ‘experimental conditions’ approximates the confining
pressure^temperature conditions of the performed deformation tests considering friction of up to 15%.
Eclogitized oceanic crust is primarily composed of
omphacite and garnet. Natural omphacite is a solid solution of the sodic endmember jadeite (jd: NaAlSi2O6) and
the calcic endmember diopside (di: CaMgSi2O6) with
up to 10% of acmite (NaFeSi2O6), some hedenbergite
(CaFeSi2O6) and Ca-eskolaite (Ca0·5œ0·5AlSi2O6), the
last of these being restricted to high pressures and temperatures (Konzett et al., 2008). Microstructures of eclogites
exposed at present at the Earth’s surface reveal that
omphacite controls their bulk rheological properties at natural conditions. In many cases, undeformed garnet crystals
are embedded in a continuous matrix of omphacite with
shape- and crystallographic-preferred orientation. In
places, omphacite microstructures are indicative of recovery and recrystallization; mechanical twins and kink
bands are also observed (e.g. van Roermund & Boland,
1981; Philippot et al., 1992; Godard & Van Roermund,
1995; van der Klauw et al., 1997; Sto«ckhert & Renner,
1998; Aoya, 2001; Piepenbreier & Sto«ckhert, 2001). With
increasing temperature, omphacite undergoes a transformation from the ordered structure with space group P2/n to
the disordered C2/c structure (Fig. 1a). The effect of the
order^disorder transition on the activated glide systems
and rheological properties of omphacite has been a matter of controversy (Brenker et al., 2002; Ulrich &
Mainprice, 2005).
Given the challenges associated with performing
deformation experiments at upper mantle conditions it is
not surprising that the availability of mechanical data on
clinopyroxenes correlates with the pressure necessary to
stabilize a specific composition (Fig. 1b). The more sodic
the composition, the higher are the required pressures.
The calcic endmember, diopside, stable even at atmospheric pressure, has been extensively investigated as single
crystals (Ave Lallemant, 1978; Ingrin et al., 1991, 1992;
Raterron & Jaoul, 1991; Jaoul & Raterron, 1994; Raterron
et al., 1994), natural rocks (Kirby & Kronenberg, 1984;
Bystricky & Mackwell, 2001; Chen et al., 2006), and synthetic aggregates (Dimanov et al., 2003; Dimanov &
Dresen, 2005; Hier-Majumder et al., 2005). Jin et al. (2001)
and Zhang et al. (2006) performed deformation experiments at a pressure of 3 GPa on aggregates synthesized
from natural omphacite crystals with compositions of
di73jd27 (73 mol % di; 27 mol % jd) and di58jd42, respectively. Synthetic aggregates of jadeite were examined by
Orzol et al. (2006).
Currently, it is difficult to deduce the effect of composition on flow properties, as different experimental techniques and samples of different origin were used in the
studies performed so far. Early work suggested that hedenbergite (CaFeSi2O6) is stronger than chromian diopside
(Kolle & Blacic, 1982). At face value, diopside aggregates
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MOGHADAM et al.
SYNTHETIC OMPHACITE AGGREGATES
appear to be stronger than jadeite aggregates at laboratory
conditions yielding dislocation creep (Orzol et al., 2006).
In contrast, it has been concluded that jadeite is stronger
than diopside at low to intermediate temperatures
where crystal plasticity controls deformation (Dorner &
Sto«ckhert, 2004). The lack of a profound understanding
of the rheological characteristics of solid-solution series of
silicates is not restricted to pyroxenes. Only limited work
has been performed on important rock-forming minerals
such as plagioclase (e.g. Rybacki & Dresen, 2000) and
olivine (Zhao et al., 2007).
To determine the relevant deformation mechanisms and
quantitatively constrain the rheology of omphacite, we
conducted deformation experiments on synthetic polycrystalline clinopyroxene aggregates with five compositions
along the omphacite solid-solution series, including endmembers jadeite and diopside. The use of synthetic samples
prepared following identical procedures and tested in a
single deformation apparatus at identical conditions has
the benefit of providing readily comparable data.
Currently, experiments at the pressure mandated by the
stability of sodic omphacite can be performed only in
Griggs-type apparatus. These are limited in their stress
resolution, even in their modified versions, because of
uncertainty in friction associated with the external measurement of axial loads (e.g. Tingle et al., 1993; Rybacki
et al., 1998). Clearly, mechanical data of higher quality can
be gained for the calcic endmember in experiments at
pressures 5500 MPa using a gas apparatus. However, the
benefits of a systematic study in a single apparatus outweigh the shortcomings of solid-medium apparatus in
stress resolution when the prime interest is the effect of
composition on the rheology of omphacite.
M AT E R I A L A N D M E T H O D S
Experimental procedures
Synthesis
Synthetic polycrystalline samples can be produced using
gel, glass, or oxide powders of appropriate composition as
starting materials. Building upon previous experience by
Orzol et al. (2006), we used glass powders with five compositions (nominally jd100, di25jd75, di50jd50, di75jd25 and
di100) along the solid-solution series between diopside and
jadeite. First, powders were pressed into gold capsules
with a length of 18 mm, a wall thickness of 0·2 mm, and
an outer diameter of 6 mm by uniaxial cold pressing at
100 MPa in several steps. After drying the cold-pressed
samples at 1508C for at least 12 h, the capsules were mechanically sealed and stored in an oven, again at 1508C.
A piston cylinder apparatus with a vessel of 25·4 mm
diameter was used for the synthesis experiments. It comprises a servohydraulic control unit for confining pressure
and a programmable temperature controller (Renner
et al., 1997). Dry sodium chloride (NaCl) served as the
pressure-transmitting medium. Heating was provided by
a graphite resistance furnace. A NiCr^Ni thermocouple
was used for measuring temperature in the upper part of
samples. Three samples were synthesized in a single run.
Several P^T^t paths in the stability field of jadeite were
tested to optimize the microstructure of the samples. We
found that a pressure P of 2·2 GPa, a temperature T
of 10008C, and a run time of 6 days are appropriate to
reach a grain size of about 10 mm and a homogeneous
microstructure. The actual chemical composition after
synthesis was determined by electron microprobe analysis (jd100: Na0·97Al1·01Si2O6; di25jd75: Na0·72Al0·76Ca0·26
Mg0·26Si2O6; di50jd50: Na0·48Al0·51Ca0·50Mg0·51Si2O6;
di75jd25: Na0·24Al0·26Ca0·76Mg0·76Si2O6; di100: Al0·01Ca1·00
Mg0·97Si1·96O6).
Deformation
Synthesized samples were machined to a cylindrical shape
of 7·3 mm length and 3·7 mm diameter, and then
dried at 1508C for at least 24 h. Platinum capsules with
a thickness of 0·1mm were used for jacketing. All deformation experiments were performed in a modified
Griggs-type solid-medium apparatus (Rybacki et al., 1998).
A eutectic mixture of CsCl^NaCl (65·5 mol % CsCl and
34·5 mol % NaCl) served as the confining pressure
medium. At 2·5 GPa, the nominal confining pressure of
all deformation tests, the melting temperature is about
10008C (Kim et al., 1972). Two NiCr^Ni thermocouples
were used for controlling and measuring the temperature
at the lower and upper ends of the sample, respectively;
a graphite furnace provided heating (Fig. 2).
Pressure and temperature were raised at 14 MPa/min
and 58C/min, respectively, to avoid damage to the graphite
furnace and to minimize pressure gradients within the
assembly. During this stage, the axial piston was periodically advanced until final conditions were reached. At
run conditions, the hit point was first determined by
advancing the deformation piston at a velocity of 1mm/h.
Subsequently, the deformation piston was retracted for
1mm before being advanced again for the actual deformation stage at specified strain rate. In some experiments,
this procedure was repeated up to three times to explore
the effect of strain rate (Fig. 3). At the end of the final
deformation stage, the piston was retracted for at least
1mm from the sample, before pressure and temperature
were decreased at rates of 83 MPa/min and 308C/min,
respectively.
Stress^strain curves were derived from the recorded
force^displacement curves assuming constant sample
volume and correcting for friction and stiffness of the
axial load column. Strain was calculated from the displacement of the deformation piston beyond the hit point
and the initial length of the sample at room temperature.
In strain rate stepping tests, the shortening achieved in
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APRIL 2010
Fig. 3. Force^displacement record of a strain-rate stepping experiment performed on a jadeite sample (omph18, Table 1) at a nominal
confining pressure of 2·5 GPa, a temperature of about 10008C, and
indicated strain rates.
Fig. 2. Sketch of the sample assembly used for deformation tests.
the previous stage(s) was taken into account. The nominal
uncertainty in strain is about 0·001, accounting for the
accuracy of the displacement transducer and uncertainty
in stiffness correction. The uncertainty in reported strain
rates of a factor of about 1·1 results mostly from the uncertainty in actual sample length during the various stages of
an experiment. The nominal uncertainty in the derived
stress is estimated to be 20 MPa, considering a load cell
accuracy of 0·02% over its range of 160 kN and an uncertainty in the sample diameter of 0·1mm. In addition, a significant methodological uncertainty for stress arises from
the determination of the contribution of friction to the
external load cell reading (Rybacki et al., 1998; Renner
et al., 2001). Here, we applied a dynamic correction; that
is, the approximately linear portion of the force^displacement curve prior to the hit point serves as the friction baseline (Fig. 4). A major source of uncertainty is the friction
on the alumina piston on top of the samples (Fig. 2). Only
when the tungsten carbide piston and the alumina piston,
which are glued to each other before an experiment,
actually stay in contact during the entire experiment was
the total friction measured with the external load cell.
Stress on the sample is overestimated by the friction on
the alumina piston in subsequent deformation cycles when
the alumina piston rests on top of the sample during nominal piston retreat; that is, tin penetrates between the tungsten carbide piston and the alumina piston rather than
salt between the alumina piston and the sample. The magnitude of the overestimation depends on the experimental
conditions. The higher the temperature and the lower the
piston velocity, the more subordinate will be the viscous
friction on the alumina piston. Uncertainty associated
with the extrapolation of the friction baseline, the position
of the baseline varying from test to test, and the issue of
piston separation add up to an uncertainty of 50^100 MPa
and therefore hinder reliable determination of the differential stress below about 100 MPa.
The confining pressure may be lower than the nominal
pressure by about 10% as a result of friction at the sealing
of the outer piston and the finite strength of the pressure
transmitting assembly parts. The actual temperature is
believed to be within about 208C of the readings of the
two thermocouples (see Rybacki et al., 1998). The thermocouple readings were not corrected for the effect of pressure (see Getting & Kennedy, 1970).
Analytical techniques
The cylindrical samples were cut into two parts along their
axes. Doubly polished thin sections were prepared from
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MOGHADAM et al.
SYNTHETIC OMPHACITE AGGREGATES
Fig. 4. The processing of force^displacement curves involved a
dynamic friction correction as exemplified for a di50jd50 sample
deformed at a temperature of 9008C and a strain rate of 2 105 s1.
one of these parts for optical and scanning electron microscope (SEM) inspection, as well as for electron backscatter
diffraction (EBSD) and electron microprobe (EMP) analysis (Table 1). For EBSD analysis, the thin sections were
chemically polished using a colloidal silica suspension
(15^20 min). For a number of samples, the second half was
used for the preparation of thin foils for transmission electron microscopy (TEM) and polished slabs with a thickness of 150^200 mm for Fourier-transform infrared
microspectrometry (FTIR). The phases in the synthesized
samples were determined by powder X-ray diffraction analysis (XRD). The crystallographic orientations and microfabrics were analyzed by SEM, using a LEO 1530
instrument with field emission gun and forescatter detector. For EBSD analysis, the SEM was operated at an accelerating voltage of 20 kV, the thin section being tilted at an
angle of 708 with respect to the beam, and with a working
distance of 25 mm. The HKL CHANNEL5 software was
used for indexing and processing of the diffraction data.
Automatic EBSD measurements were performed to obtain
orientation maps and grain geometry. Reference files with
reflectors optimized for jadeite and diopside were applied
for jadeite samples and for all other samples, respectively.
Reflector choice is not critical for different compositions
along the omphacite solid-solution series (Mauler et al.,
2000). Six bands were used for indexing. Patterns indexed
with a mean angular deviation (MAD) of 41·0 were discarded. Further processing included the automatic removal
of isolated points that were incorrectly indexed and the
replacement of non-indexed points by the most common
orientation of a specified number (six or seven) of neighbouring indexed points. The original percentage
of indexing is 45^70%. Processing of raw data resulted
in an indexing percentage of 50^86%. The percentage of
correct indexing increased with increasing grain size of
the samples. A misorientation angle of 108 was chosen as
the threshold between low- and high-angle grain boundaries. Grain size is given as the diameter of a circle of
equivalent area. The shape preferred orientation (SPO)
was gained from the orientation of the long axes of the
grains in 2D section. Crystallographic preferred orientations (CPOs) are displayed as density plots in equal area
projections of the lower hemisphere. The contours in the
density plots are scaled by multiples of a uniform distribution (MUD). The maximum scaling contour was set to
2·0^2·4 MUD. Forward scattered orientation contrast
(OC) imaging was performed to investigate the substructure within the grains, in particular the occurrence of
twins. OC imaging provides qualitative information on
the crystallographic orientation (e.g. Prior et al., 1999).
Regions of different crystallographic orientation appear in
different gray shades.
Arrangement and density of dislocations were analyzed
by TEM with a Philips EM301 microscope operated at
100 kV. For preparation of foils a GATAN PIPS ion mill
was used. All diffraction contrast images were produced
using bright-field conditions. Type and concentration of
water and water-related defects were studied by FTIR
(e.g. Kronenberg & Wolf, 1990) at room temperature
using a Bruker IFS 48 spectrometer with an attached
IR-microscope (see the Electronic Appendix, available for
downloading at http://www.petrology.oxfordjournals.org).
The IR microscope permits spectrometric analysis with a
spatial resolution of about 0·1mm using polished sections
of 0·1^0·3 mm thickness. The hydrogen concentration in
samples was calculated from the absorption spectra in the
wavenumber range 3000^4000 cm1 following the method
proposed by Paterson (1982).
Microfabric of synthesized samples
Within the limit of detection, XRD analysis confirmed
that the synthesized samples are composed of a single crystalline phase. All aggregates are in the disordered state
(space group C2/c) in accord with the phase diagram
(Fig. 1a). Only in the jd100 and di25jd75 samples did SEM
images reveal the presence of a second phase (Fig. 5d).
The volume percentage of this second phase is less than
about 4% in jd samples, and even lower in di25jd75 samples
(Table 1). The distribution of this phase at triple junctions
and the gently curved interfaces suggest it to be derived
from a melt present at run conditions. The second phase is
characterized by a higher SiO2 and a lower Na2O content
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Table 1: Experimental conditions and results for the deformation experiments on the synthetic polycrystalline clinopyroxene
aggregates at a nominal confining pressure of 2·5 GPa
Composition
jd
Sample
di50jd50
Strain (%)
rec.
total
16·8
4·8 106
50
20·1
omph16
1013 13
1·1 106
182
11·2
1010 10
5·0 106
296
12·1
1012 12
2·0 105
435
11·0
1018 18
1·1 106
227
14·2
1020 20
5·0 106
325
14·1
Micro-analysis
Grain
Melt
size (mm)
content (%)
SEM, EBSD
7·5 3·0*
54
SEM, EBSD
5·2 4·0
52
SEM, EBSD
9·3 4·4
52
1015 15
2·0 105
432
15·0
omph54
1080 20
2·0 105
100
12·1
omph10
908 8
3·0 106
399
9·2
908 8
1·5 105
704
10·8
907 7
7·5 105
1037
12·1
omph25
908 8
5
1481
14·0
32·0
SEM
omph14
1012 12
5·0 106
305
6·5
19·1
SEM, EBSD
3·3 1·2
1009 9
8·5 105
1203
3·1
omph17
998 2
1·1 106
265
8·1
15·0
SEM, EBSD
2·1 1·2
omph51
982 18
2·0 105
590
27·6
omph59
1117 17
5·2 106
293
15·1
omph52
omph11
omph12
omph30
omph28
omph19
di
Stress (MPa)
1108 8
omph22
di75jd25
Strain rate (1/s)
omph57
omph18
di25jd75
Temperature (8C)
1·6 10
5y
6
501
7·3
2·3
1·1 106
1280
23·2
31·4
900
3·0 106
1356
6·1
900
1·5 105
2445
7·3
900
7·5 105
2700
8·2
1004 4
5·0 106
416
4·2
1003 3
2·0 105
1024
11·6
1003 3
8·5 105
1741
6·1
1105 5
1·1 106
410
18·8
4·8 106
1534
11·3
3·1
SEM, EBSD, TEM
10·0 6·0
1·1 106
450
8·1
13·0
SEM, EBSD, TEM
10·1 6·1
20·0
SEM, EBSD
1100 1
912 12
887 13
1109 9
4·9 10
SEM
SEM, EBSD, TEM
7·3 4·7
1114 14
5·2 106
632
14·0
912 12
1·1 106
1450
18·0
omph09
907 7
3·0 106
1588
16·1
907 7
1·5 105
2024
8·2
omph29
995 5
5·0 106
818
11·3
omph32
905 5
3·8 106
1601
22·8
25·8
SEM, EBSD
5·0 4·0
omph38
990 10
4·6 106
867
30·6
27·7
SEM, EBSD
7·5 4·0
omph47
1069 31
4·8 106
544
20·1
18·8
SEM, EBSD, TEM
omph50
993 7
2·0 105
1088
26·2
20·1
SEM, EBSD
omph53
882 18
3·0 106
1060
22·1
6·8
8·4 4·8
11·0 7·0
8·7 5·4
11
omph61
902 2
7·9 105
1647
15·6
0·2
SEM
11
omph62
1127 27
2·0 105
497
12·6
13·9
SEM
11
omph08
890 10
1·5 105
1217
10·5
omph15
925 25
1·1 106
1091
10·3
omph27
910 10
7·9 105
1487
11·7
omph26
980 20
5·0 106
853
6·3
2·0 105
672
14·1
omph63
1092 8
52
9
omph21
SEM
54
The reported temperature is the average of the upper and lower thermocouple. In the test on sample omph11 the upper
thermocouple failed. Melt content was estimated from SEM images. Axial strain is reported as derived from the recorded
force–displacement curves (rec.) and from length measurements on samples recovered after quenching to ambient
conditions (total).
*Grain sizes based on automated EBSD measurements (quoted as average standard deviation).
yGrain sizes based on line intercept determinations on SEM images.
926
MOGHADAM et al.
SYNTHETIC OMPHACITE AGGREGATES
Fig. 5. (a) SEM image (secondary electron signal) shows irregular to prismatic grain shape of synthesized jadeite. Arrows point to an unspecified phase, interpreted to represent a quenched hydrous melt between the jadeite crystals. (b) SEM image (back-scatter electron signal) shows
equant grains of synthesized diopside. (c, d) TEM bright-field micrographs of starting material (di75jd25, sample L11) showing grains devoid
of dislocations with smoothly curved to straight boundaries.
compared with a jadeite composition, as indicated by
energy-dispersive X-ray analysis.
The typical grain size of the starting material is in
the range of 2^20 mm (Fig. 6a^c). In 2D sections perpendicular to the cylinder axis, grains do not show an SPO
(Fig. 6a and b). In 2D sections parallel to the cylinder
axis, grains with diameters 410 mm can exhibit a weak
SPO (Fig. 6c). A significant crystallographic preferred
orientation (CPO) was not found anywhere in the starting
material (Fig. 6a^c). Grain-size distribution and grain
shape vary systematically with composition. In diopside
samples, grains are mostly equant and exhibit a narrow
size range (Fig. 5b); in contrast, grain shape in jadeite samples is more anisometric and grain size varies (Fig. 5a).
Samples with intermediate composition show microstructures that are transitional between those of the two compositional endmembers. Samples with the composition
di25jd75 tend to show the smallest grain size. Generally, all
grains appear internally strain-free under the optical
microscope and the TEM (Fig. 5c and d).
In atom fractions, the estimated hydrogen contents
range from slightly above to slightly below 102 H/Si from
jadeite to diopside aggregates, respectively. Water is most
probably incorporated during glass powder preparation
(Orzol et al., 2006). Intermediate compositions tend to contain the least amount of hydrogen. A large portion of the
hydrogen is expected to be present in water inclusions
along interfaces and in the quenched melt observed in the
sodic samples. Spectra of undeformed samples indeed exhibit prominent broad band absorption between 3000 and
3700 cm1, but the jadeite and diopside samples also show
distinct absorption peaks at 3614 and 3650 cm1, respectively, in agreement with previous studies (Bromiley &
Keppler, 2004; Bromiley et al., 2004). Such peaks are not
observed for intermediate compositions. Compared with
experimentally determined water-solubility in synthetic
Na-rich clinopyroxene and Cr-diopside (Bromiley &
Keppler, 2004; Bromiley et al., 2004), all our samples
contain excess H2O. These observations qualify the experimental conditions as ‘wet’.
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Fig. 6. Selected crystallographic and shape orientation patterns obtained by automated EBSD measurements (2 mm step size) of starting materials (a^c) and deformed samples (d^g). Left: density plots (equal area projection of the lower hemisphere) are displayed for the poles to (100)
and (010) planes and the [001] direction. Shortening direction is vertical for all diagrams. Right: 2-D orientation of long axes of grains with
diameters 510 mm and 410 mm are displayed in a rose diagram. In (a) and (b), the section perpendicular to the cylinder axis of the samples is
displayed, in (c^g) the section parallel to the cylinder axis.
928
MOGHADAM et al.
SYNTHETIC OMPHACITE AGGREGATES
R E S U LT S
Mechanical data
We performed deformation experiments on 32 samples
at temperatures between 900 and 11008C, a confining pressure of 2·5 GPa, and constant strain rates between 1·1 106 and 8·5 105 s1. A total of 24 tests were conducted
at a single set of conditions, allowing an unequivocal correlation of the mechanical data and the microstructure of
the deformed samples. In the eight step tests, a single
sample was axially shortened at up to three strain rates.
Typical examples of the derived stress vs strain curves are
presented in Fig. 7. The majority of samples exhibit deformation at fairly constant stress. The occurrence of limited
softening or hardening behavior does not systematically
correlate with experimental conditions, indicating uncertainties in the friction correction as the potential cause,
rather than actual material behavior. Indeed, the vast
majority of samples appear homogeneously shortened and
thickened after deformation. The reported flow stresses
(Table 1), simply referred to as strength in the following discssion, correspond to the maximum stresses attained in a
deformation step. The reproducibility of mechanical data
obtained in tests performed at similar conditions on different samples is of the order of 10%.
For all compositions, strength systematically decreases
with increasing temperature and decreasing strain rate
(Fig. 8a and b). The apparent effect of composition
on relative strength evolves with temperature (Fig. 9a).
At 9008C, the lowest investigated temperature, clinopyroxenes of intermediate composition are stronger than either
endmember, of which jadeite is systematically the weaker
one. At 11008C, we find an almost monotonous decrease
in strength from diopside to jadeite aggregates. Notably,
Fig. 7. Stress^strain curves for omphacite samples of indicated compositions deformed at a temperature of 9008C, a confining pressure
of 2·5 GPa, and a strain rate of 2 105 s1 (continuous lines) and
10008C, 2·5 GPa, and 5 106 s1 (dashed lines).
Fig. 8. Relation between stress and strain rate for (a) jadeite and (b) diopside aggregates. Error bars represent uncertainty in stress as a result
of uncertainty in friction baseline and sample dimensions. It should be noted that the two experiments on jadeite aggregates with nominal
flow stresses below 100 MPa fall below the methodological uncertainty. Dashed and continuous lines represent the fit by a power-law creep equation (Table 2) and the considered micromechanical models (Table 3), respectively. The box indicates the results of the micro-indentation experiments performed by Dorner & Sto«ckhert (2004), for which the strain rates could actually be as high as 102 s1.
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Fig. 9. (a) Variation of measured strength with composition at indicated temperatures and strain rates. The grey fields represent the relations
between strength and composition when relying on scaling with melting temperature and shear modulus [see equations (5) and (7)]. Flow
stress determined for synthetic jadeite aggregates by Orzol et al. (2006) is given for comparison (shifted slightly to the left for presentational purposes). (b) Relation between sample composition and apparent activation energy for creep, Q (top), and apparent stress exponent, n0
(bottom), gained from fitting a power law [equation (1)] to the data reported in Table 1.
the range in stress experienced by samples of the two
compositional endmembers hardly overlaps. At the
imposed strain rates, jadeite and diopside aggregates essentially deformed at stresses below and above 500 MPa,
respectively.
We determined best-fit parameters of a power-law
relation
Q
0
"_ ¼ A n exp ð1Þ
RT
for the mechanical data, using a non-linear inversion
method (Sotin & Poirier, 1984). This method accounts for
the reported uncertainties in experimental data, strain
_ stress difference s, and absolute temperature T
rate ",
(Table 1). Here, A, n’, Q, and R denote the pre-exponential
factor, apparent stress exponent, apparent activation
energy of creep, and universal gas constant, respectively.
The values for n’ vary between 3 and 10, and generally
seem to increase from sodic to calcic composition
(Table 2, Fig. 9b). The values for Q vary between 200 and
600 kJ/mol, without obvious relation to composition.
Microfabric of deformed samples
In some cases, the difference between axial finite strain,
determined from the length of the cylindrical sample after
removal from the apparatus, and permanent strain, calculated from the record of the displacement transducer,
exceeds e ¼ 5%, the maximum uncertainty in the
Table 2: Parameters of a power-law [equation (1)] fit to
the data reported inTable 1
Composition
ln(A) (MPan s1)
n’
di
29·6 4·8
8·1 1·5
391 79
di75jd25
10·3 3·2
5·5 1·9
414 136
Q (kJ/mol)
di50jd50
4·7 1·1
3·5 0·2
323 18
di25jd75
4·3 3·6
3·0 0·5
281 63
jd
3·5 6·3
4·2 0·8
413 79
recorded strain resulting from the uncertainty in hit-point
determination (Table 1). Deformed samples often show
tensile cracks oriented normal to the sample axis. These
cracks, which originate during quenching and release of
the axial load, increase the apparent final sample length,
causing the determined permanent strain to fall short of
that recorded during the experiment. Samples that, in
contrast, appeared shorter than the estimated recorded
strain probably experienced non-hydrostatic quenching;
these were therefore excluded from the microstructural
analysis.
On the optical and the SEM scale, there is no marked
difference in grain size between starting material and
deformed samples (Table 1, Fig. 10). The deformed clinopyroxene samples show an SPO and a weak CPO (Fig. 6d^g).
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SYNTHETIC OMPHACITE AGGREGATES
Fig. 10. OC images of (a) jadeite (sample omph57) deformed at 11088C; (b) jadeite (sample omph16) deformed at 10108C; (c) di50jd50 (sample
omph52) deformed at 11008C; (d) di50jd50 (sample omph22) deformed at 9128C; (e) di75jd25 (sample omph19) deformed at 11148C; (f) di75jd25
(sample omph28) deformed at 8878C; (g) diopside (omph50) deformed at 9938C; (h) diopside (omph53) deformed at 8828C. For all samples,
the confining pressure during deformation was 2·5 GPa (for details see Table 3). Subgrains are widespread in jadeite (a,b), whereas mechanical
twins tend to be more frequent in diopside-rich compositions (e^h). High-angle grain boundaries between grains of low twin density and
grains with a high twin density are strongly sutured (g).
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The long axes of the grains, as seen in axial 2D sections,
are oriented normal to the direction of shortening. The
CPO is very weak with the maximum MUD typically
52. The strongest CPO with a maximum MUD of 2·4 is
recorded in the sample of intermediate composition
di50jd50 (Fig. 6e). Samples of the diopside endmember
composition show the least pronounced CPO (Fig. 6g).
For all compositions, the normal to the (010) planes tends
to be oriented parallel to the shortening direction, defining
a point maximum. The [001] direction is oriented perpendicular to the shortening direction, forming a girdle in the
foliation plane.
Mechanical twins are found in almost all deformed samples (Figs 10d^h and 11). The overall twin density decreases
with decreasing maximum differential stress experienced
by the sample. As such, twins are widespread in deformed
samples of calcic composition and rare or absent in those
of more sodic composition. EBSD measurements and
TEM observations reveal that the twin plane is (100) and
the glide direction is [001]. Most twin lamellae are less
than 0·5 mm wide (Figs 10d^h and 11). Some are also visible
on the optical scale. In inhomogeneously deformed grains,
the twins can be wedge-shaped and taper out in the grain
interior (Fig. 10f and g). At serrated high-angle grain
boundaries, the diameter of bulges corresponds to the
thickness of the twin lamellae (Fig. 10g).
Dislocation density reaches about 1013 m2. The highest
dislocation densities are observed for intermediate compositions (Fig. 11e and f) in accord with the observation of
their highest strength. Subgrains are widespread in the
more sodic compositions and rare in the more calcic compositions (Fig. 10a and b). The low-angle grain boundaries
are frequently oriented normal to (001) (Fig. 11a and g).
Recrystallized new grains with a diameter of the order of
1 mm are observed on the TEM scale within deformed samples of jadeite (Fig. 11c and d). The small recrystallized
grains show very few dislocations. The FTIR spectra of
deformed samples are similar to those obtained for the
starting material, indicating that water speciation is not
significantly affected by deformation (see the Electronic
Appendix).
DISCUSSION
The mechanical data indicate thermally activated creep.
The strength of the synthetic clinopyroxene aggregates
decreases with decreasing strain rate and increasing temperature. Although the sodic endmember jadeite is consistently weaker than the calcic endmember diopside, the
details of the variation in strength with composition
change with temperature. At 900 and 10008C, some intermediate compositions are significantly stronger than
either compositional endmember. Only at 11008C does
strength increase almost monotonically from jadeite to
diopside aggregates. In the following discussion, we first
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compare our observations with those of previous studies.
Then, the likely active deformation mechanisms are identified by combining key mechanical and microstructural
observations to fit micromechanically motivated flow laws
to the data. The dependence of strength on composition is
discussed in the light of first-order scaling relations.
Finally, the geological implications of these data are
considered.
Comparison with previous work
In direct comparison, stress exponent and activation
energy values derived for our jadeite aggregates agree
well with those reported by Orzol et al. (2006). However,
the absolute strength values determined in this study are
systematically lower than those previously determined
(Fig. 9a). At a given stress, the difference corresponds to
less than an order of magnitude in strain rate. Compared
with Orzol et al. (2006), we used a different glass precursor
and we did not employ seeding for synthesis. Nevertheless,
grain sizes achieved by the two approaches are comparable. Bulk water contents of our samples estimated by
FTIR spectroscopy exceed the contents quoted by Orzol
et al. (2006) by a factor of 3^5. Microscopic inspection
reveals that samples rich in the jd-component (jd100 and
di25jd75) contain up to 4% of an unspecified phase
(Table 1), probably a quenched hydrous melt. The presence
of this melt phase is thought to be responsible for the
observed difference in strength between this study and
that of Orzol et al. (2006). Such a moderate effect of melt
on strength is comparable with findings by Dimanov et al.
(2003) for diopside, and by Cooper & Kohlstedt (1986)
and Hirth & Kohlstedt (1995) for olivine aggregates. The
correspondence in the stress exponent and activation
energy between melt-free and melt-containing samples
shows that the rate-controlling mechanism remains
unaffected but that the average load-bearing area of the
grains changes.
The mechanical data for omphacite aggregates of intermediate composition compare fairly well with the results
of Zhang et al. (2006), who investigated synthetic aggregates prepared from crushed natural omphacite crystals
(di58jd42; FeO ¼1·74 wt %; about 102 H/Si). Combined,
the two datasets yield an apparent stress exponent of 6·5
at a temperature of 11008C (Fig. 12a); that is, a value
higher than that deduced by Zhang et al. (2006) on the
basis of their data alone. Jin et al. (2001) deformed synthesized Fe-bearing omphacite aggregates (di73jd27; FeO/
MgO ¼ 0·17; about 102 H/Si) at a temperature of 1500 K
and a confining pressure of 3 GPa. At a strain rate of
4·6 104 s1, a strength of 300 MPa was derived from
two stress^strain curves. The conditions of these experiments are too different from ours for a direct comparison.
Absolute strength determined for our synthetic
fine-grained, wet diopside aggregates compares well with
the results of Kirby & Kronenberg (1984) obtained for
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Fig. 11. TEM bright-field micrographs of deformed clinopyroxenes. (a) Deformed jadeite shows mechanical (100) twins and a high dislocation
density. A low-angle grain boundary is marked by an arrow (jd100; sample omph57). (b) Bulges in high-angle grain boundaries correlated with
twins indicate that grain boundary mobility is affected by orientation relations (black arrow, jd100; sample omph57). (c, d) Recrystallized
micron-sized grains are free of dislocations and show various crystallographic orientations, as indicated by the inserted diffraction pattern in
(c) (jd100; sample omph57). (e, f) Slightly curved twin lamellae in an inhomogeneously deformed omphacite grain with high dislocation density
(di50jd50, sample omph22). (g) Wide twin lamella and a low-angle grain boundary marked by an arrow (di75jd25; sample omph28). (h) High
density of narrow twin lamellae in diopside (di100; sample omph47).
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Fig. 12. (a) Comparison of results for intermediate compositions from this study (labeled di50jd50; filled symbols) and the study of Zhang et al.
(2006) (labeled di58jd42; open and crossed symbols). The dashed lines represent an apparent stress exponent of 6·5. (b) Comparison of our results
for synthetic diopside aggregates () with single crystal data of Raterron & Jaoul (1991) (5) and data for natural Sleaford Bay clinopyroxenite
reported by Kirby & Kronenberg (1984) () and Chen et al. (2006) (i). The lines represent the indicated stress exponents.
samples of coarse-grained Sleaford Bay clinopyroxenite at
1·5 GPa confining pressure (Fig. 12b). At face value, this
agreement suggests a modest effect of compositional
details, confining pressure, and grain size on the flow
behavior of diopside aggregates at the explored conditions.
The strength observed by Chen et al. (2006) at a confining
pressure of 0·3 GPa for this rock is, however, slightly lower
than that observed in the two studies at higher pressure
and as such is close to the strength of diopside single crystals (Fig. 12b). A direct one-to-one comparison of the
stress^strain rate relations obtained by deformation experiments on single crystals with those obtained on polycrystalline samples is not possible. However, our observations
on di100 and di75jd25 samples agree well with the strain
rate dependence of strength obtained for single crystals at
temperatures above 10008C, where the glide system
{110}½ 5a b4 was activated (Raterron & Jaoul, 1991;
Fig. 12b). Between 800 and 10008C, the glide system
(100)[001] dominates deformation of single crystals in the
tested orientation (Raterron & Jaoul, 1991).
The CPO patterns observed in our deformed samples
are weak (Fig. 6d^g), probably because of the limited total
strain. Qualitatively, the weak CPO is similar for all our
deformation experiments, independent of temperature
and clinopyroxene composition. The [001] orientation
describes a girdle perpendicular to the shortening direction, whereas the poles to the (010) plane define a point
maximum parallel to the shortening direction. Bascou
et al. (2001, 2002) obtained a similar CPO of omphacite in
numerical simulations, and interpreted the pattern to
result from the combined activation of the slip systems
f110g½51104
, {110}[001], and (100)[001]. These slip systems are consistent with TEM observations on naturally
deformed omphacite (e.g. Buatier et al., 1991; Godard &
Van Roermund, 1995; Ulrich & Mainprice, 2005).
According to Bascou et al. (2001, 2002), slip on the {110}
planes is responsible for the orientation of (010) planes parallel to the foliation, whereas the predominant glide direction [001] defines a girdle in the foliation. These patterns
are comparable with the S-type of Godard & van
Roermond (1995) observed in naturally deformed eclogites
that is also consistent with axial symmetric compression
(Ulrich & Mainprice, 2005). Similar patterns were
observed in experimentally deformed clinopyroxene
(Mauler et al., 2000) and jadeite (Orzol et al., 2006) aggregates. For natural eclogites, a second type of CPO pattern
was reported, the L-type, characterized by a point maximum of [001] parallel to the lineation and a girdle of
[010] perpendicular to the lineation (e.g. Helmstaedt
et al., 1972). Based on TEM observations, Brenker et al.
(2002) postulated that cation ordering in omphacite controlled the development of either L-type (ordered structure) or S-type patterns (disordered structure) in natural
eclogites. A subsequent study on three mantle eclogite
xenoliths did not support this hypothesis (Ulrich &
Mainprice, 2005). Omphacite is disordered in our experiments and the CPO patterns are consistent with the predictions (Bascou et al., 2001, 2002; Ulrich & Mainprice,
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MOGHADAM et al.
SYNTHETIC OMPHACITE AGGREGATES
Table 3: Results of micromechanical modeling
Composition
Climb-controlled creep according to equation (2)
n
jd
di
ln(Aclimb)
Hcreep
(MPan s1)
(kJ/mol)
4·5
4·8 5·6
443 57
3·5
0·7 4·3
341 43
6·5
25·7 2·6
313 27
Glide-controlled creep according to equation (3)
di
M
sP
Qglide
acalc
bcalc
ln(Aglide)
(m3)
(GPa)
(kJ/mol)
(nm)
(nm)
(MPa2·5 s1)
7·4 0·9
528 23
0·23 0·02
0·74 0·02
15·4 3·1
16·5 1·6
3·4 10
Italic font indicates parameters that were prescribed in the inversion [stress exponents in (a) and the range of dislocation
source density in (b)]. Bold values are actual inversion results whereas normal font indicates parameters calculated from
the inverted ones using mdi 56 GPa and s,di 4140 m/s.
2005) for axial symmetric compression. With respect to the
activated glide system, the orientation of (010) normal to
the shortening direction suggest {110}[001] to be predominant, as proposed by Bascou et al. (2001, 2002). Activation
of the glide system (100)[001], observed by Raterron &
Jaoul (1991) and Raterron et al. (1994) to be the most
active between 800 and 10008C in experiments on diopside
single crystals, is not reflected by the CPO. The Burgers
vector [001] common to both glide systems leaves the possibility of cross slip.
Mechanisms controlling the plastic
deformation of omphacite
Inferences from the microfabric
Observed microfabrics and relations between strength and
strain rate or temperature indicate that samples of all compositions are plastically deformed by a combination of
mechanical twinning, dislocation glide and climb. Twin
lamellae parallel to (100) have previously been observed
in clinopyroxenes from eclogite-facies rocks (Carter &
Raleigh, 1969; Wenk, 1970; Ave Lallemant, 1978; Mu«ller,
1993; Godard & Van Roermund, 1995; Trepmann &
Sto«ckhert, 2001; Mu«ller & Franz, 2008) and in samples
deformed in the laboratory (Kirby & Christie, 1977; Kolle
& Blacic, 1982; Orzol et al., 2003; Mu«ller et al., 2007). The
mechanisms of twinning in pyroxenes have been discussed
in some detail (Kirby & Christie, 1977; Kolle & Blacic,
1982). Kolle & Blacic (1982) proposed a compositioninsensitive critical resolved shear stress (CRSS) of about
140 10 MPa for mechanical (100)[001] twinning of clinopyroxene. Experiments on natural polycrystalline jadeite
(Orzol et al., 2003) indicated a CRSS of 150 25 MPa, in
accord with the suspected negligible influence of composition. Based on these results, mechanical twinning
should occur with increasing frequency in clinopyroxene
aggregates subject to a differential stress exceeding about
300 MPa and the observed higher frequency of mechanical
twins in deformed samples of more calcic composition
is probably a result of the generally higher differential
stresses attained in these experiments compared with
those on samples of more sodic composition (Table 3).
Low-angle grain boundaries form by rearrangement of
geometrically necessary dislocations and recrystallized
grains by migration and/or formation of high-angle grain
boundaries driven by the reduction of stored strain energy
during high-temperature creep or static annealing (e.g.
Nicolas & Poirier, 1976; Poirier, 1985). Geometrically necessary dislocations are introduced during inhomogeneous
crystal plastic deformation, dominated by dislocation
glide. Although the importance of dislocation glide is
also reflected by the CPO, the low-angle grain boundaries
indicate dislocation climb. Strain-induced grain boundary
migration is documented by serrated high-angle grain
boundaries (Fig. 10g). Here, the diameter of bulges corresponds to the thickness of twin lamellae, indicating that
grain boundary mobility is affected by the orientation relations between adjacent grains and strain gradients as a
result of accommodation processes at the interface.
The observation that low-angle grain boundaries and
recrystallized grains are more widespread in the sodic pyroxenes suggests that dislocation climb is more effective in
those compositions compared with the calcic pyroxenes at
the conditions of the deformation experiments. This assertion is consistent with the observation that the CPO is
more pronounced in the more sodic and less pronounced
in the more calcic compositions. CPOs observed in this
study are generally weak, probably as a result of the limited finite strain. A weak CPO in fine-grained
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experimentally deformed clinopyroxene aggregates has
been taken as an indication of diffusion creep (Mauler
et al., 2000). The observed dislocation densities, subgrains,
recrystallized grains, and serrated grain boundaries (Figs
10 and 11) as well as the mechanical behavior (large apparent stress exponents, Fig. 9b), however, clearly argue for
dislocation-accommodated deformation in our samples.
Inferences from the mechanical data
Two state variables, temperature and stress, are central for
the mechanics and kinetics of deformation of crystals by
motion of dislocations. The relative magnitude of these
two variables with respect to melting temperature Tm and
shear modulus m are associated with regimes dominated
by different dislocation processes (Frost & Ashby, 1982).
Dislocation climb accommodated by the formation and
migration of jogs is a thermally activated process realized
by thermally activated vacancy diffusion with an activation enthalpy that roughly scales with the melting temperature of a substance (e.g. Poirier, 1985; Karato et al.,
1995). For a dislocation to glide, it has to be lifted out of its
potential trough in the lattice. Overcoming this potential
barrier that scales with the shear modulus can be achieved
by mechanical force.
Whereas melting temperatures increase from jadeite to
diopside (Bell & Davis, 1969), the relation between shear
modulus and composition is opposite (Duffy & Anderson,
1989). Thus, the explored conditions yield separate ranges
of scaled temperature and strength for the compositional
endmembers (Fig. 13). The tests on jadeite aggregates correspond to the highest homologous temperatures and the
lowest normalized strengths whereas those on diopside
aggregates represent the lowest homologous temperatures
and the highest normalized strengths. The boundary
between the conditions for diopside and jadeite aggregates
is close to a temperature of 2/3 Tm, considered to separate ‘low-temperature glide’ and ‘high-temperature climb’
for metals (Weertman, 1999).
The notion that the deformation of aggregates with a
calcic composition is controlled by glide kinetics is supported by further details of the mechanical data.
Apparent stress exponents as large as those derived for
calcic clinopyroxene (Table 2) are considered unlikely for
cases where dislocation climb is the rate-controlling step.
Furthermore, the increase in strength from diopsidic to
intermediate compositions found here for 900 and 10008C
(Fig. 9a) correlates with the composition dependence of
the shear modulus, suggesting glide as the relevant dislocation mechanism. Dorner & Sto«ckhert (2004) found such a
correlation between shear modulus and strength of pyroxene single crystals by indentation hardness measurements
at temperatures between 300 and 7508C. In these comparatively low-temperature and high-stress experiments,
climb cannot be significantly activated at laboratory
strain rates, leaving glide as the predominant mechanism
Fig. 13. Normalized results for the compositional endmembers,
jadeite and diopside. Determined strength values are given in relation
to shear modulus and temperatures of experiments are given in relation to melting temperature. Melting temperatures of omphacite
increase fromTm,jd ¼13808C for jadeite to Tm,di ¼16508C for diopside
at a pressure of 3 GPa (Bell & Davis,1969). Thus, the homologous temperature ( ¼ T/Tm) of our experiments on jadeite aggregates, ranging from 0·71 to 0·83, consistently exceeds that of the diopside
aggregates. The relation between shear modulus and composition is
opposite to that between melting temperature and composition. We
calculated the ratio s/m using published temperature and pressure
variations of shear moduli (Duffy & Anderson, 1989).
of plastic deformation. The correlation between the trends
of shear modulus and observed strength with composition
near the calcic endmember suggests that the deformation
of the calcic pyroxenes is still controlled by dislocation
glide, despite the higher temperatures of our experiments.
The relative weakness of the aggregates with the most
sodic composition indicates that for these the transition to
climb-controlled deformation occurs at temperatures
between those attained in the present study and those
explored by Dorner & Sto«ckhert (2004). Considering the
deformation of the more sodic aggregates to be controlled
by dislocation climb is in perfect agreement with the
observed apparent stress exponents (Table 2), the ubiquity
of subgrain boundaries (Figs 10 and 11), and the relation
between melting temperature and composition.
Constraints on flow law parameters
Experimentally constrained flow laws can be extrapolated
to geological conditions with some confidence only when
based on micromechanical models (e.g. Paterson, 1987). In
particular, the power-law parameters quoted in Table 2
constitute purely empirical fits unsuitable for reliable
extrapolation. Therefore, we made an effort to choose flow
laws whose underlying models are in agreement with our
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MOGHADAM et al.
SYNTHETIC OMPHACITE AGGREGATES
mechanical and microstructure data and to derive physically reasonable parameters for them. As shown in the
Appendix, jadeite samples are found to closely obey the
flow law for dislocation climb
45
mDeff
ð2Þ
"_W,climb ¼ 35 05
b M kT m
proposed by Weertman (1968) for physically reasonable
magnitudes of the parameters involved (Table 3).
Although the diopside data can, in principle, also be modeled by (2), the required magnitude of the parameters
involved appears unreasonable (see the Appendix), and
the microstructures and the observed relation of strength
to composition suggest dislocation glide-controlled deformation instead. We employed a flow law
Qglide
25
1
ð3Þ
"_glide ¼ Aglide exp P
RT
as proposed by Weertmann (1957) and found interestingly
reasonable parameters (Table 3). In contrast, the
power-law parameters quoted in Table 2 are purely empirical. In particular, the estimates of apparent activation
energies elude a substantial discussion of their relation to
composition or independent kinetic data as they correspond to different apparent stress exponents.
Starting from the endmember flow laws, simple scaling
relations can be used to derive a zero-order description
of the behavior of omphacite aggregates of intermediate
composition within the two separate deformation regimes
of glide and climb. The lack of understanding of how the
sequential processes of glide and climb can be combined
in a unifying flow law hinders a more comprehensive quantitative modeling of the compositional dependence of
the rheology of omphacite aggregates across the regime
boundaries.
In the case of climb-controlled creep [equation (2)],
the thermal activation of the effective diffusion coefficient,
presumably scaling with homologous temperature, dominates the effect of composition (see the Appendix). A flow
law involving a parameter for bulk composition (X ¼ 0
corresponds to jadeite, X ¼1 corresponds to diopside) is
thus potentially
Qclimb,jd þ 1 X
_"W,climbðXÞ Ajd 45 exp ð4Þ
RT
with 1 (72 10) kJ/mol. For our experimental conditions
the relation between strength and bulk composition
leads to
@ ln climb
1 1 0 2:
@X ð5Þ
_
",T
For glide-controlled creep [equation (3)], the
dependence of shear modulus on composition
dominates the scaling relation. A composition-dependent
flow law is
"_glideðXÞ Aglide,di 25
Qglide,di
exp ½1 þ 2 ð1 X Þ
1 P,di
RT
ð6Þ
with 2 ¼ m/mdi 0·2. For our experimental conditions
the expected effect on strength amounts to
@ ln glide
ð1 3 0 6Þ:
ð7Þ
@X ",T
_
The strength scaling according to equations (5) and (7) is
in reasonable qualitative agreement with our observations
(Figs 8 and 9a). In the glide regime, predictions and observations are fairly close, whereas the trend with composition in the climb regime appears underestimated.
This discrepancy may be an artifact of transitional behavior (Fig. 14) or may be related to the finite melt content of
those samples with the most sodic compositions.
Geological implications
Extrapolation of flow laws derived from laboratory
experiments to natural conditions requires addressing two
central issues. Are the same deformation mechanisms
operative in nature and the laboratory? Which geological
environments are characterized by thermodynamic conditions comparable with those in the experiments? At face
value, our tests apply only to temperatures at which disordered omphacite is stable (T47008C, Fig. 1) and to
‘wet’ high-pressure environments as probably found in subduction zones. The previously suggested difference in
CPO-relevant glide systems for ordered and disordered
omphacite (Brenker et al., 2002) was questioned subsequently (Ulrich & Mainprice, 2005) and our results may
well apply to ordered omphacite, too.
Our diopside samples qualify as ‘wet’; however, the kinetics of dislocation glide are intrinsically much less sensitive to point defect chemistry than the kinetics of
dislocation climb and are probably less sensitive to the
presence of water-related defects in the nominally anhydrous pyroxenes. Thus, the parameters (Table 3) of the
Peierls-type flow law [equation (3)] used to model the
glide-controlled deformation of diopside aggregates probably apply also to less hydrogen-rich environments.
Numerical studies revealed that the Peierls Law may play
a crucial role for plastic deformation to localize in shear
zones (Karato et al., 2001; Kaus & Podladchikov, 2006).
It is probably impossible to judge whether glide was rate
limiting during ancient deformation of omphacite in rocks
exposed at present. Glide-controlled deformation may be
inferred from the microstructural record, where inhomogeneous crystal plastic deformation is evident, whereas
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NUMBER 4
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Fig. 14. Schematic illustration of compositional dependence of (a) scaled and (b) absolute strength at indicated homologous and absolute temperatures, respectively. The combination of the opposite dependence of melting temperature and shear modulus on composition and of a
switch in deformation regime leads to the increase in strength from either compositional endmember towards intermediate compositions at
intermediate temperatures (9008C T2 11008C; this study). The increase from the sodic side where deformation is controlled by dislocation
climb results from the increase in melting temperature with increasing calcic component, whereas the increase from the calcic side results
from the increase in shear modulus with decreasing calcic component (the illustrated iso-temperature lines represent observations at
T1 7508C by Dorner & Sto«ckhert (2004) and at T3 11008C).
features indicating climb and dynamic recrystallization are
absent. Microstructures observed in omphacite in eclogitic
rocks are variable (e.g. van Roermund & Boland, 1981;
Boland & Tullis, 1986; Buatier et al., 1991; Godard & Van
Roermund, 1995; Mu«ller & Franz, 2008). In most studies,
deformation by dislocation creep is inferred from the presence of a CPO, without being supported by a characteristic
microstructural record. Only recently, Mauler et al. (2001)
suggested instead that the CPO in natural eclogites may
be related to anisotropic growth rather than deformation.
Piepenbreier & Sto«ckhert (2001) appear to have given
the only report of unequivocal evidence for dynamic
recrystallization of omphacite in eclogite rocks. The
inferred temperature at deformation is slightly below
5008C. In accord with the constraints from this field
study, the extrapolation of the flow laws derived in this
study to geological strain rates indicates a switch from
glide to climb between 400 and 5008C (Fig. 15). It should
be noted that the predicted transition temperature is the
lowest possible because our laws for the deformation of
melt-bearing jadeite aggregates kinetically controlled by
climb (Table 3, upper part) and diopside aggregates controlled by glide (Table 3, lower part) constitute lower
bounds for the two deformation mechanisms in the system
diopside^jadeite according to the discussed scaling relations. When allowing for the presence of minor Fe as typical for natural omphacites, however, some weakening
compared with the current predictions is possible because
of (1) the reduction in shear modulus (Duffy & Anderson,
1989) and (2) the acceleration of diffusion kinetics resulting
938
MOGHADAM et al.
SYNTHETIC OMPHACITE AGGREGATES
Fig. 15. Extrapolation of the determined climb and glide flow laws for jadeite and diopside aggregates (Table 3) to geological strain rates of
1015 (dotted lines), 1014 (dot-dashed lines), and 1013 s1 (continuous lines).
from the relation between the multivalence of iron and the
incorporation of hydrogen (e.g. Chen et al., 2006; Costa &
Chakraborty, 2008). The weakening associated with the
presence of Fe is probably limited considering the good
quantitative correspondence of our results to the study of
Zhang et al. (2006) on aggregates prepared from crushed
natural, Fe-bearing omphacite crystals.
Although constituting lower bounds for Fe-free systems,
the predicted stresses are high, of the order of 1GPa, for a
temperature of about 5008C. Geological observations suggest that stresses of this magnitude are not characteristic
for long-term deformation in high-pressure subduction
zone environments (for a review see Sto«ckhert, 2002). We
therefore propose that crystal plastic deformation, mechanical twinning, and dynamic recrystallization of natural
omphacite at high stresses, as reported by Piepenbreier &
Sto«ckhert (2001), indicates localized and episodic
high-stress cycles related to seismic activity in the subduction zone. This hypothesis is in accord with the geological
record, experimental results, and methodological considerations. The structural record of such cyclic deformation with high peak stresses was previously identified in
several crustal environments (Ku«ster & Sto«ckhert, 1999;
Trepmann & Sto«ckhert, 2003; Birtel & Sto«ckhert, 2008;
Nu«chter & Sto«ckhert, 2008), and also was studied in
laboratory experiments by Trepmann et al. (2007). The
observation of recrystallized grains being restricted to the
immediate vicinity of garnet crystals (Mu«ller & Franz,
2008) is in accord with plastic deformation driven by local
stress concentration.
Finally, considering that our samples qualify as ‘wet’and
that the incorporation of hydrogen into silicates tends
to cause ‘hydrolytic weakening’, it is unlikely that drier natural omphacite deforms extensively by dislocation creep
anywhere inside the Earth unless temperatures exceed
about 7008C.
CONC LUSIONS
Triaxial compression experiments were performed on
synthetic pyroxene aggregates with five compositions
along the diopside^jadeite solid-solution series, in a solidmedium apparatus at a confining pressure of 2·5 GPa, temperatures between 900 and 11008C, and strain rates
between 1·1 106 and 8·5 105 s1. The mechanical
data indicate thermally activated creep and the microfabrics of the deformed samples show evidence of various
deformation mechanisms involving dislocations. The
observed magnitude of the stress exponent and the microstructural features, notably the abundance of subgrains,
suggest that the deformation of the most sodic aggregates
is controlled by dislocation climb. In contrast, the large
apparent stress exponent and the positive correlation
between strength and shear modulus near the diopside
endmember strongly suggest that dislocation glide is the
rate-controlling mechanism for the most calcic aggregates
at the explored conditions. The apparent insensitivity of
the deformation behavior of diopside aggregates to Fe and
H content as deduced from a comparison with previous
studies is in accord with these conclusions; in contrast to
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JOURNAL OF PETROLOGY
VOLUME 51
climb-controlled deformation, glide-controlled deformation should be insensitive to impurity concentrations that
affect diffusion kinetics. Our micromechanical modeling
yields physically reasonable flow-law parameters. The
extrapolation of the derived flow laws predicts stresses
probably exceeding those actually realized on long time
and space scales in subduction zones. As suggested previously for intermediate to lower crustal environments, localized and episodic high-stress stages related to Benioff
zone seismicity may be responsible for the microstructural
record of crystal plastic deformation of omphacite
observed in natural eclogites.
AC K N O W L E D G E M E N T S
The German Science Foundation (DFG) provided funding
for this study within the scope of the Collaborative
Research Centre 526 ‘Rheology of the Earthçfrom the
upper crust into the subduction zone’. We greatly appreciate the incredible patience which the reviewers nominated
by the DFG granted the slow progress of this challenging
experimental study. We thank Rolf Neuser for SEM,
Thomas Fockenberg for XRD, and Klaus Ro«ller for
FTIR analyses and kind advice, Frank Bettenstedt and
Ralf Bleiwei for dedicatedly machining sample assemblies
and preparing samples, and Michael Burchard for performing the thermodynamic calculations underlying
Fig. 2b. Constructive reviews by U. Faul, F. Heidelbach,
and an anonymous reviewer, and the input by the editor,
J. Hermann, stimulated considerable thinking about our
research results.
S U P P L E M E N TA RY DATA
Supplementary data for this paper are available at
Journal of Petrology online.
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432^443.
Zhao, Y. H., Zimmerman, M. E. & Kohlstedt, D. L. (2007). Influence
of Fe content on the creep properties of olivine. Lunar and Planetary
Science XXXVIII, 1800.
A PPEN DI X: SELECT ION OF
F L O W L AW S A N D PA R A M E T E R
D E T E R M I N AT I O N
Various mechanisms have been considered rate-controlling
in the deformation of polycrystalline silicates. Here, the
small amount of melt in the sodic aggregates is thought
to cause moderate weakening of our jadeite samples in
comparison with the results of Orzol et al. (2006), as a
result of its effect on the load-bearing area (e.g. Cooper &
Kohlstedt, 1986) rather than by altering the rate-limiting
deformation mechanism, as indicated by the agreement of
the observed apparent stress exponents and activation
energies between the two studies. The large stress exponents also exclude substantial contribution of diffusion
creep. A contribution of grain boundary slidingçalthough
unlikely considering the serrated grain boundariesç
cannot be excluded per se, but the grain-shape accommodation has to be realized by dislocation processes rather
than by diffusional transport, again because of the high
stress exponents.
Twinning cannot be considered as rate controlling in
steady-state flow, nucleation being quasi-instantaneous at
a certain threshold of resolved shear stress, and growth by
motion of partial dislocations generally being controlled
by strain accommodation at the interfaces with adjacent
grains of a polycrystal. Moreover, twinning is restricted to
grains with a suitable crystallographic orientation, and
these sites will be rapidly exhausted at a given stress level.
Mechanical data and observed microstructures strongly
suggest deformation related to dislocation processes. The
models adopted below rest on the notion that the task of
quantifying the overall kinetics of a large number of conceivable processes affecting the motion of a huge number
of dislocations in a large number of grains constituting a
polycrystalline aggregate can be reduced to identifying
the slowest of sequential and the fastest of parallel processes. The motion of a dislocation comprises stages of
unhindered glide and stages during which obstacles are
overcome by climb of edge segments or cross-slip of screw
segments (e.g. Hirth & Lothe, 1982; Reed-Hill &
Abbaschian, 1994). After discussing potential rate-controlling mechanisms, we make the simplifying assumption
that the available data for the two compositional endmembers, jadeite and diopside, represent two distinct
deformation regimes kinetically controlled by two different
dislocation mechanisms, dislocation climb and glide,
respectively. We neglect transitional behavior although
deformation of diopside samples may involve dislocation
climb at the highest temperature of 11008C as indicated
by the near absence of a peak in strength for the intermediate compositions (Fig. 9a). Flow-law parameters are
_ strength or differential
derived from sets of strain rate ",
stress s, and absolute temperature T using a non-linear
inversion method that accounts for the reported uncertainties in experimental data (Sotin & Poirier, 1984).
Is cross-slip relevant?
Cross-slip is described by rate equations (Poirier, 1985; de
Bresser, 2002) that are formally similar to the glide equation employed below and, thus, cross-slip can be considered to be implicitly included in our approach. Cross-slip
was explicitly discussed for olivine (Poirier & Vergobbi,
1978; Raterron et al., 2007) and calcite (de Bresser, 2002),
yet its role as a rate-controlling step in silicates remains
uncertain, climb being probably more effective at high
temperatures because of fewer crystallographic constraints.
Cross-slip requires the association of dissociated dislocations for the switch onto an alternative glide plane.
Whereas dissociation of dislocations in clinopyroxene has
been observed by TEM (e.g. Skrotzki, 1994), the role of
cross slip in crystal plastic deformation of clinopyroxenes
has not been explored in detail. In principle, for a glissile
screw dislocation to overcome a discrete obstacle by
changing the glide plane depends on the availability of a
suitable plane in zone with the Burgers vector and in
appropriate orientation, and the stress state to overcome
the Peierls potential in the new system. The availability of
a suitable plane is easily conceivable for metal phases with
high symmetry and an according number of potential
glide planes (e.g. Puschl, 2002; Lu et al., 2004; Wang et al.,
2007). For clinopyroxenes, various glide systems have been
invoked from CPO patterns and the post-mortem analysis
of dislocations by TEM, in materials that have been
deformed both naturally (Buatier et al., 1991; Godard &
Van Roermund, 1995) and experimentally (Raterron &
Jaoul, 1991; Raterron et al., 1994). In experiments at temperatures below 10008C the glide system (100)[001] appears
to be dominant, and at higher temperatures this is accompanied by {110}[001] and {110}1/251104, among others.
The increased number of active glide systems may render
cross slip a feasible mechanism at higher temperatures, at
which climb also becomes more effective, as documented
by the formation of low-angle grain boundaries in our
experiments.
Climb-controlled deformation
(jadeite aggregates)
The velocity with which dislocations climb is limited by
the formation and mobility of jogs realized by vacancy
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JOURNAL OF PETROLOGY
VOLUME 51
diffusion. Weertman (1968) derived a general rate equation
for climb-controlled creep:
45
mDeff
"_W,climb ¼ 35 05
ð8Þ
b M kT m
here given according to Poirier (1985) (m is shear modulus,
is atomic volume, and b is the Burgers vector).
The dimensionless factor is of order unity and may
differ for the two diffusion processes considered to contribute to the effective self-diffusion coefficient:
ð9Þ
Deff ¼ Dl þ pR20 d Ddc
for volume diffusion through the lattice (Dl) and shortcircuit diffusion along dislocation cores (Ddc) of width R0.
Equation (8) covers popular special cases of ‘power-law
creep’ with stress exponents in the range of 3^6·5, when
the stress dependence of the density of dislocation sources,
M, and dislocation density, d, are incorporated appropriately (see Poirier, 1985). The formulation of the effective
diffusion coefficient assumes that the fastest of the alternative paths for the slowest constituent controls the overall
kinetics. The exact formulation of the contribution from
core diffusion is a matter of debate (e.g. Frost & Ashby,
1982; Balluffi et al., 2005). In this model, the temperature
dependence of creep is determined by the activation
enthalpies of the involved diffusion coefficients; for example, Dl ¼ D0 expðHl =RT Þ.
The apparent stress exponent of close to four observed
for jadeite aggregates (Table 2, Fig. 8a) is most obviously
explained by a combination of a moderate dependence of
dislocation density on stress and lattice diffusion as the
rate-controlling parameter in equation (8). Specifically, for
M ¼ const. and Deff ¼ Dl the stress exponent assumes a
value of 4·5. We consider M 1016·5 1·5 m3 to represent a
reasonable range for the dislocation source density; that
is, 1^1000 sources in a grain of a diameter of 10 mm. The
lower bound of this range was quoted by Weertman (1957)
when modeling the deformation of zinc. With this estimate
and mjd 75 GPa, b 5 1010 m, and jd 1028 m3, we
jd
jd
gain an order of magnitude
of Dl ¼ Aclimb
35
05
jd
2115
ðmbÞ M kT exp Hcreep =RT = 10
m2/s
jd
at 10008C where we used the pre-exponential factor Aclimb
jd
and an activation enthalpy for creep Hcreep inverted for
a prescribed stress exponent of 4·5 (Table 3). Alternatively,
for M ¼ M0(/0)2 and Deff Dl the stress exponent
amounts to 3·5. With M0 1016·5 m3 at s0 100 MPa, we
jd
gain almost the same estimate for Dl . These two estimates
compare well with directly measured lattice diffusion
coefficients for Si in various silicates (Bejina & Jaoul,
1997), in particular considering (1) that it may be overestimated by up to an order of magnitude becuse of the weakening effect of melt and (2) the relatively low melting
temperature of jadeite. Direct determinations of diffusion
kinetics are lacking for jadeite crystals. A dominance
of dislocation core diffusion (i.e. Deff pR20 d Ddc) is in
NUMBER 4
APRIL 2010
contrast unlikely for jadeite aggregates, because reaching
a stress exponent of about four then requires an
unrealistically strong dependence of the dislocation source
density on stress of M / 4 respecting the well-established
^ Þ2 with ^ of the order of
scaling relation b2 d ¼ ð=m
0·1^3 (Taylor, 1934; Takeuchi & Argon, 1976; Kohlstedt
et al., 1995).
In principle, the diopside data can also be modeled by
equation (8) which allows for stress exponents as high as
6·5 that reach the lower limit of the determined range
(Table 2). An exponent of 6·5 results for a dominance
of dislocation core diffusion (i.e. Deff pR20 d Ddc ^ Þ2 Ddc ), and a stress-independent denpb2 d Ddc ¼ pð=m
sity of dislocation sources. For this scenario, we gain an
order of magnitude estimate of the coefficient for disloca55 35
05
^ 2 Adi
tion
core diffusion
of Ddi
dc ¼ climb m b M kT exp
di
=RT =p < 10165 m2 =s that appears too
Hcreep
low compared with the order of magnitude of interface diffusion coefficients in silicates (Farver & Yund, 2000;
Milke & Heinrich, 2002) that, in the absence of appropriate data, may serve as a lower bound for dislocation
core diffusion coefficients (Balluffi et al., 2005). An estimate
too low in comparison with independently determined diffusion coefficients is in accord with our reasoning that
climb is not rate-controlling in diopside aggregates at the
explored conditions. Climb could yield a faster than the
observed rate, but glide as the slower sequential step is
rate-controlling. Because cross-slip is operating parallel to
climb, it can also be discarded as rate-limiting.
Glide-controlled deformation (diopside
aggregates)
The lattice resistance represented by the Peierls stress sP
constitutes the central mechanical parameter for dislocation glide. An explicit flow law for glide-controlled creep
25
Qglide
a
exp "_glide ¼ 12 25 05
1
b M
m
P
RT
Qglide
1
Aglide 25 exp P
RT
ð10Þ
was proposed by Weertmann (1957), which formally
coincides with that given by Frost & Ashby (1982) when
the dislocation source density M is assumed to be linearly
proportional to stress (a is the distance between glide
planes). The frequency of lattice vibrations
rffiffiffiffiffiffiffiffiffiffiffiffiffi
5 P
ð11Þ
¼ vs
4ab m
(shear-wave velocity v2s ¼ m=) and the apparent activation energy
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
32a3 b3 mP
0 57NA a3 b3 mP
ð12Þ
Qglide ¼ NA
2
10p
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MOGHADAM et al.
SYNTHETIC OMPHACITE AGGREGATES
both scale with the square root of the Peierls stress. Relying
on the estimate of the Peierls stress for an edge dislocation
from elastic theory (e.g. Hirth & Lothe, 1982), the prefactor 12a=b25 M 05 in equation (10) can be written as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5a
pa
12vs
exp
pð1 pÞMb6
ð1 pÞb
ð13Þ
11=12 5=6
vs NA
:
Oð1Þ pffiffiffiffiffi P
M m1=12 Q 5=6
glide
Because Peierls stress, apparent activation energy, and preexponential factor are not independent of each other, we
can either continue with the traditional parameterization of
the flow law using Aglide, sP and Qglide or alternatively take
the perspective that the three central, purely geometrical
characteristics of the gliding dislocations, a, b and M, control
strain rate. It is important to emphasize that in equations
(11)^(13) derived from elastic models the parameters a and b
have phenomenological rather than a simple crystallographic character for multi-constituent minerals with low
symmetry and complex bonding (Hirth & Lothe, 1982).
Furthermore, the representation of the elastic behavior of a
solid by only two elastic parameters (here, shear modulus
and Poisson’s ratio) rests on the assumption of isotropy,
which is clearly not true for pyroxenes.
Equation (10) fits our mechanical data for synthetic diopside aggregates at least as well as the simple power-law relation (1) (see Fig. 8b). Constraining the dislocation source
density to a reasonable range of M 1016·5 1·6 m3, the
inversionyields a Peierls stress (Table 3) that nearly coincides
with the value of 6·5 0·8 GPa found by Dorner &
Sto«ckhert (2004) for diopside. It shouldbe notedthat evaluating the hardness data relying on equation (10) rather than
the slightly modified version used by Dorner & Sto«ckhert
(2004) yields a somewhat lower value of 5 1GPa.
Furthermore, a consistency check using the characterization
of Frost & Ashby (1982) demonstrates that, by holding
Qglide50·2mb3, the apparent activation energy qualifies as a
case of ‘weak obstacles’. The corresponding obstacle spacing
l mb=P 10b indeed suggests that lattice or Peierls resistance constitutes the principal obstacle for glide. Finally, the
values of a and b calculated from the inversion results (Table
3) are obviously of the right order of magnitude for lattice
parameters.Thus, all the parameters determined for the dislocation glide model are fully consistent with physical considerations or independent data.
Scaling relations for the dependence
of strength on composition
Constraining the rheology of the mechanical endmembers,
jadeite and diopside, is the necessary first step towards
our ultimate goal of quantitatively describing the behavior
of omphacite aggregates of intermediate composition.
However, the lack of understanding of how the sequential
processes of glide and climb can be combined in a unifying
flow law currently hinders this quantitative modeling.
Therefore, we rely on simple scaling relations expected to
be applicable for the two considered deformation mechanisms. In the case of climb-controlled creep, the thermal
activation term associated with the effective diffusion coefficient in equation (8) suggests a scaling with homologous
temperature, T=Tm, such that D / expðH=RTÞ
expð=Þ (e.g. Poirier, 1985; Karato et al., 1995).
It should be noted that, although omphacites of the diopside^jadeite solid-solution series melt incongruently, the
difference between solidus and liquidus is moderate
(530 K) at elevated pressure (Bell & Davis, 1969), permitting calculation of ‘unique’ homologous temperatures
for intermediate compositions. The activation enthalpy
observed for jadeite aggregates (Table 3) yields an estimate
of 32·3 4·1, a value well within the expected range
(see Poirier, 1991).
For our analysis, we express bulk composition such
that X ¼ 0 and X ¼1 correspond to jadeite and diopside,
respectively. We assume @b3 =@X @a3 =@X @=@X;
that is, the geometrical characteristics of the glide system
vary with composition as the unit cell does, and estimate
(1) @ ln =@X ¼ @ ln =@X 0 02 using density values
and
di ¼ 3400 kg/m3,
(2)
jd ¼ 3260 kg/m3
@ ln m=@X 0 25 using mjd ¼ 87·8 GPa ^ 0·013 GPa/
K T þ1·7p and mdi ¼ 70 GPa ^ 0·01GPa/K T þ1·7p
(Duffy & Anderson, 1989), and (3) @=@X ^0·16 with
melting temperatures of Tm,jd 1653 K and Tm,di 1923 K
at the pressure of our experiments (Bell & Davis, 1969).
Tracking explicitly the dependence of all parameters
involved in equation (8) on composition, we find
@ ln "_climb @ ln m 1 @ ln @ ln D
@
3
5
2
þ
þ
@X
6 @X
@X T
@X
@X
,T
ð14Þ
demonstrating that the homologous temperature scaling
of the diffusion coefficient contributes dominantly to the
dependence of climb-controlled strength on composition.
The dependence of shear modulus on composition is subordinate and the relation of lattice parameters on composition is negligible.
In contrast, the dependence of shear modulus on composition dominates the scaling relation for glide-controlled
creep, as @ ln m=@X @ ln =@X. From the full analysis,
we find that the variation of strain rate with composition
is actually controlled by the relation between the apparent
activation energy and shear modulus (12)
@ ln "_glide Qglide
@ ln m
:
ð15Þ
1
P @X
@X ,T
RT
945