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Anisotropic high‑temperature creep in hydrousolivine single crystals and its geodynamicimplications

Masuti, Sagar; Karato, Shun‑ichiro; Girard, Jennifer; Barbot, Sylvain Denis

2019

Masuti, S., Karato, S., Girard, J., & Barbot, S. D. (2019). Anisotropic high‑temperature creepin hydrous olivine single crystals and its geodynamic implications. Physics of the Earth andPlanetary Interiors, 290, 1‑9. doi:10.1016/j.pepi.2019.03.002

https://hdl.handle.net/10356/107511

https://doi.org/10.1016/j.pepi.2019.03.002

© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CCBY‑NC‑ND license (http://creativecommons.org/licenses/BY‑NC‑ND/4.0/).

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Physics of the Earth and Planetary Interiors

journal homepage: www.elsevier.com/locate/pepi

Anisotropic high-temperature creep in hydrous olivine single crystals and itsgeodynamic implicationsSagar Masutia,⁎, Shun-ichiro Karatob, Jennifer Girardb, Sylvain D. Barbota,1a Earth Observatory of Singapore, Asian School of Environment, Nanyang Technological University, SingaporebDepartment of Geology and Geophysics, Yale University, New Haven, CT, USA

A R T I C L E I N F O

Keywords:OlivineSingle crystalSlip systemsTransient creepSteady-state creepFabric transition

A B S T R A C T

We compile the experimental data on deformation of hydrated olivine single crystal to determine the influence ofwater and pressure. We analyse the data from low pressure (P=0.1MPa and 0.1–0.3 GPa) high-resolutionexperiments as well as the results from higher pressures (2 to 6 GPa) containing larger uncertainties, using a flowlaw of the form = +i i

dryiwet with (i:slip system). The data are normalized to a common stress (150MPa) and

temperature (1573 K) and we determine the parameters water content exponent (r) and activation volume (V∗).We found largely different values of r and V∗ for different orientations (slip systems) implying that the influenceof water and pressure is highly anisotropic providing largely different values of r and V∗. For the [110]c or-ientation where the [100](010) slip system is activated, we obtain r[100]= 0.35 ± 0.08, andV∗[100]= 11.0 ± 3.0 cm3/mol, while for the [011]c orientation where the [001](010) slip system is activated,r[001]= 1.3 ± 0.30, and V∗[001]= 5.6 ± 3.6 cm3/mol (for a fixed water content). The highly anisotropic ef-fects of water and pressure suggest that creep in olivine is not controlled solely by diffusion but also controlledby the density of jogs (or kinks) on dislocations that is controlled by water and pressure. We find that the easierslip system changes from the a-slip (slip along the [100] direction) at low water content and low pressure to thec-slip (slip along the [001] direction) at high water content and high pressure, suggesting that the c-slip is softerthan the a-slip in most of the asthenosphere, whereas the opposite is true in most of the lithosphere. Implicationsof these results are discussed using a model of deformation of a polycrystalline aggregate made of anisotropiccrystals. It is suggested that in the asthenosphere where most of plastic deformation occurs, short-term time-dependent small strain deformation associated with post-seismic and post-glacial deformation is dominated bythe c-slip and strongly dependent on the water content, whereas the long-term large strain deformation asso-ciated with mantle convection is dominated by the a-slip that is only weakly dependent on water content.

1. Introduction

The effect of water (hydrogen) on the plastic deformation of olivinehas been well appreciated (e.g., Blacic, 1972; Mackwell et al., 1985;Karato et al., 1986; Mei and Kohlstedt, 2000a, 2000b; Karato and Jung,2003). In the simplest model of high-temperature creep, the rate ofdeformation is directly controlled by diffusion of the slowest movingspecies (e.g., Karato, 2008; Weertman, 1968). Kohlstedt (2006) adoptedthis model to explain water weakening in olivine. However, Mackwellet al. (1985) reported a high degree of anisotropy in plastic flow ofhydrous olivine. These findings contradict the simplest version of dif-fusion-controlled model because diffusion of Si and O in olivine isnearly isotropic (Costa and Chakraborty, 2008). A report by Fei et al.(2013) on the influence of water on Si diffusion in Mg2SiO4 are marked

contrast with others studies (e.g., Mackwell et al., 1985; Karato et al.,1986; Mei and Kohlstedt, 2000a, 2000b; Karato and Jung, 2003), pos-sibly due to the anisotropic effect of water on olivine deformation thatis caused by some factors other than diffusion.

The plastic anisotropy in hydrous olivine has never been char-acterized quantitatively. This is partly due to the difficulties in theexperimental study of the influence of water under high-pressure con-ditions (e.g., Karato and Weidner, 2008) but also due to the compli-cations caused by the combined effects of water and pressure (Karatoand Jung, 2003). In this paper, we focus on characterizing the aniso-tropic effect of water and pressure based largely on the experimentalresults on single crystals under hydrous conditions. In the followingsections, we review the available laboratory data at low and highpressure that constrain plastic anisotropy of hydrous olivine. We

https://doi.org/10.1016/j.pepi.2019.03.002Received 7 May 2018; Received in revised form 6 March 2019; Accepted 10 March 2019

⁎ Corresponding author.E-mail address: sagarshr001@e.ntu.edu.sg (S. Masuti).

1 Now at University of Southern California, 3651 Trousdale Pkwy, Los Angeles, CA 90089-0740, USA.

Physics of the Earth and Planetary Interiors 290 (2019) 1–9

Available online 18 March 20190031-9201/ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

T

characterize the potential errors associated with bringing all availableresults to a single standard of temperature and pressure condition.Using these data we identify the flow law parameters individually forthe two dominant slip systems. Finally, we discuss the implications forthe fabric transitions and the rheological properties of polycrystallineolivine aggregates.

2. Errors in the experimental study on plastic deformation

In order to characterize the water and pressure effect, it is importantto analyse the data that cover a broad range of water content andpressure. As any single apparatus cannot cover the range of pressureconditions found in the lithosphere-asthenosphere system, results ob-tained using different apparatus and methods must be compared. Inorder to combine these data, we evaluate the uncertainties associatedwith different apparatus and sample preparation methods. The char-acterization of these uncertainties will allow us to describe the activa-tion volume and power exponent of water content for a broad range ofpressure and water content, while mitigating potential biases fromusing data gathered in different laboratory settings.

2.1. Errors in stress, strain rate and temperature measurement

In this section, we summarize the uncertainties in each method withthe focus on the measurements of macroscopic quantities (stress, strain-rate and temperature). Errors associated with characterizing the che-mical environment (water content, oxygen fugacity, oxide activity) willbe discussed in the next section.

Plastic deformation of single crystal minerals can be studied underroom pressure (P=0.1MPa), high temperature conditions (e.g.,Kohlstedt and Goetze, 1974). The influence of water on plastic de-formation can be studied at room pressure if a gas mixture with dif-ferent water fugacities is used (Karato, 1989; Poumellec and Jaoul,1984). Under these conditions, the water fugacity is small and hencethe water effect is small. However, because both stress and strain ratecan be measured precisely under these conditions (errors in stressmeasurements are< 0.1MPa, errors in strain-rate measurementsare< 10%), these studies provide an important data set to understandthe influence of water at low water fugacities (∼10 kPa correspondingto the water content of 5×10−4 ppmwt). The temperature in theseexperiments is estimated with an error of± 5 K.

Results for higher water content can be obtained using a high-pressure deformation apparatus. Up to ∼0.3 GPa pressure, a high-re-solution gas-medium deformation apparatus (e.g., Paterson, 1990) canbe used. Under these conditions, substantially larger effects of watercan be observed because the water fugacity increases with pressure (atP=0.3 GPa, water fugacity is ∼100–150MPa at 1573 K for Ni-NiOoxygen fugacity buffer). With the Paterson apparatus where an internalload cell is used to measure the stress, one can obtain results with smallerrors (∼1MPa) and small errors in strain rate (< 10%) (Paterson,1990). Temperature in these experiments is estimated with an errorof± 10 K. Indeed, it is the study by Mackwell et al. (1985) on olivinesingle crystals using this apparatus in which a markedly different ani-sotropic water weakening effect was reported (see also Karato, 1989).Although the mechanical results by such a study are of high quality, therange of water content and pressure is so narrow that it is difficult tocharacterize the anisotropic flow laws applicable to a broad range ofupper mantle condition from their data alone.

The Griggs-type solid medium apparatus is adequate at pressuresfrom 0.5 to 3 GPa. With this type of apparatus, a conventional methodof stress measurements using an external load cell has large errors(Gleason and Tullis, 1993), but with an improved method of stressmeasurements such as the use of dislocation density, one could mini-mize the errors in stress measurements to± 10% (±∼20–30MPa)(Karato and Jung, 2003). Blacic (1972) conducted experiments onsingle and polycrystalline olivine samples using Griggs apparatus at

high pressures (1.5 GPa). However these experiments were at lowtemperatures (< 1300 K) and the corrections for the friction were notapplied. Therefore, the results by Blacic (1972) are not used in thepresent study. No studies have been conducted on olivine single crystalsusing a Griggs apparatus at high temperatures (> 1500 K).

Beyond ∼3GPa, those conventional deformation apparatus (a gas-medium deformation apparatus and the Griggs apparatus) cannot beoperated. New types of deformation apparatus such as the rotationalDrickamer apparatus RDA (Yamazaki and Karato, 2001) or deformationDIA (D-DIA) (Wang et al., 2003) must be used (see also Durham et al.,2002). At these high pressures, much higher water fugacity can berealized, and the influence of pressure can be determined. In theseexperiments, stress is measured using X-ray diffraction and the errors instress estimate are relatively large due to the complex relationshipbetween the microscopic stress on each grain (that is measured by X-raydiffraction) and the macroscopic stress acting on sample (Chen et al.,2006; Karato, 2009; Singh, 1993). The data we use are from Girard(2011) and Girard et al. (2013) in which strength of a single crystalolivine was estimated from the X-ray diffraction on polycrystallineAl2O3 using several diffraction planes. Each diffraction plane providesan estimate of strength, and the average strength is taken as an ar-ithmetic average and the error is one standard deviation (Table 1).Temperature measurements in these experiments have large un-certainties caused by the temperature gradient and in these cases, weestimate an error of± 60 K.

2.2. Errors in characterizing the chemical environment

Poumellec and Jaoul (1984) conducted deformation experiments atatmospheric pressure using single crystal olivine on two different or-ientations: [110]c and [101]c. They controlled the water fugacity andoxygen fugacity using different gas mixtures (H2/H2O and CO/CO2).The water effect was determined by comparing the results for the sametemperature and oxygen fugacity but at different water fugacity.However, Poumellec and Jaoul (1984) did not measure the watercontent of their samples. Indeed, the water contents in their sampleswould be so small that one would not be able to measure them with thecurrently available methods. We estimate the water content in theirsamples based on the thermo-chemical environment in their experi-ments using the experimental results of water solubility by Zhao et al.(2004). In addition, the oxide activity was not controlled in their ex-periments. However, we can infer the oxide activity from their ex-perimental procedure. In order to avoid chemical reaction, they in-serted a platinum (Pt) foil between their samples (San Carlos olivinesingle crystals) and an Al2O3 piston. Platinum reacts with olivine toremove FeO, and therefore we assume that their sample was under thesilica-rich conditions equivalent (or close) to the opx buffer.

In an experimental study with a gas apparatus, the oxygen fugacityis controlled by metal-metal oxide buffer (Fe-FeO, Ni-NiO). Oxide ac-tivity is opx buffered. Regarding the water content, we used the re-ported values of water content of samples determined by infrared ab-sorption spectroscopy and also saturation values at 300MPa (i.e.,∼16 ppmwt). There is some evidence for water escape in some ex-periments (e.g., Mackwell et al., 1985). However, using saturation va-lues or the reported values does not affect the results significantly andwe report the results of using the saturation values.

Other important data sets are those from Girard (2011) and Girardet al. (2013). These are deformation experiments on olivine singlecrystals at pressures 4.7–6.1 GPa using D-DIA apparatus. Girard et al.(2013) published the results of high pressure deformation experimentson olivine single crystal for the [110]c orientation. Results for the[101]c and [011]c orientations are also available from Girard (2011).These are important data because the pressure range is substantiallyhigher than that in other olivine single crystal studies.

However, there are several limitations on these data. First, watercontent during an experiment is not well constrained. They added water

S. Masuti, et al. Physics of the Earth and Planetary Interiors 290 (2019) 1–9

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in the early stage of an experiment through the dehydration of talc.After dehydration at ∼1100 K, temperature was increased to 1473 Kand deformation experiments were performed. The water content wasmeasured only in the samples after the experiment. The water contentsreported in their papers are less than the saturation limits under thoseconditions. Consequently, some water loss occurred during these ex-periments. Therefore, we use the reported water content and the stressmeasurements at the later stage of deformation experiments in ouranalysis. In addition, the measured stress values in their experimentsdid not change by>20% with time (strain), which is much less thanthe stress uncertainties in these experiments. Therefore, this assumptionwill not have a large effect on the results of our analysis.

The oxygen fugacity is not well constrained in these high pressureexperiments (4.7–6.1 GPa). They used rhenium (Re) foil to separate thesample from alumina, but they also used a W-Re thermocouple close tothe sample. Oxygen fugacity is likely between W−WOx/2 (x= 4–6;Cottrell et al. (2009)) and Re−ReO2 buffer. The oxygen fugacity cor-responding to the Fe-FeO is in between them, and we consider that theuncertainty in oxygen fugacity is ∼±2 orders of magnitude. Assumingq∼0.2 (Keefner et al., 2011), this leads to an uncertainty in strain-rateof a factor of ∼±2.5. As to the oxide activity, their samples are likelycontrolled by the opx buffer because the samples were in contact withtalc that produces pyroxene upon dehydration.

3. Method of analysis

In studying deformation of olivine single crystals, three differentorientations are used to activate different slip systems: [110]c, [101]cand [011]c ([110]cmeans the orientation where normal stress is appliedin a direction bisecting the [100] and the [010] orientations). For the[110]c orientation, the [100](010) slip system is activated, for the[011]c orientation, the [001](010) slip system is activated and for the[101]c orientation both the [100](001) and the [001](100) slip systemsare activated.

One of the most important factors to characterize a dislocation in a

crystal is the Burgers vector, a displacement vector by which the peri-odicity of the underlying lattice structure is preserved. As the energydensity of a dislocation line is proportional to b2, (b is the total slip), thelength of the Burgers vector b controls the energetics of deformation(Hirth and Lothe, 1982, Ch.1). The length of the Burgers vector is dif-ferent between b=[100] (0.48 nm) and b=[001], (0.63 nm). Thisleads to ∼50% difference in dislocation energy. Therefore, it is im-portant to compare the creep behaviour of different slip systems withdifferent Burgers vectors. Because [110]c and [011]c orientations acti-vate the slip systems whose slip plane is common ((010) plane) but slipdirections are different ([100] versus [001]), a comparison of creepproperties of samples with [110]c and [011]c orientations provide in-sight into the plastic anisotropy and such anisotropic effects have alsobeen observed in the dislocation recovery experiments (Yan, 1992). The[101]c orientation is expected to give the results in between because forthis orientation, two slip systems involving b=[100] or b=[001] areactivated (we call them a-slip and c-slip respectively).

We analyse these data assuming a flow law of the form

= + +A C f E pVRT

expi in r

Oq i

idry

Wi i i2 (1)

for each orientation, where i is the strain rate for the i-th orientation(such as [110]c), i

dry is the strain rate for the i-th orientation under drycondition, Ai is the pre-exponential constant, σ is the stress, ni is thestress exponent, E∗ is the activation energy, R is the universal gasconstant, T is the temperature, p is confining pressure, V∗ is the acti-vation volume, CW is the water content, ri is the water content ex-ponent, fO2

is the oxygen fugacity and qi is the oxygen fugacity ex-ponent. The correction for i

dry is important only for the room pressureexperiments by Poumellec and Jaoul (1984) where the influence ofwater is small (this correction was made based on the data presented byPoumellec and Jaoul (1984)).

The pre-factor Ai includes the effect of oxide activity, but we con-sider only the opx buffered case, and treat this as a constant in-dependent of thermodynamic conditions. The influence of oxygen

Table 1Experimental conditions and results of previous studies used in our study. The influence of oxygen fugacity is corrected using the pressure dependence of oxygenfugacity. The results shown correspond to the oxygen fugacity buffered by Fe-FeO. The water content reported in this Table is based on the (Paterson, 1982)calibration.

Source Orientation Pa

(GPA)Strain rateb

(10−5 s−1)Stress(MPa)

T(K)

Water contentc

(ppmwt)

Poumellec and Jaoul (1984) [110]c 10−4 0.017 30 ± 0.1 1564 5×10−4

Poumellec and Jaoul (1984) [101]c 10−4 0.018 43 ± 0.1 1564 5×10−4

Mackwell et al. (1985) [110]c 0.3 1 58 ± 1 1573 6.87Mackwell et al. (1985) [110]c 0.3 10 141 ± 1 1573 6.87Mackwell et al. (1985) [011]c 0.3 1 135 ± 1 1573 6.87Mackwell et al. (1985) [101]c 0.3 1 48 ± 1 1573 6.87Girard et al. (2013) [110]c 4.7 3.06 541 ± 100 1473 18.3 ± 4.5Girard et al. (2013) [110]c 4.8 1.28 390 ± 140 1473 21.7 ± 5.4Girard et al. (2013) [110]c 6.1 2.4 280 ± 100 1473 37.5 ± 25Girard (2011) [011]c 4.7 1.74 541 ± 100 1473 18.3 ± 4.5Girard (2011) [011]c 6.1 6.1 280 ± 130 1473 37.5 ± 25Girard (2011) [101]c 4.9 3.9 249 ± 110 1473 18.1 ± 6.25Girard (2011) [101]c 4.9 2.42 292 ± 80 1473 18.1 ± 6.25Tielke et al. (2017) [101]c 0.3 2.3 100 ± 1 1473 12.5Tielke et al. (2017) [101]c 0.3 6.2 122 ± 1 1473 12.5Tielke et al. (2017) [101]c 0.3 1.2 150±1 1473 12.5Tielke et al. (2017) [101]c 0.3 2.3 204 ± 1 1473 12.5Tielke et al. (2017) [101]c 0.3 1.3 86 ± 1 1473 12.5Tielke et al. (2017) [101]c 0.3 5.6 133 ± 1 1473 12.5Tielke et al. (2017) [101]c 0.3 2.8 206 ± 1 1473 12.5Tielke et al. (2017) [101]c 0.3 4.4 250 ± 1 1473 12.5Mei and Kohlstedt (2000a) All 0.3 6.7 120 ± 1 1473 16.25Mei and Kohlstedt (2000a) All 0.1 2.74 90 ± 1 1473 6.87

a For the gas-medium apparatus the uncertainties in the pressure are assumed to be 0.5MPa and for high pressure (> 4GPa) they are assumed to be 0.5 GPa.b Calculation of errors in strain rate including the influence of error propagation is given in the Supplementary material.c Uncertainties in measurements such as water content are assumed to be 10% (when not given).

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fugacity can be large because for a given oxygen fugacity buffer, cor-responding oxygen fugacity changes with pressure (and temperature)(e.g., Karato, 2008, Ch.2). The influence of pressure (and temperature)on oxygen fugacity is included in our analysis.

Table 1 summarizes the experimental data used in this study. Thetable includes all the data including those from high-resolution mod-estly high-pressure results (Mackwell et al., 1985; Tielke et al., 2017),room pressure (Poumellec and Jaoul, 1984), and high-pressure (ex-ceeding 4 GPa) data (Girard, 2011; Girard et al., 2013).

In order to do determine water content sensitivity (“r”) and thepressure sensitivity (“V∗”), we normalize all the data to common tem-perature (1573 K), common oxygen fugacity (corresponding to that ofthe Fe-FeO buffer), common oxide activity (opx buffer) and a commonstress (150MPa). In doing this normalization, E∗ =470 kJ/mol, n=3and q=0.2 are assumed (e.g., Bai et al., 1991; Keefner et al., 2011; Meiand Kohlstedt, 2000b; Poumellec and Jaoul, 1984).

One limitation of the currently available data set is that there areonly two direct deformation experiments for the [011]c orientation(Girard, 2011; Mackwell et al., 1985). Consequently, we add two con-straints based on the experimental observations of polycrystalline oli-vine. First, we use the observations of fabric transitions (e.g., Jung andKarato, 2001; Karato, 2008) at P=0.5–2 GPa. These studies showedthat fabric transitions from [100] slip (A-/E-type fabric) to [001] slip(C-type fabric) at a water content of ∼80 ppmwt at P∼2GPa(T∼ 1500 K). This implies that the creep strength for the a-slip ([110]c)and the c-slip ([011]c) should agree at this condition (e.g., Kaminski,2002; Lister, 1979).

Second, we also use the results from Mei and Kohlstedt (2000b),who performed experiment on polycrystalline olivine at pressures from0.1 to 0.45 GPa. In a polycrystalline olivine deformation experiment atsteady-state, more than one slip system is active at any given time.Based on the results suggesting that the c-slip is the rate-controlling slipsystem at low pressures with a modest water content, we assume thattheir results are controlled by the c-slip. In order to use these data, weneed to determine the proportionality constant between single crystaland polycrystal strength using the single crystal experimental datacarried out at similar thermodynamic conditions (such as confiningpressure). In this context we use single crystal experiments fromMackwell et al. (1985), which were under similar conditions to that ofMei and Kohlstedt (2000b), to determine the proportionality constant,and then use this constant to determine the data point corresponding toconfining pressure of 0.1 GPa. Using these additional constraints, weare able to determine all the necessary parameters for the flow law forthis orientation.

4. Data analysis

Because the magnitude of errors is considerably different amongdifferent studies, we analysed the data using the weighted least squaresmethod, where the weights are calculated as the reciprocal of thevariance of each data point. The key in this step is the error estimatethat includes the estimate of error in each data set as well as the errorscaused by the normalization to a reference temperature and stress.

The errors in each measurement were discussed above. Now we

estimate the errors associated with normalization. We normalize theresults of various studies to a common condition of T=1573 K andstress= 150MPa (and oxygen fugacity of Fe-FeO buffer). In doing so,we assumed a single stress exponent and activation energy. In reality,these parameters have some uncertainties. For example, n=3 ± 0.5and E∗ =470 ± 50 kJ/mol (Mackwell et al., 1985; Tielke et al., 2017).We take these errors in the parameters n and E∗ and propagate theuncertainties during normalization (see Supplementary material).

As all of experimentally derived strain-rate, water content, andpressure have uncertainties, we estimate the water content stress ex-ponent r and the activation volume V* using a generalized least-squaresmethod that account for errors in multiple axes (Ghilani and Wolf,2006) and weighted least squares method. The details and comparisonof these methods are given in Supplementary materials. As can be seenin Table S2, different methods give essentially the same results. Thedifference between the mean values of water content exponent andactivation volume is not too high when comparing the weighted least-square to that of generalized least-squares. However, the uncertaintiesare slightly higher in case of generalized least-squares than theweighted least-squares as expected. We conclude that the uncertaintiesin water content and pressure of fabric transition data point mentionedabove is not too large and hence we report/discuss values from theweighted least-squares approach in the main paper. The parameters A,r, and V∗ determined for [110]c, [101]c, [011]c are shown in Table 2,and the data and model predictions are shown in Figs. 1 and 2.

5. Results

A marked difference in the values of flow law parameters amongdifferent orientations can be noted. Particularly important is a largedifference in the water content exponent, r. For the [110]c orientationr=0.35 ± 0.08 whereas that for the [011]c orientationr=1.32 ± 0.3. For the [101]c r=0.63 ±0.06. Since b=[100] dis-locations are involved for the [110]c orientation and b=[001]

Table 2Flow laws for three different orientations corresponding to the Fe-FeO buffer.Water content sensitivity and activation volume estimated using the singlecrystal experimental data for different slip systems using weighted least squares(errors are for one standard deviation).

Orientation [110]c [101]c [011]c

Pre-exponential factor (A ins−1MPa−n ppmwt−1)

104.88± 0.03 105.54±0.01 102.63±0.24

Water content exponent (r) 0.35 ± 0.08 0.63 ± 0.06 1.32 ± 0.3Activation volume (V∗ in cm3/mol) 11.0 ± 3.0 14.2 ± 3.9 5.63 ± 3.6

Fig. 1. Strain-rate versus water content relationships for three different or-ientations. Strain-rates are normalized to T=1573 K and P=1GPa. Pressureeffects are corrected using the results of the weighted least square fit to thedata. Color of the symbol and the profile indicate different slip systems.Different symbols correspond to the different studies. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web ver-sion of this article.)

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dislocations are involved for the [011]c orientation, we interpret thatthe creep involving b=[100] and creep involving b=[001] havedifferent sensitivity to water content. The [101]c orientation involvesboth b=[100] and b=[001], and the estimated value of r for thisorientation is in between those for the [110]c and the [011]c orienta-tions. Similarly, the estimated activation volumes are different for creepinvolving different Burgers vectors. However, the activation volume for[101]c is not in between [110]c and [011]c as expected.

Based on these results, we propose that flow law of olivine singlecrystals can be classified into two classes: one involving b=[100]dislocations and another b=[001] dislocations. The results from the[110]c correspond to b=[100] dislocations, and the results from the[011]c to b=[001] dislocations, and the results from the [101]c areexpected to be in between [110]c and [011]c. Relative easiness of de-formation by these two classes of slip systems changes with variousconditions particularly with pressure and water content (Fig. 2). In thefollowing, we will use this simplified model and discuss the relativecontributions of these slip systems to plastic flow in Earth's uppermantle.

6. Discussions

6.1. Interpretation of the anisotropic water and pressure effects

The marked difference in the water content dependence for differentslip systems supports the notion that high-temperature creep of olivinemight not be controlled solely by diffusion as some of the simple modelsof suggest (e.g., Kohlstedt, 2006; Weertman, 1968; Weertman, 1999).Yan (1992) reported similar results, namely, the anisotropy in dis-location recovery in olivine (stronger dependence of dislocation re-covery on the water content for the c-slip than for the a-slip). In con-trast, diffusion of Si (and O) in olivine is remarkably isotropic (Costa

and Chakraborty, 2008).Consequently, we consider an alternative model where dislocation

creep is controlled by recovery with the recovery rate depending on thejog density as well as on the diffusion coefficient. In this model, whenjog formation is difficult (due to high Peierls stress, for example), thendislocation climb rate depends not only on diffusion coefficient but alsoon jog density viz., (e.g., Hirth and Lothe, 1982; Karato, 2008, Ch.17 &9, respectively),

D T P C c T P C( , , )· ( , , )W j W (2)

where D is the relevant diffusion coefficient and cj is the jog density. Notonly diffusion is water content dependent but also the jog density couldbe water content dependent if a jog is hydrated (Karato, 2008, Ch.10).The energy of a jog and hence the water content dependence of jogdensity depend on the crystallographic nature of a dislocation (e.g.,Hirth and Lothe, 1982) and hence the influence of water on jog densitylikely varies among different slip systems leading to anisotropic plasticdeformation.

The inferred small water content dependence for the [110]c or-ientation (b=[100] slip) similar to that on Si diffusion suggests thatcreep by this slip system is directly controlled by diffusion. This impliesthat for this type of dislocation, jog formation is easy and hence thedislocation line is saturated with jogs (Hirth and Lothe, 1982). Incontrast, for slip systems involving b=[001] dislocations, jog forma-tion energy is larger and hence the dislocation line is not saturated withjogs. This contrast in the water content dependency between the twoslip systems is likely due to different jog density between the two slipsystems.

The jog density at thermal equilibrium is given by c expjH

RTj

where Hj∗ is jog formation energy that is given by (e.g., Hirth and Lothe,1982, Ch.8)

=H µ b a·1

·j2

(3)

where b is the length of the Burgers vector, a is the length of the atomicdistance in the direction normal to the glide plane, β is a non-dimen-sional constant that may contain a variable that depends on the prop-erty of a dislocation, μ is shear modulus and ν is Poisson's ratio. For the[100](010) and the [001](010) slip systems, a is common but b is dif-ferent, and the formation energy of a jog Hj∗ is different by ~50%: cj ishigher for the [100](010) dislocations than for the [001](010) dis-locations.

The first-order conclusion from the relations (2) and (3) is that thelength of the Burgers vector makes an important influence on the jogdensity. Given the difference in the Burgers vector between b=[100](0.48 nm) and b=[001] (0.63 nm), we expect that the jog density forthe b=[100] dislocations is much higher than that for the b=[001]dislocations. In such a case, it is possible that the b=[100] dislocationsis saturated with jogs, whereas the b=[001] dislocations are under-saturated with jogs. This provides a possible explanation for the ani-sotropic effect of water and pressure and there could be other models.

Jogs of a dislocation may react with hydrogen (e.g., Karato, 2008;Karato and Jung, 2003; Mei and Kohlstedt, 2000b). This will introduceadditional water sensitivity such a way that water fugacity sensitivity ofcreep will be given by (Karato, 2008, Ch.10)

+fH Or r

2diff jog (4)

where fH2O is water fugacity and rdiff and rjog are the water fugacityexponents for diffusion coefficient and jog density respectively. Usingthe empirical relation, CW∝fH2O (e.g., Kohlstedt et al., 1996), this relationcan be translated to a relation between strain-rate and water content( +CW

r rdiff jog). For the slip system involving b=[100] dislocations,rjog= 0 if it is saturated with the jogs. But for the slip system involvingb=[001] dislocations, dislocations are not saturated with jogs and ifjogs are hydrated, then rjogs > 0. This explains the stronger sensitivity

Fig. 2. Strain-rate versus pressure relationships for three different orientations.Strain-rates are normalized to T= 1573 K and CW=1000 ppm H/Si or62.5 ppmwt. Water effects are corrected using the results of the weighted leastsquare fit to the data. Color of the symbol and the profile indicate different slipsystems. Different symbols correspond to the different studies. (For inter-pretation of the references to color in this figure legend, the reader is referred tothe web version of this article.)

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of the c-slip system to water content.A similar explanation may be presented for the anisotropic pressure

effect. From Eq. (2), one has,

= +V V V ,creep diff jog (5)

where Vcreep∗, Vdiff

∗ and Vjog∗ are activation volume for creep, diffusion

and jog formation respectively. For the slip system involving b=[100]dislocations, the pressure dependence of creep rate is the same as thatof diffusion, Vcreep

∗ = Vdiff∗. In contrast, for the slip system involving

b=[001] dislocations, pressure dependence include that of jog den-sity, consequently, Vcreep

∗ = Vdiff∗ + Vjog

∗. There is no study on Vjog∗,

but if we used the results of theoretical modelling on the pressure de-pendence of dislocation mobility (e.g., Skelton, 2018) as a guide, wemay speculate that Vjog∗ < 0 for the b= [001] dislocations that willexplain our observations. However, we admit that this explanation ishighly speculative because there is no direct calculation jog formationenergy in olivine as a function of pressure.

Anisotropy in kink formation (and migration) energy is an alter-native model, but based on the marked anisotropy in dislocation re-covery reported by Yan (1992), we prefer a model involving jogs.

6.2. Creep in olivine for different slip systems

Using the flow laws for the [001](010) and the [100](010) slipsystem(s) (a- and c-slip system), we calculate the viscosity-depth pro-files for two slip systems (Fig. 3). We used the oceanic geotherm cor-responding to the age of 60Myrs where a cooling half-space model fortemperature is assumed. All profiles show a marked minimum in visc-osity at ∼120 km. From this plot, we can see that the relative strengthof these two slip systems changes with depth.

To illustrate the transition in the softer slip system with depth moreclearly, we calculated conditions where the strain rates by two slipsystems ( [110] (the a-slip) and [011] (the c-slip)) agree (Fig. 4). The re-sults are shown in the pressure-water content space for a few assumedtemperatures. We find that under low pressure and/or low water con-tent, the [100](010) slip system with Burgers vector b=[100] is softer(higher strain rate for a given stress) than the [001](010) slip systemwith Burgers vector b=[001], whereas under high pressure and/orhigh water content, the [001](010) slip system with Burgers vectorb=[001] is softer (higher strain rate for a given stress) than the [100]

(010) slip system with Burgers vector b=[100]. This is consistent withthe results on deformation of polycrystalline olivine based the resultsfrom P=0.1 to 2 GPa showing the water content exponent is ∼1.2close to the water content exponent for the [011]c orientation(b=[001]) (Karato and Jung, 2003). Similar results were also obtainedusing low pressure data by Hirth and Kohlstedt (2003). However, in thedeeper regions of the upper mantle with high water content (e.g., theasthenosphere), the strong slip system likely changes to the one invol-ving b=[100] (Fig. 3).

6.3. Creep of a polycrystalline olivine and some geophysical implications

When there are multiple slip systems with largely differentstrengths, plastic deformation and deformation microstructures of apolycrystalline aggregate are connected to different slip systems indifferent ways for different properties. There have been few experi-mental studies on the transient creep (Chopra, 1997; Hanson andSpetzler, 1994; Thieme et al., 2018) and steady-state creep of olivineaggregates (e.g., Karato and Jung, 2003; Mei and Kohlstedt, 2000b),and some models have been proposed. However, here we discuss amodel based on our results and observation.

Creep by the softest slip system and creep by the hardest slip systemprovide the upper and the lower limit of real strain rate of an aggregatethat contains crystals oriented in various directions and the actualstrain-rate is in between these two limits. The creep behaviour of such apolycrystal evolves with strain caused by the evolving interaction be-tween the two slip systems.

Ashby and Duval (1985) provided a theoretical model of deforma-tion of a polycrystalline material that is made of highly anisotropicsingle crystal. Assuming homogeneous strain that includes elasticstrain, they derived a flow law equation for a polycrystal in terms of theflow laws for the soft and the hard slip systems that contains internalstress caused by the interaction between the soft- and hard-slip systems.Because internal stress evolves with strain, deformation behaviour alsoevolves with strain. At low strain (comparable to or lower than theelastic strain), internal stress is small and deformation of a polycrystal isdominated by the soft slip system, whereas at high strain, internal stressbecomes large and deformation is dominated by the hard slip system.The experimental studies on ice (highly anisotropic crystal) supportthese notions (e.g., Duval et al., 1983; Kocks, 1970).

In a geophysical context, low strain deformation is relevant to mostof time-dependent deformation that are used to infer mantle viscosity,

Fig. 3. A plot of effective viscosity (for a fixed stress (10 kPa)) for all threeorientations. An oceanic geotherm corresponding to 60Ma old oceanic platewith cooling half-space model for temperature is assumed (red profile). (Forinterpretation of the references to color in this figure legend, the reader is re-ferred to the web version of this article.)

Fig. 4. Diagram showing the dominant slip system as a function of watercontent and pressure with 95% confidence intervals at temperatures of 1273 Kand 1673 K. The a-slip is the slip system that is activated for the [110]c or-ientation (with b=[100], (010) slip plane) and the c-slip is the slip system thatis activated for the [011]c orientation (with b=[001], (010) slip plane).

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e.g., post-glacial rebound (Peltier et al., 1980) and post-seismic de-formation (Freed et al., 2010; Masuti et al., 2016; Qiu et al., 2018; Sunet al., 2014), where strain magnitude is comparable to or less than theelastic strain (Karato, 1998). In contrast, deformation associated withmantle convection is large. Consequently, in the former, the soft slipsystem dominates whereas in the hard slip system dominates in thelatter (for more details of transient creep, refer to Karato (1998).

Fig. 4 shows that in most of the deep, water-rich regions, the a-slipsystem is stronger than the c-slip system. This change in A-type to C-type fabric through E-type can be further supported by seismologicalobservation of the change in strength of radial anisotropy (Visser et al.,2008) (see also Karato, 2008). In such a case, steady-state creep isdominated by the a-slip system. For such a slip system, water contentdependence is weak, and therefore steady-state creep strength of theseregions is likely not very sensitive to water content. In contrast, tran-sient creep in these regions is highly sensitive to water content.

Note that, even though steady-state creep in the asthenosphere maynot be sensitive to water content, the dehydration that occurs beneaththe mid-ocean ridge will strengthen a rock substantially. This is becauseas one removes water, the rate controlling slip system for the steady-sate creep will change from the a-slip system to the c-slip system andthe net change in the strength will be substantially higher than a casewhere the a-slip system alone controls the creep strength (Fig. 5).

6.4. Implications for the fabric transitions in the deep upper mantle

Deformation fabric such as the lattice-preferred orientation (LPO) ofminerals is controlled by the weakest slip system (e.g., Karato, 2008;Lister, 1979). Some studies have shown that the LPO changes withchange water content under low pressure conditions (Jung and Karato,2001), while others have suggested that it is pressure induced under dryconditions (Jung et al., 2008; Raterron et al., 2009). However, from ourresults we suggest that the weakest slip system in olivine changes withwater content and pressure. Therefore, deformation fabric (LPO) ofolivine will change both with water content and pressure.

Up to ∼3GPa (∼100 km depth), the difference in pressure effects

are minor and water effects dominate. The present results are consistentwith the fabric transitions in olivine reported by Karato et al. (2008)that summarizes the experimental results below P=2GPa. Using ournew results, we further examine the likely LPO in the deeper regionsdown to ∼200 km (P=6GPa). An inspection of Fig. 3 shows thatdown to these pressure conditions, high water content corresponds tothe olivine LPO controlled by the c-slip (e.g., the C-type or the B-typefabric), whereas low water content corresponds to the a-slip (e.g., the A-type or the E-type fabric). The reported trend from the A-type to the C-type fabric transition through the E-type fabric at relatively low stressesis consistent with the present results. We conclude that the resultssummarized by Karato et al. (2008) are valid to P∼ 6GPa (∼200 km).

Applicability of our results to higher pressures is not clear. Similarto our results, there are reports showing that activation volume for thea-slip is larger than for the c-slip under the dry (or water-poor) condi-tions (Girard, 2011; Raterron et al., 2007). These results suggest that, atlow water content, the c-slip becomes easier than the a-slip at pressureshigher than ∼6–8 GPa at the laboratory strain rate. If one were to applyour results showing larger water content sensitivity for the c-slip thanfor the a-slip, then one would expect that at pressures of 7–11 GPa andat high water content, the c-slip is easier than the a-slip. This inferencedoes not agree with the results by Ohuchi and Irifune (2013) who re-ported the A-type fabric (due to the a-slip) dominates under the deepupper mantle conditions (P= 7–11 GPa) with high water content. Wedo not know how to explain this discrepancy. Further experimentalstudies are needed to solve this puzzle.

7. Summary and concluding remarks

We analysed the experimental data on plastic deformation of olivinesingle crystals with the presence of water (hydrogen). The data includethose obtained at P= 0.1MPa to P= 6GPa. Assuming that the stressexponent and activation energy are the same for all slip systems, wedetermined the water content and pressure dependence of flow laws forthree different orientations assuming the flow law of the form,

( )C texpiwet r PV

RTWi i . We show a marked difference in these para-

meters (ri, Vi∗) among different orientations involving different slipsystems. The slip system involving a-slip has a weak water contentdependence (r=0.35) and strong pressure dependence (V∗ =11 cm3/mol), and the slip system involving c-slip has a strong water contentdependence (r= 1.3) and weak pressure dependence (V∗ =5.6 cm3/mol).

The water content exponent that we determined is consistent withthe lab results at relatively low pressures showing r∼ 1.2 (e.g., Karatoand Jung, 2003) where the [001](010) slip controls the rate of steady-state creep. Similarly, our conclusion is consistent with those obtainedby low pressure data suggesting r∼ 0.2–0.3 (e.g., Poumellec and Jaoul,1984) in single crystals where the a-slip system dominates. However,the estimates of activation volume contain large errors. This is due tothe large experimental errors in high-pressure studies that are currentlyavailable.

We must emphasize that our analysis contains a few assumptions.One of them is the assumption that the sensitivity of deformation totemperature, stress exponent and oxygen fugacity is the same for all slipsystems. This is an over-simplification. If deformation with the c-slipsystem involves jog formation but that of the a-slip system does not,then the activation energy for the former will be somewhat higher thanthe latter. However, this will cause no major change as far as we con-sider deformation at high temperatures close to the normalized tem-perature (T=1573 K) that we used. Similarly, the influence of thedifferent oxygen fugacity is not large because the influence of oxygenfugacity is modest (exponent is ∼0.2). Potentiallyimportant is thepossible difference in the stress exponent. There are some observationssuggesting that the stress exponent for the b=[001] slip system islarger than that for the b=[100] slip system (e.g., Goetze, 1978). Since

Fig. 5. Change in viscosity upon dehydration. Dehydration due to partialmelting will strengthen a rock (Hirth and Kohlstedt, 1996; Karato, 1986). De-hydration depth of 65 km is assumed where T=1473 K corresponding to de-hydration beneath the mid-ocean ridge. Initial water content is assumed to be200 ppmwt. The slip system controlling the steady-state rheology is the oneinvolving b=[100] at the beginning. Consequently, the initial stage of hard-ening due to dehydration is limited. However, as the water content is reduced,the dominant slip system changes to the one involving b=[001] and moresubstantial weakening will occur.

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we need large extrapolation in stress (to a lower level) to apply the labdata to Earth, this has a large effect. Indeed based on the results sum-marized by Goetze and Poirier (1978) and Karato et al. (2008) sug-gested that the stress exponent for the a-slip system is smaller than thatof the c-slip system. If this is the case, a comparison of the relativestrength of these slip systems at a geological stress level needs to bemade with great care.

Implications of our results to deformation of a polycrystalline ag-gregate of olivine are discussed based on a theoretical model developedby Ashby and Duval (1985). This model predicts evolving flow beha-viour: small strain deformation dominated by the softslip system butlarge strain deformation by the hard slip system. This is a general be-haviour of deformation of a polycrystalline aggregate as far as homo-geneous strain is assumed, but it is confirmed only for ice where ani-sotropy is large. Similar tests for olivine and other mantle materials willbe important.

Our results predict that water content sensitivity will be differentbetween steady-state creep and transient creep, and will change withdepth (under the assumption that the weakest slip system controls thetransient creep and strongest controls the steady-state creep). Our re-sults also explain the apparent discrepancy on the water sensitivitybetween creep and diffusion. However, the applicability of our resultsto pressure exceeding 6 GPa remains to be tested. Also important is thepossible difference in stress exponent and the activation enthalpyamong different slip systems. High-resolution deformation experimentsunder high pressures with well-constrained chemical environment areneeded to address these issues.

Acknowledgements

The visit of SM for a month to Yale University and the research waspartly funded by Dr. Stephen Riady Geoscience Scholars Fund, 2016.This research is supported by the National Research FoundationSingapore under the NRF Fellowship scheme (National Research FellowAwards No. ~NRF-NRFF2013-04) and by the Ministry of Education -Singapore under the Research Centres of Excellence initiative. Thispaper is also supported partially by the National Science Foundation ofUnited States (EAR-1082622 to SK). We thank the editor Prof KeiHirose. The reviews by Greg Hirth and anonymous reviewer, andcomments by Anwar Mohiuddin helped clarify several issues. This workcomprises Earth Observatory of Singapore contribution no. 172.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.pepi.2019.03.002.

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