High-pressure yield strength of rocksalt structures using quartz Raman piezometry

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quartz Raman piezometry

Bruno Reynard, Razvan Caracas, Herve Cardon, Gilles Montagnac, Sébastien

Merkel

To cite this version:

Bruno Reynard, Razvan Caracas, Herve Cardon, Gilles Montagnac, Sébastien Merkel. High-pressure

yield strength of rocksalt structures using quartz Raman piezometry. Comptes Rendus Géoscience,

Elsevier Masson, 2019, 351 (2-3), pp.71-79. �10.1016/j.crte.2018.02.001�. �hal-02347201�

Internal

Geophysics

(Physics

of

Earth’s

interior)

High-pressure

yield

strength

of

rocksalt

structures

using

quartz

Raman

piezometry

Bruno

Reynard

a,

*

,

Razvan

Caracas

a

,

Herve´ Cardon

a

,

Gilles

Montagnac

a

,

Se´bastien

Merkel

b

a

Universite´ deLyon,E´colenormalesupe´rieuredeLyon,Universite´ Claude-BernardLyon-1,CNRSUMR5276,46,alle´ed’Italie,69007Lyon, France

b

Universite´ deLille,CNRS,INRA,ENSCL,UMR8207,UMET,Unite´ Mate´riauxetTransformations,59000Lille,France

1. Introduction

Understanding plastic deformation and mechanical propertiesofoxidesandmetalsunderextremepressures isessentialformodellingthedynamicsofconvectionthat generateplatetectonics,volcanism,andmagneticfieldson Earthandtelluricbodiesinsolarandextrasolarsystems. Theformationandthemobilityofpointandlineardefects attheoriginofplasticdeformationinthosesolidsathigh pressurearecurrentlystudiedwithstate-of-the-art first-principlesandatomisticsimulations(Cordieretal.,2012; Kraychetal.,2016)thatrequirevalidationoranchoringon

experimental data (Merkel et al., 2002, 2003,2007), in comparisonwithmesoscalemodels(Amodeoetal.,2016; Linetal.,2017).StaticpressuresintheEarthreachmore than300GPa,andcanonlybereproducedexperimentally usingdiamondanvilcells(DAC)withverysmallsamples. Plastic deformation under high stresses also occur in mechanicaldeviceswithpartsmovingathighspeeds(e.g., boundaryconditionlubrication)orformaterialssubmitted toimpacts. Thusfar,mostinformation onhigh-pressure plastic deformation mechanisms and yield strengthsof oxides,silicatesandmetalshavebeenobtainedfromX-ray diffractiontechniquesonsynchrotronsowingtothesmall samplevolumesinthediamondcell(Dorfmanetal.,2015; GleasonandMao,2013;Merkeletal.,2002,2003;Miyagi etal.,2006;Shiehetal.,2002;Singhetal.,2008).Someof

ARTICLE INFO Articlehistory:

Received12December2017

Acceptedafterrevision22February2018 Availableonlinexxx

HandledbyJamesBadro Keywords: Highpressure Yieldstrength Halides Oxides ABSTRACT

TheRamanfrequenciesofquartz areusedtoevaluatedeviatoricstressesin rocksalt-structure media in diamond-anvil cell experiments to pressures up to 20GPa. The piezospectroscopic effect in quartz is modeled by first-principles calculations. Non-hydrostaticstressesmeasuredinhalogensaltsgiveyieldstrengthestimatesof3GPainthe B1structure(NaCl),and4.5GPafortheB2structure(KClandKBr).Ramanmeasurements inMgOshowthattheyieldstrengthisreachedataround61GPa.Measurementson quartzaloneindicatesimilaryieldstrength.TheestimatedyieldstrengthinMgOisthuslikely alowerbound,inconsistencywithformerradialX-raydiffractionmeasurementsthatgavea yieldstrengthof81GPa,andlower-pressurelarge-volumepressexperimentsindicatinga yieldstrengthof6GPaat5GPa.Formervaluesoftheyieldstrengthbelow2GPadetermined bypressuregradientmeasurementswereunderestimatedduetounverifiedassumptionsin boundaryconditions.Theyieldstrengthincreaseswithincreasingcoordinationofionicsolids, likelycontributingtoincreaseviscosityatphasetransitionsneartheupper–lowermantle boundary.

C 2018PublishedbyElsevierMassonSASonbehalfofAcade´miedessciences.Thisisan

openaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/ by-nc-nd/4.0/).

* Correspondingauthor.

E-mailaddress:bruno.reynard@ens-lyon.fr(B.Reynard).

ContentslistsavailableatScienceDirect

Comptes

Rendus

Geoscience

w ww . sc i e nce d i re ct . co m

https://doi.org/10.1016/j.crte.2018.02.001

1631-0713/C_{2018PublishedbyElsevierMassonSASonbehalfofAcade´miedessciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense}

these X-ray based determinations of strength are in contradictionwithestimatesbasedonsimpleanalysisof stressfieldandpressuregradientintheDAC(Meadeand Jeanloz,1988a,1988b).

WeinvestigateheretheapplicationofRaman spectros-copyfor in-situpiezometry in theDACon simpleionic compounds.TheRamanspectraofcrystallinematerialsare sensitivetoappliedstressandvibrationalmodefrequency variations can be calibrated to estimate hydrostatic pressure(Goncharovetal.,1985; Hanflandetal.,1985). For determining yield strength of materials at high pressure, it is necessary to measure both hydrostatic pressure and deviatoric stress. Deviatoric stresses are knowntoliftthedegeneracyofRamanpeaksbelongingto specific symmetry species in high-symmetry crystals (Cerdeira et al., 1972; Grimsditch et al., 1978). This is especiallytrueinpiezoeletricnon-centrosymmetric crys-tals such as quartz where splitting occurs both due to symmetrybreakingandtocouplingoflongitudinalmodes of vibrations with the electrical field imposed by the deviatoricstress(Grimsditchetal., 1977;Tekippeetal., 1973).Wethereforechosequartztotestthemethodona seriesof simpleionicsolidswitheithertheB1rocksalt structure(NaCl, MgO)or itshigh-pressuremodification, theB2(

a

-CsCl)structure(KCl,KBr).Forthatpurpose,we combined experimental measurements and first-princi-ples calculations of Raman spectra. We discuss the reliability of stress analysis using different methods, Raman piezospectroscopy, radial X-ray diffraction, and measurementsofpressuregradients.

2. Methods

2.1. DFTcalculationsofRamanspectra

We obtained the stress effect on the vibrational frequenciesofquartzfromfirst-principlescalculationswith thedensity functionaltheory(DFT)in theABINIT imple-mentation (Gonze et al.,2002,2005), whichis basedon planewavesandpseudopotentials.We employedregular high-symmetry666gridsofspecial k-points( Monk-horstandPack,1976)anda34Ha(1Ha=27.2116eV)cutoff for the kinetic energy of the plane waves. Previous simulations (Caracas and Bobocioiu, 2011) showed that theseparametersensureaccuracyinenergyontheorderof 1mHaperunitcellandapressureerrorbarontheorderofa fewhundredMPa.In thepseudo-potential approach,the coreelectronsarereplacedbypseudopotentialswhilethe outer electrons are treated explicitly. The considered valenceelectronsare2s2_2p2 _and2s2_2p4_{,respectively, for}

SiandO(FuchsandScheffler,1999).

First,werelaxthequartzstructureatinthe0–20GPa pressurerangein2.5GPasteps,i.e.weallowtheatomsto moveundertheactionoftheinteratomicforcesuntilthey disappear,andthevaluesofbothaandclatticeparameters to change to yield purely hydrostatic pressures of the desiredvalues.Thenoneachoftheserelaxedstructures, weapplyboth positive andnegativeuniaxial andshear stressesof1%,2%,and5%.Inthesesimulations,weallow onlytheatomstorelax, whilekeeping thestrainedcell

shapefixed.Eventuallyontheresultingstrained structu-res,wecomputethefrequencies,theatomicdisplacement pattern,andtheRamantensorsofthenormalmodesinthe center of the Brillouin Zone. For this task, we usethe density functional perturbation theory in the ABINIT implementation(Baronietal.,2001;Gonze,1997;Gonze andLee,1997;Gonzeetal.,2005).Asthecomputationof the Raman tensors is limited to the Local Density Approximation (LDA) fortheexchangecorrelationterm, wechosetokeepLDAforallthesimulations.

2.2. High-pressureRamanexperiments

RamanspectrawerecollectedwithLabRamHRconfocal systemsdescribedelsewhere(Auzendeetal.,2004;Zhang etal.,2013).Ofspecificrelevancetothisstudy,thespotsize was less than 2 microns and optical depths less than 10microns,whichallowedustocheckforverticalgradients in the less compressed samples, but not at the highest pressures,forwhichthicknessisaround20microns.

Allsampleswerereagent-gradecommercialpowders. Quartz powder with small grain size (0.5–2

m

m) was preparedbyannealingareagent-gradepowderat8708C for24hinthe

b

-quartzstabilityfield,thenat5508Cfor 72htoensurefullconversiontothepiezoeletric

a

-quartz structure. Full conversion and absence of measurable stressonthecrystalswascheckedbyRamanspectroscopy. Anatase(TiO2)isaminorimpurityinthepowderwitha

strongRamanbandat144cm1_{intheregionofinterest,}

butanegligibleproportion(<2%)ofRamanspectradisplay contaminationbyanatase.

High-pressure experimentswerecarriedoutinMao– Bell-and Almax–Boehler-type DACequippedwith500– 600-

m

mculet low-fluorescence diamonds.The samples consistedoffine-grainedpowdersofsalts(0.5–10

m

m)or MgO(0.5–2

m

m)mixedwithabout10%ofquartz.Mixtures weredried overnight at 1208Cbefore loadingthe cells. Withthismixingratio,saltsandMgOservedaspressure mediumtoquartz,andwereloadedeitherasloosepowder thatwascompresseddirectlybythediamondsoraspellets ofprecompressedmaterial.Thesampleswereloadedina stainless-steelgasketinwhicha250-

m

mholewasdrilled. Experiments with smaller (150

m

m) and larger gasket holes were performed on NaCl. Experiments with pre-pressed pellets (1 to 4GPa precompression) were per-formedonMgO.Deviatoricstressesobtainedfromtheruns withdifferentgasketholediametersdifferatlowpressure (<5–7GPa)andconvergeathigherpressure.Anadditional experimentwasperformedupto20GPatocalibratethe pressuredependenceofthequartzRamanspectrumina methanol–ethanol–water 16:4:1 mixture (MEW) that remainsliquidupto11GPa.

Becauseofthecylindricalsymmetryoftheexperiment, thestressonthesampleisdescribedas:

s

¼

s

1 0 0 0

s

1 0 0 0

s

3            ¼

s

p 0 0 0

s

p 0 0 0

s

p             ¼ t=3 0 0 0 t=3 0 0 0 2t=3             (1)

B.Reynardetal./C.R.Geosciencexxx(2018)xxx–xxx 2

where

s

P is the normal mean stress or hydrostatic

pressure, and t=

s

3–

s

1 the uniaxial stress component

ordifferentialstressisameasureof,andreferredtointhe followingasthedeviatoricstress.

To mapthepressure and deviatoricstressfields,the samplewasmappedatintervalsrangingfrom10to20

m

m, yieldingafewtenstoafewhundredusabledatapoints, dependingon pressure. Note thatthepressure medium becomes more transparent at higher pressures, thus increasing spectral quality. Also, note that, because of the small grain size, the transparency of the pressure medium,thespotsizeofabout2

m

m,andthedepthoffield ofafewmicrons,itispossiblethatafewgrainsofquartz contributetotheRamansignaloneachmeasurementspot. 3. Results

3.1. Ramanpiezospectroscopyofquartzathighpressure Non-hydrostaticstressesaffectRamanmode frequen-ciesbycausingdistortionsanddecreasingthesymmetryof theunit celldependingon both themagnitudeandthe orientationofstresseswithrespecttothecrystallographic axes.Inpiezoelectriccrystalssuchasquartz,Ramanmodes belonging toE symmetry species have transverseoptic (TO) and longitudinal optic (LO) component whose splitting is affected by non-hydrostatic stresses both becauseofsymmetryreductionandbecauseoftheelectric field caused by the permanent elastic deformation (Tekippeetal.,1973).AmongtheEmodesofquartz,the lowest frequency one(128cm1 _{atambientconditions)}

hasnegligiblesplittingwhenthecrystalisunstressedand issplitundertheapplicationofastressorelectricfield. Splittingofthe128cm1_{modecomponentsoccursdueto}

deviatoric stresses(

s

)appliednormal tothec-axis and shearstressesalongitaccordingto:

Dn

E¼2 c0

s

xx

s

yyþd0

s

xz 2 þ 2c0

_s

xyþd0

s

zx  2 h i1=2 (2) where

Dn

EistheTO–LOsplitting,xthedirectionalongthe

a-axis,zthedirectionalong thec-axis,y theorthogonal direction.c’andd’areconstantswiththenotationusedin anearlierwork(BriggsandRamdas,1977;Tekippeetal., 1973).

Thevaluesofc’andd’wereexperimentallydetermined atuniaxialstressesupto0.8GPa,77K,and295K(Briggs andRamdas,1977;Tekippeetal.,1973).Forapplicationsat higher pressures, the vibrational frequencies of the strained crystals were calculated using DFT up to 20GPa.The valueofc’is obtainedby strainingtheunit cellalongainthebasalplaneandd’byapplyingashear strain

e

xz. At ambient pressure, the values of c’ and d’

obtained fromDFT of2.5and–0.6cm1_{/GPaareingood}

agreement with the experimental values of 2.2 and– 0.6cm1_{/GPa,respectively,whenonlylinearvariationsof}

1–2%instrainorabout1–2GPainstressareconsidered.At high pressure, c’ varies in a complex manner and d’ is constantwithinerrorbars(Fig.1).Sincevaluesofd’are low, the overall contributionof this termto theactual valueofthesplittingcanbeneglected,andcalculationsof

d’wherenotextendedtothefullpressure/strainrange.On the opposite, c’ varies with pressure, and additional calculationswith5%ofstrainalongashowthatitvaries withstress.Thusuniaxial

e

xxstrainsalongaof1,2and5%

yieldTO–LOsplittingandstressesintherange1–1.5,2–3, and 5–8GPa, respectively, which were fitted with the expression:

Dn

E¼2 c0

s

xx

s

yyþc00

s

xx

s

yy 2

 

(3) ateachpressure.Thevariationsofc’andc’’withpressure werefittedwithpolynomialexpressions(Fig.1): c0_{¼2:44ð26Þ0:17ð3ÞP3:2ð8Þ10}4

P3 (4)

c00_{¼0:06ð3Þ0:017ð5ÞP3:5ð8Þ10}5_P3 ₍₅₎

where the numbers in parentheses are the standard deviationonthelastdigitoftheparameter.Thesesimple interpolationswereusedtoconvertthemeasuredTO–LO splittingofthe128cm1_{modeintostressundergoneby}

quartz crystals on the Raman analytical spot. More complexextrapolationschemesarenotnecessary,because quartzundergoesphasechangesandamorphizationabove 20GPa(Kingmaetal.,1993a,1993b).

Theuncertaintyonindividualdeviatoricstress measu-rements is estimated as 20%, mostly due to peak broadening at high stresses. Measurements on several hundreds of spots at high pressure show a standard deviationoflessthan10%,aproxytotheglobaluncertainty (Fig.2).Uncertaintiesinpressureincreasewithpressure anddeviatoricstressbecauseofpeakbroadening,andare estimatedas10%.Peakbroadeningisattributedtostress gradientsintheindividualquartzcrystalsthatcannotbe furtheraccountedfor,andarenotdealtwithinthepresent piezospectroscopicanalysis.

Pressurewasestimatedfromthepositionofthestrong A1modeat464cm1atambientconditions,becauseDFT

calculations show that its position depends mostly on hydrostaticpressure(

s

P),andisnegligiblyaffectedbythe

deviatoricstress.Aleast-squarefittoexperimentalvalues inMEWpressuremediumuptoabout20GPagives: P¼0:112ð3Þð

n

ðPÞ464Þþ4:3ð3Þ104_ð

_n

_ðPÞ464Þ2

(6) with

n

ðPÞthefrequencyatpressureincm1_,anddigitsin

parenthesesarestandarddeviationsestimatedfromthefit. Maps of the strong A1 mode position, i.e. pressure

distribution, as well as typical Raman spectra at the center and rim of the diamond culet illustrate the distribution of pressure and stress in an experiment witha MgOpressuremedium(Fig.3).Atlowpressure (at5GPa), pressureis homogenous.Athigher pressu-res,pressureismaximumatthecenterofthegaskethole anddecreasesstronglybutsmoothlytowardstheedges of the hole. The deviatoric stress shows the opposite behavior,itisveryheterogeneousatlowcentralpressure, i.e. low compaction near P5GPa, and becomes very homogeneous at high pressure or higher compaction, above10–15GPa(Fig.2).

Because we use a population of quartz grains, a random orientation with respect to the applied stress

field in the DAC results in stresses along the a-axis rangingfromamaximumvalueof

s

xx–

s

yy=twhenthe

c-axis is perpendicular to the compression axis to zero whenitis parallelto it.Atthesame time,therelative intensityof theLOmode variesfromthemaximum to zero in the vertical polarization geometry used here, resultinginhighersignalfromgrainswithorientations suchthatboththeTO–LOsplittingandLOintensityare maximum. This is confirmed by statistics of TO–LO measurements(Fig.2).Athighpressures,themeasured stressdistribution isverypeakedwhencompared with the theoretical distribution with randomly oriented quartz grains. This suggest that only grains with an anglebetweenthec-axisandthecompressionaxishigher than60 or even708 contribute to thestress measure-ments. Forthat populationof quartz grains,the stress measurementgiven by TO–LO splitting is close to the macroscopic stress field imposed by the pressure medium. Hence, the maximum values of stresses deduced from TO–LOsplittings are a goodestimateof thesamplemacroscopicstresst.Similardistributionsare observedinthestudiedsalts,eventhoughtheabsolute stressesarelower,asdescribedbelow.

3.2. Non-hydrostaticstressesinsaltsandMgO

Frequencies of the TO and LO components of the 128cm1 _{E mode recorded in methanol–ethanol–water}

(MEW),NaCl,andMgO(Fig.4)showtheexpectedsplitting according to their nature and state (liquid or solid). In MEW,thesplittingisnullwhencompressionishydrostatic intheliquidstate(<11GPa)andbecomesmeasurableas soonasthetransitiontoaglassiscrossed.Splittingsare largeandnon-nullatallpressuresinsolid-statemedia.In NaCl,theyprogressivelyincreasewithincreasingpressure toreachvaluessimilartothoseoftheDFTfor2%strain alongthea-axis,correspondingtostressesintherange2– 3GPa.InMgO,theyspanabroadrangeatlowpressuresto finallyreachanarrowrangeofvaluessimilartothoseof theDFTfor 5%strainalongthea-axis,correspondingto stressesintherange5–8GPa.LargerstressesinMgOthan inNaClareconsistentwithstrongerionicinteractioninthe oxidethaninthehalide.Theexcellentagreementbetween DFTandexperimentalfrequenciesvalidatestheuseofTO– LOsplittingtointerpolatestressesfromEqs.(3–5).

Thebroaddistributionofstressvaluesatlowpressures, especially in MgO (Figs. 2–4), is likely related to a

Fig.1.PressuredependenceoftheconstantsrelatingtheTO–LOsplittingofthelowestfrequencyEmodeofthequartztotheappliedstress. Ambient-pressuremeasurementsofc’gaveavalueof2.2cm1_{/GPa,andofd’of0.6cm}1_{/GPa(Grimsditchetal.,1977;Tekippeetal.,1973).(a)Absolutevaluesofthe}

constantsincreaseupto15GPabeforerapidlydropping,possiblyinrelationtoinstabilitiesandphasetransformationsprecedingamorphizationinthe21– 26GParange(Kingmaetal.,1993a).(b)Thesecondderivativec’’obtainedfromDFTshowsanon-lineardependenceonstressandstrainthatwasnot exploredexperimentallyduetothelimitedpressurerange(<0.8GPa).

B.Reynardetal./C.R.Geosciencexxx(2018)xxx–xxx 4

heterogeneous stress distributionin thesample.At low compression, the initial powder still has a significant porosity,asindicatedbyalowlighttransmissionunderthe microscope. At this stage, dense connected bridges of particlesformbetweenthetwodiamondanvilsandwill sustain high stresses while ‘‘protecting’’ imperfectly compacted volumes of powder that undergo lower differentialstresses,awell-knowneffectinthecompaction ofdisordered granularmedia(Guyonetal.,1990).With increasing pressure and compaction, the whole sample becomes well-compacted, more transparent to visible light,andthestressdistributionishomogeneous(Fig.2). This observation argues for stress built-updue tolocal straininthecompactingpowderratherthanapplicationof amacroscopichomogeneousdeviatoricstressfieldinthe pressuremedium.

4. Discussion

Stressesmeasuredbyquartzpiezometryinthe differ-ent pressure mediaincrease withpressure and reach a limitatpressuresabove5–10GPainMgOandNaCl(Fig.5). InKClandKBr,stressesincreasewithincreasingpressure aftertheB1–B2transitionaround0.3GPa,anditisdifficult

toconcludewhetherornotitreachesaplateaubecauseof the amorphization of quartz above 20GPa. From the analysis of stress distribution, the highest measured stresses are inferred to represent the actual deviatoric stresstwhosemaximumvaluegivesanestimateofthe yield strength of the material composing the pressure medium.

Thepresentresultsareconsistentwithyieldstrength estimatesfromradialX-raydiffraction,andtheyshowthat thepressuregradientmethodcannotbesimplyusedto estimate stress in the DAC. For MgO, we infer a yield strength of 61GPa, slightly lower than the values of 81GPadeterminedbyradialandstandardX-ray diffrac-tion(Merkeletal., 2002;Singh etal., 2004), inlinewith t5.5GPa measured at5GPa in large-volume press experiments (Lin et al., 2017), and much higher than estimated from pressure gradient measurements (Meade andJeanloz,1988a).Thepresentmeasurementsallowtesting some oftheassumptionsunderlying theinterpretation of previous methods. Quartz piezometry permits to map radially(i.e.intheplaneperpendiculartothecompression axis)boththepressureandthedeviatoricstressdistribution in the samplechamber. The vertical distribution is more difficulttoobtainsinceitreliesonthedepthoffieldofthe

Fig.2.TheoreticalTO–LOsplittingdistributioninrandomlyorientedquartzgrains(upperleft)andexperimentallymeasureddistributionsonquartzgrains inMgOatseveralvaluesofthepressure.Topleft:normalizedtheoreticaldistributionofTO–LOsplittingorstressesforarandomlyorientedgrain population.Grainswithc-axisatangleshigherthan60,65and708ofthecompressionaxisareshownwithlightgray,darkgray,andblackbars,respectively. Bottom:TO–LOsplittingdistributionsatpressuresabove10GPaareverynarrow,indicatingmostlygrainswithc-axisatahighangleofthecompression axis,contributetothesignal,andtheapplieddifferentialstressfieldishomogeneous.Atlowerpressureandcompression(upperpanel),theTO–LOsplitting distributionandstressfieldareheterogeneous,butthemaximumstressesarealreadyclosetothoseatthehighestpressureandcompressionvalues.

Fig.3.Left,low-pressurelow-compactionstageoftheDACexperiment(4–6GPa).Right,high-pressurehigh-compactionstage(8–17GPa).A.Cross sectionschemeoftheDACwithdiamondsinblue,gasketinorange,andsamplepowder.B.Topviewmicrograph,diamondculetsizeis600microns. Compactionofthesampleisillustratedbyitshighertransparencyinthegasketholenearthecenterontherightthanontheleft-hand-sidemicrograph. C.Pressuredistributionimage(600600microns,step20microns)obtainedfromtheA1peakposition;thepressureisfairlyhomogeneousontheright andshowsastronggradientontheleft.D.TypicalRamanspectraillustratingpressureanddeviatoricstressmeasurement.

B.Reynardetal./C.R.Geosciencexxx(2018)xxx–xxx 6

confocalsystem,whichisabout10micronsandonlyslightly lowerthansamplethicknessof20–30micronsatthehighest pressures. Within our resolution, no measurable vertical gradientswereobserved.

TheradialpressuregradientsinMgOmeasuredhereare consistent with those measured by Meade and Jeanloz (MeadeandJeanloz,1988a),butinspiteofthesesignificant pressure gradients (e.g., from 14GPa at the rim of the samplechamberto20GPaatthecenterinMgO,Fig.2),we findthatthedeviatoricstressesfromquartzpiezometry areremarkablyconstantacrossthewholechamber,except in thefirststepsofcompression.Withvalues oftupto 61GPaatpressuresaslowas5GPa,tisnotsmallwith respectto

s

P.ThisisonefundamentalassumptionofMeade

andJeanloz(MeadeandJeanloz,1988a)intheiruseofthe pressure gradient method that is not verified. At this pressure,the radialstressheterogeneity isalsovery high, butitdecreasesrapidlyathighercompressionvaluestoreach homogeneous values through the pressure gradient (Figs. 2and4).

Ramanpiezometry showsthatthestressdistribution becomesrapidlyuniformafterinitialcompression. Stres-sesarehighlyvariable(2to7GPa)atP6GPa,whereas theyarehomogeneous(5–6GPa)atP>10GPa.Radial X-raydiffractionexperimentsarethusperformed through-out a homogeneous deviatoric stress field within a pressure gradient. This accounts for the agreement between the present measurement of deviatoric stress and those obtained by several studies(Lin et al.,2017; Merkeletal.,2002;Singhetal.,2004).Minordiscrepancies of20%betweenstudiesaremarginallywithin uncertain-ties,andmayarisefromtheoverestimationoftbyMerkel etal.(2002)becausetheirmeasurementismadethrough theradialpressuregradient,orbecausethequartzyield strength is reached before that of MgO in the present experiments.Tocheckthelatterpossibility,anexperiment wasperformedwithquartzonly(Fig.4)anditwasverified that the maximum deviatoric stress is similar to that measuredonquartzinMgO.Thus,thepresent measure-mentislikely thatoftheyieldstrengthof quartzanda

Fig.4.ObservedTO–LOfrequencies,upwardanddownwardpointingtriangles,respectively,invariouspressuremediacomparedwiththeoretical predictions.ThesolidblacksymbolsareDFTfrequenciesforhydrostaticcompression(solidcircles,negligibleTO–LOsplitting)anddeformationsof2and5% alongthea-axis(solidtriangles).TheemptyblacksymbolsarehydrostaticcompressioninMEWto11GPa.Anon-hydrostaticstressinMEWabove11GPais observedasmoderateTO–LOsplitting.ThepeakpositionisestimatedfromthemeanoftheTO–LOfrequenciesabove11GPainMEWforthesakeof comparisonwithDFT.Thesolidlinesforhydrostaticcompressionandthedashedlinesfornon-hydrostaticcompressionareguidestotheeye.Blueandred symbolsareforcompressioninNaClandMgO,respectively.TO–LOsplittingissymmetricalwithrespecttothehydrostaticfrequencyupto2%deformation andtheLOcomponentismoreaffectedthantheTOcomponentathigherstrainorstress,bothinDFTandexperiments.

minimumestimatefortheyieldstrengthofMgO.Thisis alsoconfirmedbytheaxialdiffractionmeasurementsof

Singhetal.(2004),inwhichonlythecentralpartofthe sample,wherelittlepressuregradientsarepresent,was studied.OtherRamanprobeswillhavetobecalibratedin ordertoestimatestressesofhighyieldstrengthmaterials likeoxides,whosestrengthsarelargerthanthatofquartz. TakingmaximumRamanmeasurementsofstressasthe bestestimateofyieldstrengthgivesvaluesofupto7GPa inMgO,3GPainNaCl,and4.5GPainthehigh-pressure formsofKClandKBr.Thelargervalueofyieldstrengthin theoxidethaninthehalogenidesisqualitatively consis-tentwithsystematicabinitioestimatesofyieldstressfor easyglidesystems(Ogataetal.,2004)giving17GPafor MgOand 3.7GPa inNaCl,and largerthanexperimental valuesaround1GPaforNaCl(Xiongetal.,2014).Theyield strengthinferredforKBrintheB2structureishigherthan boththetheoretical(Ogataetal.,2004)andexperimental (Filleteretal.,2006)determinationsof2.5–2.6GPainthe B1structure.Thissuggeststhat yieldstrength increases withincreasing coordination, as already seen by shock measurementsonKClaroundtheB1–B2transition(Millett etal.,2002),anideathatrequiresfurtherexperimentaland theoreticalverification.Thisisconsistentwiththegeneral trend of increasingthe mechanical strength of silicates with depth in the Earth’s mantle, especially at the transitions from olivine–spinel structures of the upper mantletoperovskite-basedstructuresinthelowermantle (Kraychetal.,2016;Merkeletal.,2007;Ricardetal.,1993).

Onemajorassumptionofthisstudy,though,isthefact thatthestressesintheembeddedquartzgrainsaresimilar tothoseofthecontainingmatrix.Thisassumptionisvalid under purely hydrostatic conditions or with elastically anisotropic materials. When solid samples are loaded, however,stressheterogeneitiesdevelopatthelocalscale, eveniftheoverallsampleisunderhydrostaticconditions. Locally,thestressesactuallyappliedonthestressprobe grainsmaydifferfromthemacroscopicdifferentialstress, aneffectthatmayalsodependon thegrainsize.Inthe presentstudy,thestressesmeasuredinthequartzgrains showabroaddistributionateachpressure.Wefindthat themaximumstressvaluesinquartztendtoagreewith the overall strength of the sample. In thefuture, these assumptionsshouldbemodelled,usingmesoscalemodels suchascrystal-plasticitybasedfiniteelementsimulations (Amodeoetal.,2016).Thisproject,however,goesbeyond thegoalsofthepresentstudy.Thisissueiscriticalforthe calibrationof pressureand stressinextreme conditions experimentsasitisalwaysassumedthatthesamplestress statecanbeevaluatedwithastressmarkerembeddedin (orcloseto)thesample.

Acknowledgements

ThisworkwassupportedbyINSUthroughthenational Ramanfacility inLyon, bygrants fromthe‘‘Programme nationaldeplane´tologie’’andfromthe‘‘ENSdeLyon’’(to

Fig.5.Measuredstressesin(a)MgO,quartz,and(b)salts.Thethicklong-dashedlinesindicateestimatesofthemaximumyieldstrengthofthematerial. MgO,presentmeasurementsinred,radialX-raydiffractiondata(Merkeletal.,2002)inorange(a).Mechanicaldataforquartzaggregatesareshownaslight purplesquares(HirthandTullis,1994);presentmeasurementsonpurequartzpowderassmallpurplecircles.Theamorphizationregionisshownasalight purpleshadedarea.Themeasurementsonquartzpowderalone(purplepoints)suggestthattheyieldstrengthlimitofthequartzisreachedat61GPa. StressmeasurementsinMgOwithquartzpiezometryarethuslikelyalowerlimitforMgOyieldstrength.Salts,theverticaldashedlinesindicateB1–B2transition pressures(Flo´rezetal.,2002)(b).ThemeasurementsonNaClwereperformedintheB1structurestabilityfield,whiletheyaremostlyintheB2structureforK salts.YieldstrengthestimatesfrompressuregradientmeasurementsinMgOandNaCl(MeadeandJeanloz,1988a,1988b)areshownasthickshort-dashedlines. ThedeviatoricstressfromX-raydiffractioninKBrupto5.6GPa(ZhaoandRoss,2015)isshownasasolidgreenline.

B.Reynardetal./C.R.Geosciencexxx(2018)xxx–xxx 8

BR). Itis a contributionof the LABEX LyonInstitute of Origins(ANR-10-LABX-0066),withintheprogram ‘‘Inves-tissementsd’avenir’’ (ANR-11-IDEX-0007) at‘‘Universite´ deLyon’’.

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