Dislocations and dislocation creep...

1. Introduction: a) Observations of textures associated with dislocation creep b) MOVIES of dislocations in metals.

2. How do dislocations move?: single dislocations: stress, energy and motion. Orowan’s equation. Frank-Read sources, Work Hardening (Fatigue, paper clips)

3. Recovery: dislocation climb, subgrain formation

4. Recrystallization: Fabric development and grain size evolution Grain rotation (LPO development) Reduction: Mechanisms of recrystallization Growth: forces driving grain growth... ----> Localization REVISIT THE QUARTZ regimes

5. Grain Growth: static and dynamic ? Thursday: 6. Piezometers and Wattmeters Empirical Piezometers (dislocations, subgrain and grain size). Stress in the lithosphere from xenoliths (Mercier) Stress across shear zones (Kohlstedt and Weathers) The WATTMETER 1. Intro: Observations

First, ALL the complexity that we want to understand !

THEN, deconstruction: simple - - - - > complex small scale (single dislocation) ----- > larger scale (behavior of ensembles) Journal of Structural Geology, Vol. 14, No. 2, pp. 145 to 159, 1992 0191-8141/92 $05.00+0.00 Printed in Great Britain (~ 1992 Pergamon Press plc

Dislocation creep regimes in quartz aggregates Journal of Structural Geology, Vol. 14, No. 2, pp. 145 to 159, 1992 Dislocation creep regimes in quartz aggregate0s 191-8141/92 $05.00+0.00 147 Printed in Great GBritainREG HIRrH and JAN TULLIS (~ 1992 Pergamon Press plc 1400 Departmreengimest of Geolo gdetericalS cieminednces, Brow bn Uyn relativiversity,P reo vidence, RI 02912, U.S.A. rates of (Received 11 September 1990; accepted in revised form 30 June 1991) 1. dislocation productionDislocation creep regimes in quartz aggregates Abstract--Using opti2.ca l dislocationand TEM microsc oclimbpy we have determined that threei regimes of dislocation creep ocRcur. ,ml j : in experimentally deformed quartz aggregates, depending on the relative rIa- tes of grain boundary migration, / J Regime 1 dislocation climb and3. d igsloraincation boundarproduction. Wyit hmigin eacrationh regime a(GBM) distinctivGe mRiEcGro sHtrIuRctruHre aisn pdr oJdAuNce Td UduLeL IS primarily to the operation of different mechanisms of dynamic recrystallization. At lower temperatures and faster strain rates the rate of dislocation production iDs teopoa grrtemate nfotr of di Gffuesoiloong-iccoanltS rocliel1en0d c -de8i ss,l Bocrao1t;wi o"n7n Uclnimiv1b;e tr-o6s ibtey ,aP n1r ;o- sv idenc1 e,. R4 I 0291.28 , U.S1.A; - 7. 1; .8 1 ; - s 1, -4 effective recovery mtheseechanism rates. In this are regim ae functionrecovery is ac ofcom mstressodated b, y strain-induced Sgtrrain Rbaotue n(dl/as)r y Strain rate (l/s) migration recrystallization. With an increase in temperature(Received or decrea s1e1F iSeptembergn. s2t. rPailont sr ao tf1e temperature9, 9th0e; accepted rate vosf straind inis lrevisedo ratecati oshowingn form the 30 location June 1 o9f9 the1) dislocation creep regimes for quartz aggregates climb becomes suffictemperatureiently rapid to accom andmodat ew reatercovery content. In this regimdeformede dynam (a)ic 'as-is'recry andsta (b)lliz withatio 0.17n oc wctu%r swater by added. The boundaries between the regimes are gradational. Open circles progressive subgrain rotation. With a further increase in temperature or derepresentcrease iregimen stra i1n, plus rate symbols dislocrepresent ation cl iregimemb 2, open squares represent regime 3, and a plus inside a square represents remains sufficiently rapid to accommodate recovery. However, in this regime grain boundary migration is rapgradationalid, between regimes 2 and 3. thus recrystallization occurs by both grAaibns btroaucntd--aUrys imngig orapttiiocnal aanndd p TroEgMre smsivcero ssucbogprya iwne r ohtaavteio dne. tTerhme iindeedn ttihfia-t three regimes of dislocation creep occur cation of the three regimes of dislocatiionn ecxrepeepri maeyn thaallvye dimefpoormrtdaeendpt e qinmudapsr ltiozcn aa tigtohgners e dfgoearvt eetshl,oe p ddmeepetenrntm doinf gaat iosontne atohdfey f lrosetwlaa ttei vme ircarote-s of grain boundary migration, 1000 law parameters and the calibration of rdeicsrloysctaatliloizne cdl igmrabi na nsdiz ed ipsisleotzrcuoacmttiueortne .rp sr.So iIdnnuc caet ditdohinet.i oaWnx,i iathl eisn hi doeeartncethni firinecgga tiimgoeneo omaf e dati rsyti nocft itvhee microstructure is produced due particular dislocation creep regime coulpdr ibme aursielyfu tl oi nt hel poipnegr taot icooenxn psotefrr aidimnif etfnheterse c noltin mmdiitetsic ohtnhasen aitsa mmwhso iuocnhft ad oygfni vasemtnrai ncina r teuacrnardyl stthaullsi ztahtieo n. At lower tempa eratu0r0e0s a0n d faster strain rates the rate of dislocation production is too great for diffusion-controlled dislocaotioo. n clim¢bP to be an o W-377 (quartzite) deformation has occurred. amount of recrystallization possible, we also describe effective recovery mechanism. In this regime recovery is accommodated by strain-induced grain boundary microstructures from samples of the fine-grained nova- 600- migration recrystallization. With an increase in temperature or decrease in strain rate, the rate .oOof di s location o 0 climb becomes sufficientlcyu lriatep itdo tioll uasctcraotme mthoed partoec erescseosv ewrhyi.c hIn w teh ibse rleiegviem we oduylnda mic recrystallization occurs by occur in quartzite at higher strains. 400- INTRODUCTION progressive subgramine rtortyat ioofn .c-axis With ap fruerftheerrr iendc roearsie nint atetmiopnesra (tuSrceh omr didec &re aCsea isne sytr ain rate di•s location climCbQ -83 (novaculite) remains sufficientl1986)y rapid aton adc pcolamnmaord faateb reicosv deeryf.i nHeodw bevye r,e icnr tyhsista rellgiizmeed g qrauianr btozu ndary2 m00i.g Or ation is rapid, gime 1 thus recrystallization occuMicrostructuralrs by both gra ionb bseoruvnadtiaornys manigdr ainterpretationstion and prog ressive subgrain rotation. The identifi- EXPERIMENTAL studies on the de eformation of geologic grains in S - C mylonites (Lister & Snoke 1984) can be i . cation of the three regimes of dislocation creep may have important implications for the0 deter0m4 ination of flow R l I I I I law parameters anuds tehde tcoa ldibertaetiromn ionf ere tchryes tkaillnizeemd agtriacins osifz ed puicetziolme estheresa. rI nz oanddeist.i on, the identification of a materials have generally concentrated on either the Regime 1. At lower temperatures and faster strain 0 10 20 30 40 50 60 particular dislocation creep regime could be useful in helping to constrain the conditions at which a given natural determination of flow laws or the characterization of In addraitteiso nm,i cdriossltoruccatutriaoln ocbrseeervpa timonisc rsousgtgreustc ttuhraet sd i(sslou-c h as deformation has occurred. 1000. microstructures produced by different deformation recrysctaatliolinz ecldim gbr aisi nd isize)fficult haandv eth bate erenc ocvaelriyb irsa atcecdo mfomro u- se as b mechanisms. Rheological data in the form of a flow law paleopdaietezdo mbye tgerarisn b(oeu.ngd. aryTwiss migra tio1n9 7re7cr,y sKtalolihzaltsitoend. t & 800. relating strain rate, temperature and stress (and possibly WeathQeurasrt z1i9te8 0sa, mOprldes &de Cfohrmriesdt iein 1984).this re gime show the highest strengths, and they are the only ones to undergo | . . INTRODUDCeTsIpOitNe the large amount ofm pertervyi ouf sc-axis rese aprrcehf eornre d orientations (Schmid & Casey other thermodynamic variables) can be used to predict appreciable strain weakening (Fig. 3a). However, nova- CQ- 94 (novaculite) ,o0 the mechanical behavior of rocks at gR1eol o(highgic co nstressditions, loqwua T):rtzc,u l tiproductionthee sraem hplaess bdefeonrm nfast,eod cino rm erecogpimr1986)eeh e1v nserhs aoivwnye d l bi etptxyllepa enclimbstrariarmi nf eanbtr aiandcl s de fGBMined b yslo recwrystallized quartz grains in S - C mylonites (Lister & Snoke 1984) can be provided that extrapolation inE XtePmERpIeMraEtNuTreA La nstdu dsiterasi no n tshtued dye wtfeooa rkimelnluianstgit or(aFnti ego. tf3h ag)e.e porlogceics ses associated with dislo- e + materials have generally concenMtriactreosdtr uocntu reesi thfreomr tqhuea rtzitues esdam tpol eds estheorrmteinnede the kinematics of ductile sheawr- 3z3o9 (nqueasrtz.i te) rate can be proven valid. Flow laws have been used to cation creep over the entire range of conditions where it D determination of flow laws or t2h0e% cshhoawr athcatte driiszloactaitoionn colifm b iIs nno ta adbdlei ttioo anc,c odmimsloo-c ation creep0 microstructures (such as provide constraints for the modelling of a wide range of is thed adteo mrecinovaenryt idne rfeogirmmea 1.ti oTnE Mm oebcsheravnaitsiomns. fDroims ltohce ation o . geologic processes including fomldicinrgo s(ter.ugc. tuPraersri sphr eotd aulc. ed crbeyep cdoiirnfefsc elourfed onertsi g idnaa el fsgotrrraaminisna tsihpoorwno dhiugrchei cndregny ssimtiaeelslc iohzfae tndani gsgmrlead i n(dis- size) have been calibrated for use as 1976), continental rifting (e.g.m Zecuhbaenr isemt asl.. R1h9e8o6l)o agnicda l dlaotcaa tiindo itnshl ogecl aiftdiooern mso r (o,u pfi nat o sf lo-o1m0w1e 6l acmwa -s2 e)s ,wp hcaillciehm ocbpo)im eamznodnm lyae threeavrcseo ve(rey.g .| Twissc 1977, Kohlstedt & mantle convection (e.g. Christreenlsaetnin &g sYtruaeinn r1a9te8,9 t)e. mByp eramtuerceh astnnraidisg mshttr se(edsgsims (elaontncsad at npidoo nssh sociwbli lnmyo b e voiWdre negcaert ahfioenr r tshbe o1 du9env8e0dlo,a pOr-y r dm &i- Christie 1984). ment of subgrain boundaries (Fig. D4ae).s pAitt eth et hoept iclalr ge amou:n1t o f previous research on comparing microstructures preostehrevre tdh einrm noadtuyrnaallmyi cd ev-a riagbrlaetsio)s nca)al.en I tnbh eeas deu dgsiretaidion snt o,e xaph nirbeuidtm iircbrte gru olafr daynnd apmaticch yr eucnrdyuslata- lliza- formed rocks to those producethde d mureicnhga nlaibcaolr abteohrayv eiox-r of trioocnk smt oaertyc gheexaotninliocstgmioinsc c(cFoaingn.d 4iabtci).oc nTohsme, phaignqhyu d aedrnteszift,oi ertshm oeafr tetia onhgnal esid n b etehne no comprehensive experimental periments it is possible to inpvreosvtidgeatde tthhaet epxrotrcaepsoselas tiodni silno ctdeaistmlioopcnae tcrioarnetsue praen r de agtnhidem easb t(rsean.igcne. Gofsu tsiuullbdogypra etin o& bi olPluounisdrtairreaierts e 1979, the proc4e0s0s- es associated with dislo- CQ-84 (novaculite) which control the mechanical breahtea vciaonr obfe t hper ocvruesnt .v Faolird . FDloruwr ylsa uwg&gs e sUht arthavaiet b1w9eite9hin0n )u.th sUise sdrie ngtgoim eo ptchtaiec tariaolt nea confrd ed eitsprloa ocnavstiemorni tshsieo enn tii re rang=,e= :o f condi.t.i.o ns where it production is greater than the rate of recovery by dislo- 2 0 0 - ° B ° B o B o o [] example, microstructural obseprvroatviiodnes choanvset rbaeinetns ufosre dth e emleocdterollnin mg oicfr oa swcoidpey r(aTnEgMe o) f weis h athve ddeotmerimnainet dd ethfoatr mation mechanism. Dislocation cation climb. J C0-82 (quartzite) a 0~ , , , , , to infer mechanisms responsibglee ofloorg tihce p lrooccaelsiszeast ioinnc olufd intgh rfeoel driTnehggei m(oene.sgse .t oPff a strdrraiiissnlh ow ceatk taieoln.ni n gc scrheroewepnp b oyci tnchcue lrqu udfaoertrsz itqea u asrttrza in producing mechanism (dis- 0 10 20 30 40 50 60 strain (Aydin & Johnson 1983, 1976),Segall &co Snitminpesnotna l1986). riftin g (aeg.g.r eZagta u-tb2e0es%,r dester paaieln.n (dF1i9gn.8 g36 a)o ncao nrtrdhe sep orlnoedlcsa attoti ivtoheen prgoalitinedts ew hooefrr ,e d iins loso- me cases, climb)A xaianl dS traa inr e(%c)o very recrystallization initiates at the grain boundaries. At the Such comparisons can also provmidaen ttlhee ceovnidveencctieo nne (cee.sgs.- Chcraistitoenn sperno d&u cYtiuoenn, d1i9sl8o9c)a.t Bioyn climmebc ahnadn igsrmai n( bdoisuloncdaaFtriygo. n3. Differentialclimb o rs tregsrs avis na xibalo sutranind caurryve sm fori -quartzite and ary to ensure valid extrapolation of flow laws to geologic migraotpitoicna l (sHcailret thh ee rte carly.s ta1l9li8ze9d) .g rTaihnse area tteoos somfa ltl htoe sbee prnovaculiteo- samples deformed in (a) regime 1, (b) regime 2 and (c) comparing microstructures presreesrovlevedd , inan dn athteu graralilny b oduen-d ariegsr aatpipoenar) .d Iifnfu ased d(sietei onregime, a nu 3m. Curvesber o aref donylny shownamic out re tco r4y5%st ashortening.lliza- (a) W-377, conditions (Paterson 1987). formed rocks to those prodceuscseedsa rdroeuwpr ieinng d F lioga.nb 4 obtrh).a et TotEerMym epoxeb-rs aertvutariteoio,nn s mtrsaheoicnwh ratanhtaiets mtahnse dc tahopenne a circles:ccom Heavitreepany dquartziteeform (7a00t°iCo, n10 i6n s- l,t h1.e5 GPa, 'as-is'). CQ-83, filled circles: novaculite (700°C, 10 5 s-l, 1.5 GPa, 0.17 wt% A great deal of experimentapl earnidm oebnstesr viat tiiso npaol swsoibrkle toam ionuvmneetsc tohifga nwaitsaemt e torhf p erre ecpsryresontactl eldiszusaretison ng itnhd etihs idlso ercfeoagritmimoena it sci osrtenrae.i pWn- riethgwateriinm e added).(e.g. G(b)u iWll-o33p9e, p&lu sePso: Heavitreeirier 1979, quar tzite (800°C, has been performed on quartz and it has been estab- each riendguicmede agr daiins tbionucntdivarey m miicgrroatsiotDrnu r(cFutiurgy.r e 4 c&is). pWUroer dainui tce1re-9d 9 d0u1)0e .-6 US- 1s, i1n.5g GPa, op t'as-is').ical CQ-94,and tfilledran scircles:miss novaculiteion (700°C, which control the mechanical bephraetv itoher omfic trhoset rcurcutusrte. sFhoowr n in Fig. 4(c) to indicate 10- 6 s- l, 1.5 GPa, 0.17 wt% water added). (c) CQ-82, open squares: lished that there are close seimxailmarpilteie,s m biectrwoseternu cdis-tura l opbrsiemravraitliyo ntos thhaev eo pbeereanti ouns eodf dieflfeecrternotn m mecichraonsicsompsy Blacko (fT E HMills) quartzitewe h a(9v0e0 °Cd, e10-6sterm 1,i n1.5GPa,ed th a0.17wt%t water that the grains with a low dislocation density (marked added). CQ-84, filled squares: novaculite (900°C, 10 -6 S- l , 1.5 GPa, location creep microstructurest op riondfuecr emd eicnh tahnei slmabso rreas- pondsyinbalemw ifitcoh rra estctherreyi sskltosa)c lawlilezirazeta imtoiiongr.na toinfg intoh rtheee wroerkg-ihmaredse neodf dislocation cree0.p17 wt%occ waterur fadded).or q uartz tory and those observed in nsattruarianl l(yA dyedfionr m& eJdo hroncskosn 1983,In S tehgiasl lp &ap Seirm wpes ofnir 1986).st illus traatge gtrheeg amteisc,r odsetrpuecntdurinesg on the relative rates of dislo- (Tullis et al. 1973, White 1976). SDuuceh tcoo tmhep aarbisuonndsa ncacne oafls o cphroarvaicdtee rtihseti ec voifd tehnec eth nreecee rsesg- imcesa toiof nd ipsrloodcautciotino nc,r edeipsl, ocation climb and grain boundary and illustrate the relationship between microstructural quartz in the crust, these disloacrayt itoon e cnrseuerpe mvailcirdo estxrturca-p olation of flow laws to geologic migration (Hirth et al. 1989). The rates of these pro- tures have proven to be useful for solving a wide range of evolution and mechanical behavior in each regime. We conditions (Paterson 1987). cesses depend on the temperature, strain rate and the problems in structural geology. For example, asym- then discuss the processes that control which regime is A great deal of experimental and observational work amount of water present during the deformation. Within has been performed on1 4q5u artz and it has been estab- each regime a distinctive microstructure is produced due SG 1 4 : 2 - B lished that there are close similarities between dis- primarily to the operation of different mechanisms of location creep microstructures produced in the labora- dynamic recrystallization. tory and those observed in naturally deformed rocks In this paper we first illustrate the microstructures (Tullis et al. 1973, White 1976). Due to the abundance of characteristic of the three regimes of dislocation creep, quartz in the crust, these dislocation creep microstruc- and illustrate the relationship between microstructural tures have proven to be useful for solving a wide range of evolution and mechanical behavior in each regime. We problems in structural geology. For example, asym- then discuss the processes that control which regime is 145 SG 1 4 : 2 - B Experimentally deformed: gime 2 e R

Dislocation creep regimes in quartz aggregates 147

1400

R2 (lower stress, higher T): recovery by climb moderate, but i R.,ml GBM sloj w, so: recrystallization I- occur/ J s byR subgegime 1 rain rotation (requiring climb) 10 -8 1; "7 1; -6 1 ; - s 1 .4 .8 1 ; - 7 1; .8 1 ; - s 1, -4 Strain Rate (l/s) Strain rate (l/s) Fig. 2. Plots of temperature vs strain rate showing the location of the dislocation creep regimes for quartz aggregates deformed (a) 'as-is' and (b) with 0.17 wt% water added. The boundaries between the regimes are gradational. Open circles represent regime 1, plus symbolsrepresent regime 2, open squares represent regime 3, and a plus inside a square represents gradational between regimes 2 and 3. depends on the development of a steady state micro- 1000 structure. Since the axial shortening geometry of the experiments limits the amount of strain and thus the a 0 0 0 0 aoo. ¢P o W-377 (quartzite) amount of recrystallization possible, we also describe microstructures from samples of the fine-grained nova- 600- culite to illustrate the processes which we believe would .oO o 0 occur in quartzite at higher strains. 400- • CQ-83 (novaculite) 200. O Microstructural observations and interpretations i . 0 04 l I I I I Regime 1. At lower temperatures and faster strain 0 10 20 30 40 50 60 rates microstructural observations suggest that dislo- 1000. cation climb is difficult and that recovery is accommo- b dated by grain boundary migration recrystallization. 800. Quartzite samples deformed in this regime show the highest strengths, and they are the only ones to undergo | . . appreciable strain weakening (Fig. 3a). However, nova- CQ- 94 (novaculite) ,o0 culite samples deformed in regime 1 show little strain weakening (Fig. 3a). e + Microstructures from quartzite samples shortened w-339 (quartzite) D 20% show that dislocation climb is not able to accommo- 0 date recovery in regime 1. T E M observations from the o . cores of original grains show high densities of tangled dislocations (up to -1016 m -2) which commonly have straight segments and show no evidence for the develop- | c ment of subgrain boundaries (Fig. 4a). At the optical :1 scale these grains exhibit irregular and patchy undula- tory extinction (Fig. 4b). The high densities of tangled dislocations and the absence of subgrain boundaries 400- suggest that within this regime the rate of dislocation i =,=: . . . CQ-84 (novaculite) production is greater than the rate of recovery by dislo- 2 0 0 - ° B ° B o B o o [] cation climb. J C0-82 (quartzite) a 0~ , , , , , The onset of strain weakening shown by the quartzite 0 10 20 30 40 50 60 at - 2 0 % strain (Fig. 3a) corresponds to the point where Axial Strain (%) recrystallization initiates at the grain boundaries. At the Fig. 3. Differential stress vs axial strain curves for quartzite and optical scale the recrystallized grains are too small to be novaculite samples deformed in (a) regime 1, (b) regime 2 and (c) resolved, and the grain boundaries appear diffuse (see regime 3. Curves are only shown out to 45% shortening. (a) W-377, arrow in Fig. 4b). TEM observations show that the open circles: Heavitree quartzite (700°C, 10 6 s-l, 1.5 GPa, 'as-is'). CQ-83, filled circles: novaculite (700°C, 10 5 s-l, 1.5 GPa, 0.17 wt% mechanism of recrystallization in this regime is strain- water added). (b) W-339, pluses: Heavitree quartzite (800°C, induced grain boundary migration (Fig. 4c). We inter- 10 -6 S- 1, 1.5 GPa, 'as-is'). CQ-94, filled circles: novaculite (700°C, pret the microstructure shown in Fig. 4(c) to indicate 10- 6 s- l, 1.5 GPa, 0.17 wt% water added). (c) CQ-82, open squares: Black Hills quartzite (900°C, 10-6s 1, 1.5GPa, 0.17wt% water that the grains with a low dislocation density (marked added). CQ-84, filled squares: novaculite (900°C, 10 -6 S- l , 1.5 GPa, with asterisks) were migrating into the work-hardened 0.17 wt% water added). Experimentally deformed: gime 3 e R

Dislocation creep regimes in quartz aggregates 147

1400 R3 (low stress, high T): recovery by climb fast, but Grain Boundary Migration i R.,ml j : is also fast, so complete I- / J Regime 1recr ystallization occurs by both subgrain rotation and 10 -8 1; "7 1; -6 1 ; - s 1 .4 .8 1 ; - 7 1; .8 1 ; - s GBM.1, -4 Strain Rate (l/s) Strain rate (l/s) Fig. 2. Plots of temperature vs strain rate showing the location of the dislocation creep regimes for quartz aggregates deformed (a) 'as-is' and (b) with 0.17 wt% water added. The boundaries between the regimes are gradational. Open circles represent regime 1, plus symbolsrepresent regime 2, open squares represent regime 3, and a plus inside a square represents gradational between regimes 2 and 3. depends on the development of a steady state micro- 1000 structure. Since the axial shortening geometry of the experiments limits the amount of strain and thus the a 0 0 0 0 aoo. ¢P o W-377 (quartzite) amount of recrystallization possible, we also describe microstructures from samples of the fine-grained nova- 600- culite to illustrate the processes which we believe would .oO o 0 occur in quartzite at higher strains. 400- • CQ-83 (novaculite) 200. O Microstructural observations and interpretations i . 0 04 l I I I I Regime 1. At lower temperatures and faster strain 0 10 20 30 40 50 60 rates microstructural observations suggest that dislo- 1000. cation climb is difficult and that recovery is accommo- b dated by grain boundary migration recrystallization. 800. Quartzite samples deformed in this regime show the highest strengths, and they are the only ones to undergo | . . appreciable strain weakening (Fig. 3a). However, nova- CQ- 94 (novaculite) ,o0 culite samples deformed in regime 1 show little strain weakening (Fig. 3a). e + Microstructures from quartzite samples shortened w-339 (quartzite) D 20% show that dislocation climb is not able to accommo- 0 date recovery in regime 1. T E M observations from the o . cores of original grains show high densities of tangled dislocations (up to -1016 m -2) which commonly have straight segments and show no evidence for the develop- | c ment of subgrain boundaries (Fig. 4a). At the optical :1 scale these grains exhibit irregular and patchy undula- tory extinction (Fig. 4b). The high densities of tangled dislocations and the absence of subgrain boundaries 400- suggest that within this regime the rate of dislocation i =,=: . . . CQ-84 (novaculite) production is greater than the rate of recovery by dislo- 2 0 0 - ° B ° B o B o o [] cation climb. J C0-82 (quartzite) a 0~ , , , , , The onset of strain weakening shown by the quartzite 0 10 20 30 40 50 60 at - 2 0 % strain (Fig. 3a) corresponds to the point where Axial Strain (%) recrystallization initiates at the grain boundaries. At the Fig. 3. Differential stress vs axial strain curves for quartzite and optical scale the recrystallized grains are too small to be novaculite samples deformed in (a) regime 1, (b) regime 2 and (c) resolved, and the grain boundaries appear diffuse (see regime 3. Curves are only shown out to 45% shortening. (a) W-377, arrow in Fig. 4b). TEM observations show that the open circles: Heavitree quartzite (700°C, 10 6 s-l, 1.5 GPa, 'as-is'). CQ-83, filled circles: novaculite (700°C, 10 5 s-l, 1.5 GPa, 0.17 wt% mechanism of recrystallization in this regime is strain- water added). (b) W-339, pluses: Heavitree quartzite (800°C, induced grain boundary migration (Fig. 4c). We inter- 10 -6 S- 1, 1.5 GPa, 'as-is'). CQ-94, filled circles: novaculite (700°C, pret the microstructure shown in Fig. 4(c) to indicate 10- 6 s- l, 1.5 GPa, 0.17 wt% water added). (c) CQ-82, open squares: Black Hills quartzite (900°C, 10-6s 1, 1.5GPa, 0.17wt% water that the grains with a low dislocation density (marked added). CQ-84, filled squares: novaculite (900°C, 10 -6 S- l , 1.5 GPa, with asterisks) were migrating into the work-hardened 0.17 wt% water added). TECTONICS, VOL. 16, NO. 6, PAGES 983-1000, DECEMBER 1997

DUNLAP ET AL.: THERMOMECHANICAL EVOLUTION OF A DUCFH_• DUPLEX 985

• un t a 131 oc k llarts Carrar• Thermomechanical evolution of a ductile duplex i[ Figure3 Section Line TECTONICS, VOL. 16, NO. 6, PAGES 983-1000, DECEMBER 1997 Range W. J. Dunlap1 Departmentof Earth and Space Sciences, University of California,Los Angeles Group G. Hirth Thermomechanical evolution of a ductile duplex 23022 ' McLeanLaboratory, Woods Hole OceanographicInstitution, Woods Hole, Massachusetts 135ø09'+ W. J. Dunlap1 Ruby Thrust Paradise DepartmentC. Teyssierof Earth and Space Sciences, University of California,Los Angeles Nappes 533 Departmentof Geology and Geophysics, University of Minnesota,Minneapolis G. Hirth Figure 4 988 DUNLAPET AL.: THERMOMECHANICALEVOLUTION OF A DUCTILEDUPLEX McLeanAbstract.Laboratory, TheWoods thermomechanicalHole Oceanographic evolutionInstitution, ofWoods a midcrustalHole, Massachusetts both the kinematicsand dynamicsof thrustsystems. 62 While ductile duplex in central Australia has been reconstructed considerableprogress has been made toward the understandingGap Unconformit""'""y'• •:]Bitter Springs Fm QuartzDislocation C. TeyssierthroughCreep space Regimeand time using 4øAr?9Ar thermochronology, of the developmentof uppercrustal thrust systems, knowledge '[7•Heavitree Quartzite • Unconformity flow stress estimates, cross-sectional restoration of I---] Basement Departmentof Geology and Geophysics, University of Minnesota,Minneapolis of the evolutionof midcrustal(i.e., 23033 greenschist ' ._.._ facies) ductile s N ß SampleLocation dislocation creep microstructures, and microstructural and thrustsystems has come more slowly. + The134050 'reason for this is at 10 km Abstract.structural Theanalysis. thermomechanical A critical evolution aspectof of this a midcrustal analysis is theboth theleast kinematics partiallyand due dynamics to the difficultyof thrustGiles systems.Creek of usingNappe While structures in ductileidentification duplex in of central populations Australiaof has white been micas reconstructed in quartzite considerable deformedprogress rockshas tobeen measuremade toward the variationthe understanding in time of such mylonites that have neocrystallized below their closure importantparameters Figureas strain 2. Regionalrate, tectonicstress, mapand of temperature.the southeasternArunta Block showing the Ruby Gap duplex and throughspace and time using 4øAr?9Ar thermochronology, of the developmentof uppercrustaloverlying thrustnappe complexes.systems, In knowledgethis region the two basal units of the Amadeus Basin, the Heavitree Quartzite and flowtemperature stress estimates, and which cross-sectional record the restorationtime when ofductile of the However,evolutionof it midcrustalis only therecently(i.e., Bitter greenschist Springs in using formation,facies) isotopic are interleaved ductile analysis with the in basement, allowing at least five thrust sheets to be dislocationdeformation creepceased. microstructures, In dating and these microstructural micas the myloniticand thrust systemsattempts has tocome datemore indentifiedmineralslowly. (numbers Thecrystallization reasonin circles).for The this floor isthat and at roof boththrusts of the duplex are emphasizedas bold lines. Samples microstructuraland agelocations informationfor K-feldspar has diffusion been experiments synthesizedare labeledto with a samplenumber or a site name (Carrara), whereas structuralmicrostructuresanalysis. A have critical effectively aspect of been this analysisdated. is The the time-least partially due to the difficultythosefor the whiteof using micasare structures dots. The crossin section in Figure ld traversesfrom north to souththe region shown identificationtemperatureof history populationsof the duplexof whitehas micasbeen constrained in quartzitethrough deformed reconstruct rocks to the measure tectonic inthe the history uppervariation left-hand of ductilely insetin time [after Dunlap ofdeformed such et al., 1995a]. rocks mylonitesmultidomain that havethermal neocrystallizedmodeling of K-feldsparbelow their4øAr/39Ar closure data.important [Kligfieldparameters et aI.,as strain1986;rate, Brodie stress, etand al., temperature.1989; Zingg and temperatureThe modeling and demonstrates which recordthat the a temperature time whengradient ductileexisted However, Hunziker, it is only 1990; recently Dunlap inet al.,using 1991 isotopic]. analysis in ['"""'"'-"'"'':'''":..-.":.--.•.....:••••i::JtUndeformedQuartzite ' ] BasementGneisses in principle, be documented through structural, 1991; Shaw et al., 1992; Dunlap et al., 1995a]. The structure deformationacrossthe ceased. duplex In during dating its these formation. micas the The mylonitic concept attempts of Some to datecharacteristics mineralmicrostructural, of crystallization midcrustaland isotopic analysis, thrustthat as longsystems both as the rocks are asof portionsof the duplex have been documentedby Shaw et al., microstructuresmicrostructuralMICROSTRUCTURE havecontinuity effectivelyduring beenductile dated.deformation MICROSTRUCTUREThe time-has greatmicrostructural follows: and (1) agearecord informationsignificantMICROSTRUCTUREthe dynamic hasconditionsamount been of synthesizedthe of deformation deformationand to are not is[1971] and Forman, [1971], but the structure has been reconstruct the tectonicsignificantly history ofoverprinted ductilely bydeformed static recrystallization rocks remappedduring this study. temperaturepotentialhistory for elucidatingof the duplex thehas kinematicbeen constrained evolutionthrough of ductile accommodatedby (annealing). plastic flow within the thrust sheets; (2) The two basal members of the Amadeus Basin are the Late multidomainduplexes.thermal Mapping modeling of theof K-feldspardeformation4øAr/39Ar mechanismsdata. and [Kligfield dynamic et aI., microstructures 1986; Brodie are et al.,not 1989;completely Zingg destroyedand byProterozoic Heavitree Quartzite and overlying Bitter Springs The modelingdemonstrates that a temperaturegradient existed Hunziker, 1990; DunlapRuby et al., Gap 1991 Duplex]. Formation(mostly carbonates).In the duplex the quartzitehas recrystallized grain sizes of quartzites deformed under postdeformation annealing, as is common in the lower crust;remained attached to the underlying basement and is acrossthe duplex during its formation. The concept of Some characteristicsRegime3of •midcrustal thrust systems are as Regime2 • The Ruby Gap duplex of central Australia is one of several commonly detachedfrom the overlying carbonates,allowing Regime1 '• and (3) strain is partitioned through time, dependingon the greenschist facies conditions has been used to evaluate Sheet 2/1 overlap follows: (1) a significantduplexes exposed amountin a southof directed, deformationcrustal-scale megathrustis at least five superposedthrust sheets to be recognized(Figures microstructuraltectonicoffsets continuity that occurredduring ductile after deformation microstructuralhas greatfreezing. spatial distributionand dominanceof temperatureand pressure potential for elucidating the kinematic evolution of ductile accommodatedby plasticsystem flowcontaining within both the basement thrustrocks sheets; of the Arunta (2) Block 2 and 3). Parautochthonous sheet 1, the lowermost thrust This analysis shows that the duplex formed as a forward dependent rheologies.and cover sedimentsThe partitioning of the 5Amadeus km ofBasin. strain As resultsshown in insheet and the footwall of the duplex, is composedof gneissic duplexes. Mapping of the deformationSheet mechanisms 3/2 overlap and dynamic microstructuresFiguresare 2 andnot 3, thecompletely Ruby Gap duplex destroyedconsists of an by antiformal basement unconformably overlain by mildly deformed propagating thrust system accommodating...... ~60 km of preservationof fabric-formingminerals producedduring many recrystallized grain sizes of quartzites deformed under postdeformation annealing,imbricate as stackis common composed ofin boththe basement lower and crust; cover rocks Heavitree Quartzite and strongly folded Bitter Springs convergencebetween the upper and lower plates of the different portionsand of is exposedthe deformationbetween granitic basementhistory.rocks Such of the upperfabric- Formation. The thrust sheetsstructurally overlying sheet 1 greenschist facies conditions has been used to evaluate and (3) strain is partitioned through time, dependingon the megathrust,with a significant fraction of the displacement forming minerals,andwhite lower plates micas,of the for Ruby example, megathrust.The may duplex beformed datedin bycontain a prominentnorth-south oriented mineral elongation Figuretectonic 5. offsets Cross that section occurredof after the Rubymicrostructural Gap duplexfreezing. showing spatialthe distribution distributionand ofdominance the mid-Paleozoic dislocationof during temperature creep the Alice regimesSprings and Orogenypressure in[Dunlap et lineation, and the basement is variably retrogressedin the occurringafter microstructuralfreezing. Finally, using the the4øAr/39Ar method, and the results can be interpreted in terms This analysis shows that the duplex formed as a forward dependent rheologies. al.,The 1991; partitioning Dunlap and Teyssier,of strain1995], results an intracratonic in greenschistfacies, overprinting Proterozoic amphibolite and quartzite.data asOnly input the to quartzitepublishedis flow ornamented; laws for quartzthe basement aggregatesand of carbonate cooling orare crystallization, compressionalunornamented.event depending that producedThe foldpattern mainly and thrust on issystems the peakin granulites facies peak assemblages[Dunlap, 1992]. Sheet2 is propagating thrust system accommodating ~60 km of preservationof fabric-formingminerals producedduring many constrainedby about 100 thin-section analyses. Insets show schematictemperaturecharacteristic attained.the cover In sediments microstructures this way,and nappecriticalcomplexes atinformation aboutin the basement on the composed of highly deformed Heavitree Quartzite and Bitter convergenceprovides abetween strain therate upperhistory and for lower the duplex.plates of Althoughthe different portions of the[Teyssier, deformation1985; Collins history.and Teyssier, Such1989; fabric-Stewart et al., Springs Formation and overlies sheet I above the sole thrust evolution of crustalrheology can be obtainedfrom midcrustal 500megathrust, gmuncertainties scale.with Thrust are a significant clearly duplication large, fraction theof timing ofthe the regimeof displacement highest-estimated 1-3 microstructures forming minerals,probably white micas,occurred for example,after microstructural may be dated by duplexes. freezing.occurringstrain If ratesafter this ismicrostructuralduring the case, duplexthen freezing.evolution the overlap Finally,does,in indeed, theusing microstructures thecorrelate the 4øAr/39Ar can method,be removedand the as results indicatedcan be by interpreted the arrowsin terms at thedata bottomwith as input theof thehighest to diagram.publishedrates of Thrustflow convergence laws sheet for numbers quartzbetween aggregatestheare upper shown and ofas cooling numbersIn orthiswithin crystallization, study, circles. microstructures depending mainly observed on the in peaka deformed provideslower aplates strainof ratethe megathrusthistory for systemthe duplex.(according Althoughto regional temperature quartzite attained. imbricated In thisin away, midcrustal critical thrust information system onare the correlated uncertaintiescooling historyare clearly studies) large, andthe timing with coevalof highest-estimated sedimentation evolutionin with of experimentally crustalrheology produced can be obtainedmicrostructures from midcrustalto evaluate the strainadjoining rates duringmolasse duplexbasins. evolution does, indeed, correlate duplexes.dynamic conditions of deformation. The timing of In thisdeformation study, microstructuresand thermal evolution observed havein a deformedbeen constrained oblique foliationwith the defined highest byrates the of long convergence axes of betweenthe recrystallizedthe upperand is illustrated in Figure 6g. Notice the extremely flattened lowerIntroductionplates of the megathrustsystem (according to regional quartzitethrough imbricated4øAr/39Ar in a midcrustalanalysis ofthrust white systemmicas are and correlatedK-feldspars from grains is often well developed. This texture is characteristicof recrystallized grains containing subbasaldeformation lamellae coolingFor history many studies) years and geologists with coeval havesedimentation analyzedin thewith experimentallyquartziteand basement producedmicrostructures gneisses, respectively to evaluate[Dunlap, the 1997]. regime 3 where recovery is accommodated by dislocation and the bent cleavage in the micas. These features are adjoiningmicrostructuresmolasse basins. preserved in naturallydeformed rocks to studydynamic By combiningconditionsthe of microstructural, deformation. isotopic, The timingand field of data, we climb, and recrystallization occurs dominantly by grain consistentdeformationhave withdocumentedand athermal regime evolutionthe 2thermomechanical overprint have been of constrained a andcompletely kinematic Introduction through4øAr/39Ar analysis of whitemicas and K-feldspars from boundary migration.tNow at RegimeResearch School3 microstructuresof EarthSciences, are Thealso Australian recrystallized evolution regimeof a midcrustal 3 fabric.ductile In thrust some system. cases, only the quartziteand basementgneisses, respectively [Dunlap, 1997]. present in theFor Nationalquartzite manyUniversity, throughoutyears Canberra, geologists sheetsAustralian 4 haveandCapital 5 analyzedand Territory, in the the deformationlamellae are observed(i.e., no grain flattening). By combiningthe microstructural,isotopic, and field data, we microstructurespreserved in naturallydeformed rocks to study WithinDuplex sheet Structures 1, and on the detachment fault surfaces northernmostparts of sheets 1 and 2 (Figures 4 and 5). In have documented the thermomechanical and kinematic sheets 3 and 4 Copyrightthe mica1997 grainsby the American that defineGeophysical the C-foliationUnion. are between sheets 1, 2, and 3 of the duplex, cataclastictextures tNow at ResearchSchool of EarthSciences, The Australian evolutionDuplexof a midcrustal structuresductile are thrustcommon system.in compressional orogens observed toNational pinPaper theUniversity, number migration97TC00614. Canberra, of quartzAustralian grainCapital boundaries Territory, (see overprintand the characterize dislocationthe creep zoneof microstructures. interactionbetween Thethe upperbrecciaand Figure 6e). 0278-7407/97/97TIn contrast, C-00614in sheet $12.00 5, where the quartz zonesDuplexare lower spatiallyStructures plates associated of crustal-scalewith themegathrusts major fault[e.g., surfaces Boyerand and microstructureCopyrightis also1997 characteristicby the American Geophysical of regimeUnion. 3, the oblique are not as commonwithin the thrustsheets. In the quartziteof 983Duplex structuresare commonin compressionalorogens foliation is Papernot number as well97TC00614. developed, and many smaller mica sheetand1 characterize iron-oxide-the cemented zoneof interactionbreccia betweenzones crosscut the upperthe and regime grains are enclosed0278-7407/97/97Twithin C-00614 the $12.00quartz (Figure 6f), indicating 1 microstructure.lower plates of crustal-scale In the southernmegathrusts part of[e.g., the Boyer duplex, and where that the grain boundariesmigrated through and past the micas. sheet3 overridessheet 1 directly on the sole thrust(Figure 2), In severalareas around the duplex, evidencefor higher stress 983the microstructuresare crosscutby similar breccia zones. In overprinting of the lower stress (i.e., regime 3) the detachment zone between sheets 2 and 3 the occurrence of microstructuresis observed. An example of this microstructure iron-oxide-cementedbreccias is increasinglyrare toward the

Figure6. Photomicrographsof Heavitree Quartzite from the Ruby Gap duplex. (a) Typicalundeformed mica- poorquartzite. Note outlines of originalsand grains and quartz cement. (b) Quartzitedeformed in regime1, with deformationbands and fine (<10 gm) recrystallizedgrains along grain margins. (c) Regime1 quartziteat higher magnification.The fuzzy nature of thegrain boundaries is due to thefact that the recrystallized grains are much smallerthan the thicknessof the thin section.(d) Quartzitedeformed in regime2 andcharacteristic of sheet2. Note extremelyflattened original grainswith an externalmantle of recrystallizedgrains and internal developmentof subgrainsby latticerotation. (e) Entirelyrecrystallized regime 3 quartzitetypical of sheet3. Quartzgrain size is limitedby micaspinning the migrating grain boundaries. (f) Entirelyrecrystallized regime 3 quartzitetypical of sheet5. The presenceof micasin theinteriors of quartzgrains suggests that pinning of migratinggrain boundaries was not as effectiveat highertemperatures. (g) Overprintingof regime3 microstructureby higher stress microstructure of regime 2. All photomicrographshave crossed nichols; long dimensionof frameis 600 gm, exceptfor Figure6b whichis 1200gm. from: “Atlas” Experimentally deformed: Naturally deformed gime 1 e R gime 2 e R gime 3 e R from: “Atlas” MOVIES of dislocations in metals

MOVIES of grain boundary migration in deforming ice: http://virtualexplorer.com.au/special/meansvolume/contribs/wilson/introduction.html Chapter 2

What is “High temperature” ?

Questions:

1. Why do lattices form?

2. What is the relation between temperature and lattice vibrations?

3. What is elasticity (in the context of high T) and the theoretical strength?

4. What are point defects?

2.1 Why do lattices form and maintain their stability?

Lennard-Jones potential. this is a heuristic model– useful for understanding, but the shape cannot be derived from first principles... but here is how it is useful.... This is the “anharmonic” model for the interatomic potential. Figure 2.1: Lennard-Jones potential x 12 x 6 Uia = Eaa (2.1) Figure 2.1: Lennard-Jonesr −potenr tial !" # " # $ Eaa is my way of saying that the depth of the well is proportional to an activation energy (Eaa), where the a stands for approximate, but this is not the definition of the activation energy. See below for a working definition.

2.1.1 Theoretical strength

The periodic, harmonic approPximationeriodic model,for the theoreticalinteratomic strengthpoten tial(Frenckcanelb ederivmostation...simply ) described by a sin wave. 1934: Polyani, Orowan, Taylor hypothesized dislocations.... b b %!! a FigureE2.2:= Flattice.jpgx - $!! (2.2) dE F = #!! (2.3) dx Figure 2.2: lattice.jpg ! dF τ = !#!! (2.4) dA x τ = τ sin 2π -1/340/56-2785/31/-09:7; (2.5) max b !$!! x τ τ 2π !%!! (2.6) ≈ max xb ! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # τ = τ sin 25π ,-./0123(2.5) max b for very small displacements (that are the elastic strains),x sin θ θ. * τ τ 2π ≈ <7#! (2.6) ≈ max b & for very small displacements (that are the elastic strains), sin θ θ. τ = µγ ≈ $ =(2.7) shear modulus x γ (2.8) ≈ a ! τ = µγ ./43.. (2.7) xµ b τmaxγ= !$ (2.8) (2.9) ≈ 2a6π a µ τ !& (2.10) max ≈ 2π ! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # 6 ,-./0123 The slope of the shear stress ~at 10the GPzeroa,crossing way toogives strongthe shear modulus µ. The implication is that if you decrease the atomic spacing b, the elastic modulus will increase. This predicts a relationship between density and shear modulus: a denser material will have a higher shear modulus. To see this experimentally, see Karato Fig 4.11. The relationship between elastic wave velocity V µ/ρ, therefore ∝ the effect of µ on velocity is greater than that of ρ. !

HOWEVER, there is NOT a relationship between activation energy and shear modulus because the elastic strains are very small, and do not feel the height of the potential well !

2.2 What is a thermally activated process?

Arrhenius ...

Figure 2.3: LoT-hiT.jpg

2.3 What are defects, what do they do?

Takei’s thermodynamics (Gibbs), Kohlstedt’s equilibrium concentration of vacancies.

7 2. Single dislocations

G free energy G free energy energy

concentration entropy entropy

Unlike for vacancies, there is no equilibrium concentration of dislocations, because the aligned vacancies do not increase the entropy as much as isolated vacancies do. Constitutive Equations, Rheological Behavior, and Viscosity of Rocks 395

s always move in a direction perpendicular to its line direction. The displacement resulting from glide of an edge dislocation is thus parallel to the direction of e e dislocation glide, while the displacement resulting from movement of a screw dislocation is perpendi- cular to its direction of motion defined by the s velocity vector, v, as illustrated in Figure 2. Finally, Burgers vector, b a slip system is defined for a dislocation by the Line direction, combination of a unit vector normal to the slip Figure 1 Sketch of dislocation loop indicating the plane, n, and b. As an example, important slip systems direction of the Burgers vector, b, and of the dislocation in clinopyroxene include {110} 110 , {110}[001], h i line, ,, at each point along the loop. The edge and screw and (100)[001] (Bystricky and Mackwell, 2001). Slip segments of the dislocation loop are denoted by e and s, respectively. systems in olivine include (010)[100], (001)[100], (100)[001], and (010)[001] with the dominate slip direction changes from one point to the next along system determined by stress, water concentration, as the loop, while the Burgers vector is the same at well as pressure and temperature conditions (Carter every point. Regions along the dislocation loop for and Ave´ Lallemant, 1970; Jung and Karato, 2001; which b is perpendicular to , are termed edge seg- Mainprice et al., 2005). ments, while regions for which b is parallel to , are To first order, dislocations with short displace- termed screw segments. Regions of a dislocation for ment vectors are preferred over those with longer which b and , are neither perpendicular nor parallel displacement vectors since the energy per unit length are referred to as mixed segments. of a dislocation, Edisl, is proportional to the square of Movement of a dislocation can take place by glide the Burgers vector. For an edge dislocation in an and climb. An edge dislocation can move in one of elastically isotropic material (Weertman and two ways. If an edge dislocation moves in the glide Weertman, 1992, pp. 45–52) plane defined by b ,, its motion is conservative. In  Gb2 r contrast, if an edge dislocation moves normal to its E ln 28 disl ¼ 4 1 – v r ½ Š glide plane, the process is nonconservative in that ð Þ c lattice molecules are added to or taken away from where G is the shear modulus, r is the mean spacing the dislocation line by diffusion such that the dis- between dislocations, and rc is the radius of the dis- location climbs out of its glide plane. A screw location core. In eqn [28], the contribution of the core dislocation does not have a specific glide plane of the dislocation to the elastic strain energy has been since b and , are parallel to each other. Hence, a neglected. For elastically anisotropic materials, the screw dislocation generally glides on the plane that full matrix of elastic constants must be considered, as offers least resistance to its motion. If it encounters an discussed in detail by Hirth and Lothe (1968, obstacle to its movement, a screw dislocation can cross pp. 398–440). slip off of its original glide plane onto a parallel glide Slip generally occurs on the closest packed plane plane in orderat ocrconystaltinue dislocationmoving. A dislo isca taio nslippedmust surfacethat contai, ensxactlb, thya tlikis,et hthee m roupturedst widely s surfaceeparated onConstitutive a fault Equations, Rheological Behavior, and Viscosity of Rocks 395

(a) (b) (c) s always move in a direction perpendicular to its line both achieve this: direction. The displacement resulting from glide of an edge dislocation is thus parallel to the direction of v s e e e v dislocation glide, while the displacement resulting s from movement of a screw dislocation is perpendi- e b b b cular to its direction of motion defined by the s velocity vector, v, as illustrated in Figure 2. Finally, Figure 2 Sketchedgillustratinge dislocationthat (a) the movementscreof anwedge dislocationdislocation, e, and (b) the movement of a screw dislocation, sBurgers, vector, b a slip system is defined for a dislocation by the with the same Burgers vector (c) produce the same displacement of the upper half of a crystal relative to the lower half. Line direction, combination of a unit vector normal to the slip Adapted from Kelly A and Groves GW (1970) Crystallography and Crystal Defects, 428 pp. Reading, MA: Addison-Wesley Publishing Company. Figure 1 Sketch of dislocation loop indicating the plane, n, and b. As an example, important slip systems direction of the Burgers vector, b, and of the dislocation in clinopyroxene include {110} 110 , {110}[001], h i line, ,, at each point along the loop. The edge and screw and (100)[001] (Bystricky and Mackwell, 2001). Slip segments of the dislocation loop are denoted by e and s, respectively. systems in olivine include (010)[100], (001)[100], (100)[001], and (010)[001] with the dominate slip direction changes from one point to the next along system determined by stress, water concentration, as the loop, while the Bukinksrgers vector is the same at well as pressure and temperature conditions (Carter every point. Regions along the dislocation loop for and Ave´ Lallemant, 1970; Jung and Karato, 2001; which b is perpendicular to , are termed edge seg- Mainprice et al., 2005). ments, while regions for which b is parallel to , are To first order, dislocations with short displace- termed screw segments. Regions of a dislocation for ment vectors are preferred over those with longer which b and , are neither perpendicular nor parallel displacement vectors since the energy per unit length are referred to as mixed segments. of a dislocation, Edisl, is proportional to the square of Movement of a dislocation can take place by glide the Burgers vector. For an edge dislocation in an and climb. An edge dislocation can move in one of elastically isotropic material (Weertman and two ways. If an edge dislocation moves in the glide Weertman, 1992, pp. 45–52) plane defined by b ,, its motion is conservative. In  Gb2 r contrast, if an edge dislocation moves normal to its E ln 28 disl ¼ 4 1 – v r ½ Š glide plane, the process is nonconservative in that ð Þ c lattice molecules are added to or taken away from where G is the shear modulus, r is the mean spacing the dislocation line by diffusion such that the dis- between dislocations, and rc is the radius of the dis- location climbs out of its glide plane. A screw location core. In eqn [28], the contribution of the core dislocation does not have a specific glide plane of the dislocation to the elastic strain energy has been since b and , are parallel to each other. Hence, a neglected. For elastically anisotropic materials, the screw dislocation generally glides on the plane that full matrix of elastic constants must be considered, as offers least resistance to its motion. If it encounters an discussed in detail by Hirth and Lothe (1968, obstacle to its movement, a screw dislocation can cross pp. 398–440). slip off of its original glide plane onto a parallel glide Slip generally occurs on the closest packed plane plane in order to continue moving. A dislocation must that contains b, that is, the most widely separated

(a) (b) (c)

v s e v s e b b b

Figure 2 Sketch illustrating that (a) the movement of an edge dislocation, e, and (b) the movement of a screw dislocation, s, with the same Burgers vector (c) produce the same displacement of the upper half of a crystal relative to the lower half. Adapted from Kelly A and Groves GW (1970) Crystallography and Crystal Defects, 428 pp. Reading, MA: Addison-Wesley Publishing Company. Slip system: (slip plane)[burger’s vector]

A B

b b the slip system with the shorter burger’s vector and longer atomic distance normal to the slip plane will be “easier”: A dislocations move more easily (at lower stress, or faster at the same stress) than B dislocations.

This leads to anisotropy in dislocation motion and eventually to the creation of lattice preferred orientation... dislocations

Olivine: (010)[100] is easiest, followed by (010)[001] (“Miller indexes”)

b-axis b-axis ba < bc < bb V = 7.72 m/s bb = 1.03 nm b Va > Vc > Vb (0 1 0 )

(100) 11) (1 (0 10) 01) a-axis (0 c-axis a-axis c-axis b = 0.48 nm bc = 0.63 nm a Vc = 8.43 m/s Va = 9.89 m/s

Goetze & Evans, single crystal experiments.. 1 Introduction

MAKING LARGE STRAINS WITH DISLOCATIONS: Figure 1: The Glubglub Model

1.1 Orowan equation strain ~ b/h b h ε = (1) b h if disl. only sweeps ∆L (2) one dislocation makb ∆esL a minusculeε strain..= (3) so to enable strain,h L many mustand be producedfor N dislocations (4) and move through the crystal b ∆L ε = N (5) h L (V = Llh) (6) Nl ε = b∆L (7) V (ρ = (Nl)/V ) (8) ε = ρb∆L (9) dε = ε˙ = ρbv¯ (10) dt d v¯ is the average dislocation density, and v¯ σ1+. d d ∝

v¯ (σ, P, T, X) σ1+ (11) d ∝ ρ(σ) σ2 (12) ∝ ε˙ σ3+ (13) ∝

2 1 Introduction

1 Introduction Because there is not an equilibrium concentration, and because each dislocation produces a very small strain, there must be constant production of dislocations... Figure 1: The Glubglub Model called a “Frank-Read source” (1950, Charles Frank and Thornton Read) 1 Introduction Figure 1: The Glubglub Model

1.1 FFigure1r.ank-Read1 Frank-Read1: TheSoGlubgluburSoceurce Model

The Peach-Koehler Force, fP K = σb per unit length of dislocation, or over the length of the Frank The Peach-KoehlerPeach-KForce,oehler FforPce:K = σb per unit length of dislocation, or over the length of the Frank Read source,Read source,F F=P Kσbx= σ, bxwhere, wherexxisis thethe ddisistantance cbeetwbeenetwtheeentwtheo pinningtwo pinningpoints. Thepointensionts. Theon tension on P K 2 2 the growintensiong dislo oncation growingis Gb , which must balance the FP K , such that FP K = σbx = Gb , or dislocation = 2 2 the growing dislocation is Gb , which must balance the FP K , such that FP K = σbx = Gb , or 2Gb 1.1 Frank-Read Source σF R (1) ≈ l 2Gb σF R (1) where σF R is the stress re2qGbuired to produce dislocations≈ lfor a source of the given dimensions, G is σ (1) the shear modulus,F Rand≈ b islthe Burgers’ vector. (wikipedia:Frank-Read Source) where σF R is the stress required to produce dislocations for a source of the given dimensions, G is ~ several MPa where σF R is the stress required to produce dislocations for a source of the given dimensions, G is the shear moDensitdulus,y of Disloandcationsb is the Burgers’ vector. (wikipedia:Frank-Read Source) σ 2 the shear modulus, and b is the Burgers’ vector. ρ (2) ≈ Gb Density of Dislocations ! " 1.2 Orowan equation σ 2 1.2 Orowan equation ρ (2) ≈ Gb b ! " ε = (3) b h 1.2 Orowanε eq= uation if disl. only sweeps ∆L (2) (4) h b ∆L ε = (5) if disl. only sweeps ∆L h bL (3) andε =for N dislocations (6) (3) b ∆L hb ∆L ε = ε = N (4) (7) h L if disl.h Lonly sweeps ∆L (4) (V = Llb h∆) L (8) and for N dislocationsε =Nl (5) (5) ε = hb∆LL (9) b ∆L V ε = N (andρ = (Nforl)/VN)dislocations (6) (10) (6) h L b ∆L (V = Llh) εε==ρbN∆L (7) (11) (7) dε h L Nl (V==ε˙ =Llρhbv¯)d (12) (8) ε = b∆L dt (8) v¯ is the averageV dislocation density, and v¯Nl σ1+. d ε = d ∝b∆L (9) (ρ = (Nl)/V ) V (9) (ρ = (Nl)/V ) (10) 1+ ε = ρb∆L v¯d(σ, P, T, X) σ (10) (13) ε = ρb∆L ∝ (11) dε dε 2 = ε˙ = ρbv¯d = ε˙ = ρbv¯ (11) (12) dt dt d 1+ 1+ v¯ is the average dislocationv¯d densitis the ayv,erageand v¯dislocationσ . density, and v¯d σ . d d ∝ ∝

v¯ (σ, P, T, X) σ1+ (13) v¯ (σ, P, T, X) σ1+ d ∝ (12) d ∝ ρ(σ) σ2 2 (13) ∝ ε˙ σ3+ (14) ∝

2 PAPER CLIPS deformation:

1. work hardening, 2. Bauschinger effect (hysteresis: unbending is easier than bending, because the dislocations push against each other) 3. Luder’s bands: localization before rupture 3. Recovery

Large strains cannot occur without recovery...

Poirier, 1985

Climb is a thermally activated diffusive process.. This is where the temperature dependence of dislocation creep comes from.

Recovery by sub grain formation (SGF)

-associated with lower temperature because diffusion length scale is shorter and and lower stress conditions because driving force is smaller

(GBM) T/Tm (SGF)

log stress Recovery by grain boundary migration -associated with higher temperature conditions (only because necessary diffusion length scale is greater than for SGR)

-associated with higher stress because necessary driving force is higher (at constant T)

(GBM)

T/Tm (SGF)

log stress 1 Introduction

Figure 1: The Glubglub Model

1.1 Orowan equation

b ε = (1) h 1if disl.Intronlyoductisweepson∆L (2) b ∆L ε = (3) h L Figure 1: The Glubglub Model and for N dislocations (4) b ∆L ε = N (5) h L OROWAN Equation(V (from= Ll Phoirier) , 1985, p 62): (6) 1.1 OrNlowan equation ε = b∆L (7) V b (ρ = (Nl)/V ) ε = (8) (1) h ε = ρb∆L if disl. only sweeps ∆L (9) (2) dε b ∆L = ε˙ = ρbv¯ ε = (10) (3) dt d h L and for N dislocations (4) 1+ v¯d is the average dislocation density, and v¯d σ . b ∆L ∝ ε = N (5) h L (V = Llh) (6) Nl 1+ ε = b∆L (7) v¯d(σ, P, T, X) σ V (11) ∝ disl. 2 (ρ = (Nl)/V ) (8) ρ(σ) σ density (12) ∝ ε = ρb∆L (9) ε˙ σ3+ (13) ∝ dε = ε˙ = ρbv¯d (10) most of the complexity comes in the dislocation dt avg. disl. velocity term... velocity v¯ is the average dislocation density, and v¯ σ1+. d d ∝ ρ σ2 ∝ ε˙ σ3+ ∝

2

2 letters to nature

(1) The correlation of kinetic and entropy-based thermodynamic fragilities is implied 23. Greet, R. J. & Turnbull, D. Test of Adam-Gibbs liquid viscosity model with o-terphenyl speci®c-heat by the repeated analyses of the calorimetric and kinetic behaviour of liquids (not data. J. Chem. Phys. 47, 2185±2190 (1967). polymers) of different classes that extract `ground state temperatures' from the data and 24. Magill, J. H. Physical properties of aromatic hydrocarbons. III. A test of the Adam-Gibbs relaxation a ®nd them to have very similar values. We refer to TK (where the supercooling liquid model for glass formers based on the heat-capacity data of 1,3,5-tri- -naphthylbenzene. J. Chem. Phys. 47, 2802±2807 (1967). entropy extrapolates to the crystal entropy) and To (where the viscosity extrapolates to in®nity by the Vogel±Fulcher±Tammann equation). Ref. 17 describes some 30 cases where 25. Takahara, S., Yamamuro, O. & Suga, H. Heat capacities and glass transitions of 1-propanol and 3-methyl pentane: new evidence for the entropy theory. J. Non-Cryst. Solids 171, 259±270 (1994). To ˆ TK within 6 3% despite several pronounced exceptions). As the ratio Tg / To is often 17 26. Takahara, S., Yamamuro, O. & Matsuo, T. Calorimetric study of 3-bromopentaneÐcorrelation taken as a measure of fragility , it is clear that TK < To implies that kinetic fragility and between structural relaxation time and con®gurational entropy. J. Phys. Chem. 99, 9580±9592 (1995). thermodynamic fragility Tg /TK are the same, with some pronounced exceptions. This 27. Angell, C. A. Liquid landscapes. Nature 393, 521±522 (1998). means that either the physically constrained analysis leading to To < TK is ¯awed, or that the analysis of Ngai3 and co-workers, being limited to dielectric relaxation of molecular 28. Starr, F. W. et al. Thermodynamic and structural aspects of the potential energy surface of simulated water. Phys. Rev. E. (in the press); also preprint arXiv:cond-mat/0007487 on xxx.lanl.gov (2000). liquids, and containing some misplots (see Fig. 3 legend), has distorted the overall picture. h i 29. Phillips, W. A., Buchenau, U., NuÈcker, N., Dianou, A. J. & Petry, W. Dynamics of glassy and liquid (2) T < T is also a recent result of MD computer-simulation studies on well de®ned o K selenium. Phys. Rev. Lett. 63, 2381±2384 (1989). systems11±13. Although the number of systems studied is small, the results are important 30. Wischnewski, A., Buchenau, U., Dianoux, A. J., Kamitakahara, W. A. & Zarestky, J. L. Neutron because the identity of T and T is obtained using all-amorphous phase calculations; that o K scattering analysis of low-frequency modes in silica. Phil. Mag. B 77, 579±589 (1998). is, there is no reliance on fusion entropy data. 31. Kob, W., Sciortino, F. & Tartaglia, P. Aging as dynamics in con®guration space. Europhys. Lett. 49, (3) Both of the above arguments are weakened by the fact that they depend on 590±596 (2000). extrapolations beyond the range of experimental or simulation data. To avoid this, and at the same time to obtain a more complete picture than given in refs 2 and 3, we compare wide-ranging data on some 25 liquids from the ®elds of geochemistry (liquid silicates), Acknowledgements covalent semiconductors (liquid chalcogenides), molecular liquids, and molten ionic We thank S. Sastry, R. Speedy and F. Sciortino for comments and criticisms. This work was hydrates, salts and oxides, and including all the systems analysed by Ngai and Yamamuro3. supported by the NSF, Division of Materials Research, Solid State Chemistry program. Unlike the cases in ref. 3, the present data set covers the whole known fragility range. The quantities studied are the shear viscosities and the excess of the liquid entropy over the Correspondence and requests for materials should be addressed to C.A.A. crystal entropy measured at the same temperature, and uncorrected in any manner. We (e-mail: [email protected]). note that Sex in ref. 22 is not the same quantity as in the present work. It requires quantitative knowledge of the residual entropy at 0 K for its assessment. Also, in ref. 26 an attempt was made to separate out vibrational contributions to the entropy of the liquid above Tg in order to obtain Sc (for use in the Adam±Gibbs equation (equation (1)) by using a function ®tted to the heat capacity of the glass. This yields a quantity which is ...... neither Sc nor Sex because the procedure does not consider the introduction of new low frequencies above Tg, which is central to our conclusions. Intermittent dislocation ¯ow The quality of data used in this study is variable, ranging from very high, in the case of adiabatic calorimetry data on molecular liquids (data cited in ref. 3) and some high in viscoplastic deformation temperature substances (B2O3), to moderate, in the case of high-temperature differential scanning calorimeter (DSC) and drop calorimetry data on minerals. Reference sources are available from the authors. Both data sets are presented (Figs 1 and 2) as functions of M.-Carmen Miguel*², Alessandro Vespignani*, Stefano Zapperi³, inverse temperature scaled by the calorimetric onset Tg for the substance, as measured JeÂroÃme Weiss§ & Jean-Robert Grassok during upscan at 10 K min-1 (after previous continuous cooling at the same rate). This corresponds to scaling by the temperature at which the enthalpy relaxation time is * The Abdus Salam International Centre for Theoretical Physics, PO Box 586, approximately 200 s (refs 17, 18). 34100 Trieste, Italy Received 10 November 2000; accepted 29 January 2001. ² Departament de FõÂsica Fonamental, Facultat de FõÂsica, 1. Angell, C. A. Relaxation in liquids, polymers and plastic crystals - strong/fragile patterns and Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain problems. J. Non-Cryst. Solids 131-133, 13±31 (1991). ³ INFM, UniversitaÁ ``La Sapienza'', P. le A. Moro 2, 00185 Roma, Italy 2. Ito, K., Moynihan, C. T. & Angell, C. A. Thermodynamic determination of fragility in liquids and a § LGGE-CNRS, 54 rue MolieÂre, BP 96, 38402 St Martin d'HeÂres Cedex, France fragile-to-strong liquid transition in water. Nature 398, 492±495 (1999). 3. Ngai, K. L. & Yamamuro, O. Thermodynamic fragility and kinetic fragility in supercooled liquids: a k LGIT, BP 53X, 38041 Grenoble Cedex 9, France missing link. J. Chem. Phys. 111, 10403±10406 (1999)...... 4. Sastry, S., Debenedetti, P. G. & Stillinger, F. H. Signatures of distinct dynamical regimes in the energy landscape of a glassforming liquid. Nature 393, 554±557 (1998). The viscoplastic deformation (creep) of crystalline materials 5. Speedy, R. J. Relations between a liquid and its glasses. J. Phys. Chem. B 103, 4060±4065 (1999). under constant stress involves the motion of a large number of 6. Roland, C. M., Santangelo, P. G. & Ngai, K. L. The application of the energy landscape model to interacting dislocations1. Analytical methods and sophisticated polymers. J. Chem. Phys. 111, 5593±5598 (1999). 7. Speedy, R. J. The hard sphere glass transition. Mol. Phys. 95, 169±178 (1998). `dislocation dynamics' simulations have proved very effective in 8. Sciortino, F., Kob, W. & Tartaglia, P. Inherent structure entropy of supercooled liquids. Physletters. Rev. Lett. thetostudynatuofredislocation patterning, and have led to macroscopic 83, 3214±3217 (1999). constitutive laws of plastic deformation2±9. Yet, a statistical ana- 9. Buechner, S. & Heuer, A. The potential energy landscape of a model glassformer: thermodynamics, slip system. An edge dislocation with Burgers vector b located at the activation of Frank±Read sources1 is accepted aslysisthe mostof therelevandynamicst of an assembly of interacting dislocations anharmonicities, and ®nite size effects. Phys. Rev. E 60, 6507±6518 (1999). origin gives rise to a shear stress j at a point r ˆ x; y† of the form dislocation multiplication mechanism under creep deformation. 10. Coluzzi,s B., Verrocchio, P., Mlettersezard, M. & Parisi,to G.natuLennard-reJones binary mixture: a thermodynamical has not hitherto been performed. Here we report acoustic emis- x x2 2 y2† The activation of Frank±Read sources has been observed in ice † ˆ approach to glass transition. J. Chem. Phys . 112,† 2933±2944 (2000). sion measurements19 on stressed ice single crystals, the results of js r bm 2 2 2 1 crystals, along with more speci®c multiplication processes . (1) The correlation of kinetic and entropy-based thermodynamic fragilities is implied 11.23. Greet,Scala,2pR. J.1&A.,2Turnbull,Starrn† xD.,TFest‡.,ofLayAdam-GibbsN† ave, E.,liquidSciortiviscositynomodel, Fwi. &th oStanley-terphenyl speci®c-heat, H. E. Con®gurational entropy and diffusion of data. J. Chem. Phys. 47, 2185±2190 (1967). Because an accurate multiplication mechanism cannotwhichbeindicatesimulated that dislocations move in a scale-free intermittent by the repeated analyses of the calorimetric and kinetic behaviour of liquids (not supercooled water. Nature 406, 166±169 (2000). letters to nature polymers) of different classes that extract `ground state temperatures'm from the data and 24. Magill, J. H.nPhysical properties of aromatic hydrocarbons.1 III. A test of the Adam-Gibbsin a relaxationtwo-dimensional point-dislocation model, we have implemen- where is the shear modulusmodelandfor glass isformersthebasedPoissonon the heat-capacityratiodata. Thisof 1,3,5-tri-long-a-naphthylbenzene. J. Chem. fashion. This result is con®rmed by numerical simulations of a ®nd them to have very similar values. We refer to TK (where the supercooling liquid 12. Sastry, S. Liquid limits: the glass transition and liquid-gas spinodal boundaries of metastable liquids. entropy extrapolates to the crystal entropy) and Trange(where thestressviscosityisextrapolateresponsibles to forPhys. mutual47, 2802±2807 interactions(1967). among disloca- ted various phenomenological procedures. A simple way of taking average velocity along the slip direction (parallel to the Burgers ties. In Fig. 1 we show that the probability distribution of energy o in®nity by the Vogel±Fulcher±Tammann equation). Ref. 17 describes some 30 cases where 25. Takahara,Phys.S.,RevYamamuro. Lett., O.85,& Suga,590H.±He593at capacities(2000).and glass transitions4±7 of 1-propanol and model of interacting dislocations that successfully reproduces the 1 tions which are important in all dislocation dynamics models . into account the new dislocation loops generated at different vector b) . Transmission electron micrographs of plastically burst intensities exhibits a power-law behaviourT spanningˆ T withinseveral6 3% despite several pronounced exceptions). As the ratio T / T is often 3-methyl pentane: new evidence for the entropy theory. J. Non-Cryst. Solids 171, 259±270 (1994). o K g o 13. Sastry, S. The relationship between fragility, con®gurational entropy and the potential energy deformed materials display, on the other hand, complex features decades. This fact is an indication that a large number of disloca- 17 26. Takahara, S., Yamamuro, O. & Matsuo, T. Calorimetric study of 3-bromopentaneÐcorrelation main features of the experiment. We ®nd that dislocations gen- taken as a measure of fragility , it is clear that TK U< Tndero impliesathatconstantkinetic fragilityexternaland stress je, dislocation i performs an sources within the crystal is the random introduction of opposite- such as cellular structures and fractal patterns2,3, which are the tions move cooperatively in an intermittent fashion. A similar between structural relaxation time and con®gurational entropy. J. Phys. Chem. 99, 9580±9592 (1995). thermodynamic fragility Tg /TK are the same, with some pronounced exceptions. This landscape of glassforming liquids. Nature 409, 164±167 (2001). 13 sign dislocation pairs in the cross-section under consideration. The ®ngerprint of a complex multiscale dynamics not appropriately behaviour has been observed in the Portevin±LeChateliermeans that eithereffectthe,physically constrained analysisovleadingerdatompedT T motionis ¯awed, or thatalong27.theAngell,xC. directionA. Liquid landscapes.describedNature 393, 521by±522 (1998).the follow- erate a slowly evolving con®guration landscape which coexists o < K 14. Richet, P. Viscosity and con®gurational entropy of silicate melts. Geochim. Cosmochim. Acta 48, 471± accounted for by the mean-®eld character of Orowan's relation. In a plastic instability where the intermittent ¯owthe isanalysisruledofbyNgaithe3 and co-workers, being limitedingto dielectricequationrelaxation of molecular 28. Starr, F. W. et al. Thermodynamic and structural aspects of the potential energy surfacerateof simulatedand frequency of this creation process depend solely on the 10 water. Phys. Rev. E. (in the press); also preprint arXiv:cond-mat/0007487 on xxx.lanl.gov (2000). addition, rapid slip events have been observed in thelettersplastic interactiontoofnaturedislocations and diffusing soluteliquiatoms.ds, andThecontaininginter- some misplots (see Fig. 3 legend), has distorted the overall picture. 483 (1984). h i with rapid collective rearrangements. These rearrangements 11,12 29. Phillips, W. A., Buchenau, U., NuÈcker, N., Dianou, A. J. & Petry, W. Dynamics of glexternalassy and liquidstress j . We have checked that other multiplication rules in deformation of various metals and alloys , and in the Portevin± mittency observed in our case is different as we only(2)haveT

P lated emissions from each individual dislocation, but rather to the

5. Speedy, R. J. Relations between a liquid and its glasses. J. PhysP . Chem. B 103, 4060±4065 (1999). under constant stress involves the motion of a large number of 10 6 ®cial ice single crystals, employing several steps of constant applied –2 10 6. Roland, C. M., Santangelo, P. G. & Ngai, K. L. The application of the energy landscape model to 1 Ȟ E = 1.6 10 cooperative motion of several dislocations occurring, for example, log interacting dislocations . Analytical methods and sophisticated stress. We observe an intense acoustic activity, exhibitingpolymers. J.aChem.stronPhys.g 111, 5593±5598 (1999).

log `dislocation dynamics' simulations have proved very effectiveafter thein activation of a Frank±Read source, or if a dislocation intermittent character (see Fig. 1 inset). We measure7. Speedy,theR. J. Theenerghard ysphere glass transition. Mol. Phys. 95, 169±178 (1998). ! σ = 0.0125 –5 associated with each acoustic burst and analyse its 8.statisticalSciortino, F.,prKob,operW. &-Tartaglia, P. Inherent structure entropy of supercooled liquids. Phys. Rev. Lett. the study of dislocation patterning, and have led to macroscclusteropicbreaks apart. To gain further insight into this behaviour, in (aJ) 2±9 E 83, 3214±3217 (1999). –4 ! =constitutiv 0.0250 e laws of plastic deformation . Yet, a statisticalFig. 3ana-b and c, we show the evolution of the total number of 9. Buechner, S. & Heuer, A. The potential energy landscape of a model glassformer: thermodynamics, σ lysis of the dynamics of an assembly of interacting dislocations anharmonicities, and ®nite size effects. Phys. Rev. E 60, 6507±6518 (1999). dislocations as well as that of the fast-moving dislocations, that is, 0 5 10. Coluzzi, B., Verrocchio, P., Mezard, M. & Parisi, G. Lennard-Jones binary mixture: a thermodynamical has not hitherto been performed. Here we report acoustic emis- 0 1,000 2,000 Figure 2 Snapshot of the total stress ®eld and arrangement of dislocationsapproacinhatonumericalglass transition. J. Chem. Phys. 112, 2933±2944 (2000). sion measurements on stressed ice single crystals, thedislocationsresults of moving faster than if they were moving only under the t (s) simulation with v j ˆ 0:025. We observe metastable structure 11.formationScala, A.,(dipolesStarr, Fand., La Nave, E., Sciortino, F. & Stanley, H. E. Con®gurational–6 entropy and diffusion of –10 σ = 0.030 MPa which indicate that dislocations move in a scale-free interactionmittentof the external stress, v . v (other threshold values yield walls) and the associated stress ®eld which goes from light bluee (lowsupercvalues)ooledto darkwaterblue. Nature 406, 166±169 (2000). –3 –2 –1 0 i j 0 2 4 6 8 fashion. This result is con®rmed by numerical simulations of a (high values). The complex low-stress pathways joining dislocations12. Sastrareythe, S. resultLiquidoflimits:the the glass transition and liquid-gas spinodal boundaries of metastable liquids. equivalent results). After the injection of a few new dislocations or log10 E σe = 0.037 MPa log | | anisotropic elastic interactions. Dislocations moving at low velocitiesPhys(lower. Rev.thanLett.v85,j) are590±593 (2000). b model of interacting10 !i dislocations that successfully reproduces the 13. Sastry, S. The relationship between fragility, con®gurational entropy and the potential energy the annihilation of a dislocation pair, several other dislocations start Figure 1 Statistical properties of the acoustic energy bursts recorded in ice single crystals depicted in red, while those moving at higher velocities areσedepicted = 0.067in black. MPaMost main features of the experiment. We ®nd that dislocations gen- under constant stress. The main ®gure shows the distribution of energy bursts for the dislocations in walls are moving slowly, whereas isolated ones tendlandscapeto move ofat ghigherlassforming liquids. Nature 409, 164±167 (2001).700 erate a slowly evolving con®guration landscape whichto cmovoexistse and rearrange, not necessarily in the close vicinity of the 0 =14. 0.086Richet, PMPa. Viscosity and con®gurational entropy of silicate melts. Geochim. Cosmochim. Acta 48, 471± different loading steps. The cut-offs depend slightly on the applied stress, but the power- speed. In the simulations, an initial number of N 0 ˆ 1;500σedislocations are randomly ) 483 (1984). with rapid collective rearrangements. These rearrangementstriggering event. To quantify this effect, we measure the collective ˆ : : E ˆ law exponent remains the same. The ®t yields an exponent tE 1 60 6 0 05. Inset,( a distributed on a square cell of size L 300. Their Burgers vectors are randomly chosen 15. Richet, P. & Bottinga, Y. Glass transitions and thermodynamic properties of amorphous SiO2, involve a comparatively small fraction of the dislocations and P typical recorded acoustic signal. The creep experiment is performed at T ˆ 210 8C to be +b or -b with equal probability. In the absence of external stress, we ®rst let the 650 velocity V ˆ S9jvij of the fast-moving dislocations and de®ne the NaAlSinO2n+2 and KAlSi3O8. Geochim. Cosmochim. Acta 48, 453±470 (1984). N 10 6 lead to an intermittent behaviour of the net plastic response. This 2 under different constant compression stresses ranging from j ˆ 0:58 MPa to 10system relax until it reaches a metastable arrangement. The number16. Adam,of remainingG. & Gibbs, J. H. On the temperature dependence of cooperative relaxation properties in acoustic energy as E ˆ V (ref. 17). In the inset of Fig. 4, we see Ȟ E = 1.6 basic dynamical picture appears to be a generic feature in the j ˆ 1:64 MPa. The stress is applied at an angle with respect to the c axis, giving riselog to a dislocations is then N < 700. At this point, we apply a constant shear stress and study glassforming liquids. J. Chem. Phys. 43, 139±146 (1965). 10±12 small resolved shear stress acting across the glide plane (j ˆ 0:030 MPa±0:086 MPa). their evolution. To avoid the discontinuities arising from truncating long-range elastic deformation of many other materials . Moreoverthat, it shouldthe signal E(t) consists of a succession of intermittent and e 17. Angell, C. A. Entropy and fragility in supercooled liquids. J. Res.600NIST 102, 171±185 (1997). The frequency bandwidth of the transducer was 10±100 kHz. The amplitude range interactions in equation (1), we resort to the Ewald summation 18.method.Richert,WeR.have& Angell,thus C. A. Dynamics of glassforming liquids. IV: On the link between molecular provide a framework for discussing fundamental aspects of –5 pronounced bursts, each one signalling the onset of collective between the minimum and the maximum recordable thresholds is 70 dB, that is, a range exactly taken into account the interaction of a dislocation with all thedynamicsin®niteandperiodiccon®gurational entropy. J. Chem. Phys. 108,c9016±9026 (1998).

(aJ) plasticity that goes beyond standard mean-®eld approaches that of seven orders of magnitude in energy. imagesE of another dislocation in the same cell. 19. Goldstein, M. Viscous liquids and the glass transition: a potential energy barrier picture. J. Chem. dislocation rearrangements. The slow and continuum motion of Phys. 51, 3728±3729 (1969). see plastic deformation as a smooth laminar ¯ow. dislocation structures (v , vj) is not considered as it only sets up a 20. Stillinger, F. H. & Weber, T. A. Packing structures and transitions200in liquids and solids. Science 225, Whenever dislocation glide is the dominant plastic deformation i 668 NATURE | VOL 410 | 5 APRIL 2001 | www.nature.com © 2001 Macmillan Magazines Ltd 983±989 (1984). mechanism in a crystalline material, we observe a constantbackgroundstrain- noise signal which cannot be detected experimentally. 0 21. Goldstein, M. Viscous liquids and the glass transition: sources of the excess heat capacity. J. Chem.

0 1,000 2,000 Figure 2 Snapshot of the m total stress ®eld and arrangement of ratedislocationsregimein ausuallynumericaldescribed by Orowan's relation gÇ < r bv. Here, Phys. 64, 4767±4773 (1976). Inm Fig. 4 we show that the distribution of energy bursts, obtained t (s) : N 22. Johari, G. P. Contributions tosimulationthe entropy ofwitha glassv j andˆ 0liquid,025.100andWetheobservedielectricmetastablerelaxation time.structuretheformationplastic(dipolesstrain-rateand of the material gÇ is simply related to average –10 by sampling the signal over different times and realizations, decays 0 2 4 6 J. Chem. Phys. 8 112, 7518±7523walls)(2000).and the associated stress ®eld which goes from light bluequantities(low values)suchto darkasbluerm, the density of mobile dislocations, and v, their (high values). The complex low-stress pathways joining dislocations are the result of the very slowly, spanning various decades. For intermediate values, log10 E the distribution shows an algebraic decay with an exponent NATURE | VOL 410 | 5 APRILanisotropic2001 | www.natuelasticre.cinteractions.om 0Dislocations© 2moving001 Macmillanat low velocities Magazines(lower Ltdthan vj) are 667 0 5,000 10,000 Figure 1 Statistical properties of the acoustic energy bursts recorded in ice single crystals depicted in red, while those moving at higher velocities are depicted in black. Most tE ˆ 1:8 6 0:2, in reasonable agreement with experiments (see under constant stress. The main ®gure shows the distribution of energy bursts for the dislocations in walls are moving slowly, whereas isolated ones tend to move att higher Fig. 1). The maximum number of dislocations we can handle in different loading steps. The cut-offs depend slightly on the applied stress, but the power- speed. In the simulations, an initial number of N ˆ 1;500 dislocations are randomly Figure 3 Statistical properties0 of dislocation velocities and density obtained in numerical our simulations severely restricts the maximum value of the signal law exponent remainshttp://wwwthe same. The ®t yields an.naturexponent teˆ.com/natur1:60 6 0:05. Inset, a e/journal/v410/n6829/extrdistributed on a square cell of size L ˆ 300. Their Burgers vectorsef/410667a0_S1.htmare randomly chosen E simulations. a, Probability density of the individual dislocation velocities |v | in the in the system. Consequently, the extension of the power-law regime typical recorded acoustic signal. The creep experiment is performed at T ˆ 210 8C to be +b or -b with equal probability. In the absence of external stress, we ®rst let the i under different constant compression stresses ranging from j ˆ 0:58 MPa to system relax until it reachesstationarya metastablestate. Thearrangementblack line. Theis numbera least-squaresof remaining®t of the intermediate data points. We grows with the number of dislocations, but is eventually bounded by j ˆ 1:64 MPa. The stress is applied at an angle with respect to the c axis, giving rise to a dislocations is then Nobtain< 700.anAtexponentthis point, wen ˆapply2:5a constantfor the scalingshear stressof theandintermediatstudy e velocities. b, Time the different nature of the extremes statistics. : : small resolved shear stress acting across the glide plane (je ˆ 0 030 MPa±0 086 MPa). their evolution. To avoidevolutionthe discontinuitiof the totales arisingnumberfrom truncatingof dislocationslong-rangein theelasticcell. c, Fraction of fast-moving The broadly distributed intermittency resulting from the statis- The frequency bandwidth of the transducer was 10±100 kHz. The amplitude range interactions in equationdislocations(1), we resortfortoathegivenEwaldrunsummationof the simulations.method. We have thus tical analysis can be interpreted as a nonequilibrium transport between the minimum and the maximum recordable thresholds is 70 dB, that is, a range exactly taken into account the interaction of a dislocation with all the in®nite periodic of seven orders of magnitude in energy. images of another dislocation in the same cell. NATURE | VOL 410 | 5 APRIL 2001 | www.nature.com © 2001 Macmillan Magazines Ltd 669 668 © 2001 Macmillan Magazines Ltd NATURE | VOL 410 | 5 APRIL 2001 | www.nature.com orowan to scholz: “you can only measure the displacement... for all you know, there is a guy inside playing the piano...” von mises criterion

complete strain compatibility for an arbitrary deformation requires FIVE slip systems to be active (6 strain components + constant volume constraint). However, this strict requirement can be relaxed by grain boundary sliding and diffusion creep... so the strength more closely reflects the weakest slip system than the strongest. dislocation production

motion: glide climb

recovery+ recrystallization: sub-grain rotation GB - migration

low T high T high stress low stress 4. Fabric (Lattice preferred orientation) Development When anisotropic crystals (i.e. slip systems with significantly different strengths) are sheared, grains can rotate by a range of mechanisms, including:

1. rigid body rotation 2. recrystallization by GBM, in which well- oriented grains consume poorly oriented grains, 3. subgrain rotation recystallization...

rigid body rotation R E P O R T S only currents of Sn atoms around the edge of References and Notes 12. N. C. Bartelt, in preparation. the island, then ␮(R) would drop rapidly with 1. L. P. Nielsen et al., Phys. Rev. Lett. 71, 754 (1993). 13. J. M. Wen, S. L. Chang, J. W. Burnett, J. W. Evans, P. A. Thiel, Phys. Rev. Lett. 73, 2591 (1994). island size, proportional to 1/R3 (17), and 2. A. Christensen et al., Phys. Rev. B 56, 5822 (1997). 3. F. Besenbacher et al., Science 279, 1913 (1998). 14. K. Morgenstern, G. Rosenfeld, B. Poelsema, G. Comsa, give a v(R) that would increase with decreas- 4. S. S. P. Parkin, Annu. Rev. Mat. Sci. 25, 357 (1995). Phys. Rev. Lett. 74, 2058 (1995). ͌ 15. S. C. Wang, G. Ehrlich, Phys. Rev. Lett. 79, 4234 ing radius as 1/ R. Thus, as long as Dd is 5. See, for instance, A. K. Schmid, J. C. Hamilton, N. C. nonzero, diffusion through the islands for Bartelt, R. Q. Hwang, Phys. Rev. Lett. 77, 2977 (1996). (1997). 6. E. Bauer, Rep. Prog. Phys. 57, 895 (1994). 16. W. W. Pai, A. K. Swan, Z. Y. Zhang, J. F. Wendelken, sufficiently large R values should provide the 7. J. Erlewein, S. Hofmann, Surf. Sci. 68, 71 (1977). Phys. Rev. Lett. 79, 3210 (1997). dominant contribution to their mobility, con- 8. G. Contini, V. Di Castro, N. Motta, A. Scarlata, Surf. 17. S. V. Khare, N. C. Bartelt, T. L. Einstein, Phys. Rev. Lett. sistent with our observations. Sci. 405, L509 (1998). 75, 2148 (1995). We can experimentally adjust the speed of 9. For movies of the motion of the islands in Figs. 1 18. F. D. Dos Santos, T. Ondarcuhu, Phys. Rev. Lett. 75, and 2, see Science Online (www.sciencemag.org/cgi/ 2972 (1995). the islands over several orders of magnitude content/full/290/5496/1561/DC1). 19. C. D. Bain, G. D. Burnett-Hall, R. R. Montgomerie, by changing the temperature slightly around 10. For convenience, our Cu crystals were grown as films RNatureE P372O, 414R(1994).T S room temperature. In an Arrhenius plot of the on a Ru substrate. The films were more than 20 20. M. K. Chaudhury, G. M. Whitesides, Science 256, atomic layers thick, and the spacing between surface 1539 (1992). velocitymore(Fig. 5B),symmetricthe linearity suggestswith thatrespect atomicto stepsthewasshearoften several micrometers.directionPreciserather than a [010] point maximum approximate axial symmetry in all the veloc- 21. F. Brochard, Langmuir 5, 432 (1989). the processes responsible for the island mo- x-ray crystallography measurements show that the kinematics. It has a sharp [100] maximum perpendicular 22.toP.theG. de Gennes,shearPhysicsplane.A 249, 196 (1998). ity distributions about that direction. As Vp is tion are activated with an energy of 900 meV. lattice spacing of these crystals is indistinguishable from that of bulk Cu [H. Zajonz, D. Gibbs, A. P. 23. L. E. Scriven, C. V. Sternling, Nature 187, 186 (1960). From theorientedabove analysis,parallelthis shouldto thebe halfshear Baddorf,directionV. Jahns, D.andM. Zehner, Surf. Sci.An447important, L141 24. A. goalW. Adamson,in studyingPhysical Chemistrymantleof Surfacesdy- minimum anywhere along a girdle normal to of the activationgirdles energyof [010]associatedandwith[001]the normal(2000)]. to that namics is to determine(Wiley, NewtheYork, ed.flow4, 1982),directionpp. 97–116. and the shear direction, it does not indicate the product D ␳ ␤. Because diffusion barriers, 11. Evidence for Sn-Cu exchange within islands rather 25. Lord Rayleigh, Philos. Mag. 30, 386 (1890). d d than at island edges comes from experiments in 26. Supported by the Office of Basic Energy Sciences, defect formationdirection.energies,Individualand exchangemeasurementsen- which weofcompletely50 rib-covered theshearsurface planewith the fromDivisionseismicof Materialsobservations.Sciences, U.S. DepartmentSeis-of orientation of the shear plane, by contrast to ergies arebontypicallyporphyroclastson the order of hundredsindicate that(2ϫ2) Snnearlyphase. Despiteallthe absencemicof islandpropertiesedges, ofEnergy,deformedunder contract DE-AC04-94AL85000.samples (com- previous conclusions based on experiments of meV, the activation energy we measure is the rate of creation of the (͌3ϫ͌3)R30° alloy phase was identical to the observed rate for islands. 10 July 2000; accepted 26 September 2000 reasonablehavewithina singlethis model.preferredFrom our mea-orientation, consistent pressional wave velocity Vp, shear wave conducted to lower strains (13, 14). These surementswithat 270“easyK, we slip”find thatonthe theSn-Cu(010)[100] system. splitting dVs, and polarization of fast shear results imply that seismic investigations of exchange rate is roughly one atom per 4000 s. R E P O R T S AssumingFora valuethe asymmetricof on the orderporphyroclasts,of 100 theHigh[100] Shearwave Vs1Strainpol) wereofcalculatedOlivinefrom the mea- the mantle (e.g., Vp and Vs deep seismic meV, typicalaxesdiffusionare distributedprefactors on theobliqueorder to the shear di- sured LPO (Fig. 4). The data for high shear on March 31, 2010 sounding, near vertical reflection, and teleseis- pressure12 Ϫ1and high-temperature torsion appa- scatter diffraction (EBSD). During deforma- [100] axes develop two maxima, one parallel of 10 s , and the experimentally measured Aggregates: Rheological and rection, as in the low strain2 deformation tex- strains suggest that a large Vp anisotropy mic shear wave splitting) may indicate the flow ratusactivation(17energy,), in whichwe find anythat (Dsmalld␳d␻ ␤element/ of the tion, the texture evolved through a transient to the shear direction and one oblique to the k T)1⁄2 ture.yields Theislandobservedvelocities ofhighthe ob-strain texture differs (here half of the single crystal anisotropy) can direction, but will not identify the shear plane sampleB undergoes deformation at constant SeismicdeformationConsequencestexture (␥ ϳ 0.5) into a recrys- shear direction. This LPO fits with previous served fromorder ofnumericalmagnitude. Sosimulationsthe simple with or without develop in highly sheared parts of the mantle. in highly recrystallized olivine mylonites. strainpicture rateof Fig.by4 seemssimpleto be consistentshear with(18). The shear tallization texture at high strain (Fig. 4). At low strain experimental results (13, 14) and modeling recrystallization (22, 23) by tM.heBystricky,*The VpK. Kunze,maximumL. Burlini,andJ.-P.dVsBurgminimum are Experimental deformation to high shear stresseverythingandweshearknowstrainat presentrateaboutat thethis outer surface ␥ ϳ 0.5, the [010] crystallographic axes tend with numerical simulations (22, 23). At high- system.[010]Given theandsimplicity[001]ofgirdlethe drivingnormal toHigh-pressurethe shearand high-temperatureparallel to torsionthe shearexperimentsdirection,on olivinealongaggregateswith an strain of olivine aggregates produces a stable offorcetheforcylindricalthe motion ofsamplethe reactivewereSn is-derivedinfromdislocation tocreepalignshow aboutnormal15 toto20%thestrainshearweakeningplanebeforeandsteady-the er strains, the texture is much stronger and LPO and recrystallized microstructure con- thelands,measuredthis mechanismtorqueof surfaceand alloyingtwist rate, respec-state behavior, characterized by subgrain-rotation recrystallization and a strong www.sciencemag.org might be expected to be common when sur- lattice preferred orientation. Such weakening may provide a way to focus flow sistent with observations in highly deformed tivelyface diffusion(19).isThefasterexperimentsthan exchange intowerethe performedin the upper mantle without a change in deformation mechanism. Flow laws natural peridotites (21, 24). Although sample atsubstrate.1200°C and 300 MPa confining pressure,derived from lowFig.strain1. dataShearmaystressnot be appropriateversus shearfor use strainin modeling high We make three comments about the sur- strain regions.forIn suchsamplesareas, seismicdeformedwave propagationat 1200°Cwill be anisotropicand with size, grain size, temperature, and strain rate withface bronzethe oxygenformation: fugacity(i) The trajectoriesnear the Fe/FeOan axis of approximate300 MParotationalat constantsymmetry aboutnominalthe shearsheardirection. In conditions may be different in the laboratory buffershown in(10Figs.Ϫ121 andbars).2 do notFourierconvey thetransformfull contrastinfra-to current thinking, the anisotropy will notϪ5indicateϪ1 the orientation of animation of the Sn island motion (9); the Sn the shear planestrainin highlyratesstrained,ofrecrystallized6 ϫ 10olivine-richs rocks.(gray than in nature, the correspondence in micro- Ϫ4 Ϫ1 redislands(FTIR)often appearspectroscopyto react to theirRanalysesE Psur-O R T S of de- curve) and 1.2 ϫ 10 s (black Downloaded from structures provides confidence that it is pos- only currentsroundingsof Sn atomsin arounda complexthe edge ofway (ReferencespausingandwhenNotes Seismic anisotropy12. N. C. Bartelt,inincurve).oceanicpreparation. andConversioncontinental to 1.5fromusingtorquea diagonal-cutandassembly (13, 14) theformedisland, then ␮(R)sampleswould drop rapidlyindicatedwith 1. L. P. Nielsenthatet al., Phys.theyRev. Lett. 71contained, 754 (1993). 13. J. M. Wen, S. L. Chang, J. W. Burnett, J. W. Evans, P. A. their best path is not obvious, for example) lithospheres isThiel,commonlyPhys. Rev. Lett. 73attributed, 2591 (1994). to defor- sible to scale results obtained in experiments island size, proportional to 1/R3 (17), and 2. A. Christensen et al., Phys. Rev. B 56, 5822 (1997). yielded useful microstructural and textural less than 30 molar parts3. F. Besenbacherperet al., Sciencemillion279, 1913 (1998).(ppm)14. K. Morgenstern,twistG. Rosenfeld, ofB. Poelsema,torsionG. Comsa, deformation into shear give a vand(R) thatarewouldefficientincrease withindecreas-finding4. newS. S. P. Parkin,unalloyedAnnu. Rev. Mat. Sci.mation-induced25, 357 (1995). Phys.latticeRev. Lett. 74preferred, 2058 (1995). orientations observations. However, these experiments to natural conditions. In addition, the 15 to ing radius as 1/͌R. Thus, as long as D is 5. See, for instance, A. K. Schmid, J. C. Hamilton, N. C. 15. S. C. Wang, G. Ehrlich, Phys. Rev. Lett. 79, 4234 regions. It is interestingd that such complexity (LPOs) of olivine andstresspyroxeneand(1–6)strainand is wasmay notperformedhave producedassteady-statein deforma- nonzero,H/Sidiffusion(20through). Samplesthe islands for wereBartelt, R. Q.deformedHwang, Phys. Rev. Lett. 77,to2977 (1996).differ-(1997). 6. E. Bauer, Rep. Prog. Phys. 57, 895 (1994). 16. W. W. Pai, A. K. Swan, Z. Y. Zhang, J. F. Wendelken, 20% observed weakening may provide a way sufficientlycanlargeariseR valuesfromshouldsuchprovidea seeminglythe 7. J. Erlewein,simpleS. Hofmann,situ-Surf. Sci. 68used, 71 (1977).to determinePhys. Rev.flowLett.(7919,patterns3210).(1997).Thein peakthe upperstressestion textures.agreeInwelladdition,withsuch an experimen- dominantentationcontributionamountsas surfaceto their mobility,ofalloying.bulkcon- (ii)8.shearG. Contini,SurfaceV. Distrain,Castro,freeN. Motta,mantleA.underScarlata, Surf.(7,two817.). S.ToV. Khare,giveN. C.weightBartelt, T. L. Einstein,to thisPhys.interpre-Rev. Lett. tal arrangement does not allow a detailed sistent with our observations. Sci. 405, L509 (1998). 75, 2148 (1995).dislocation creep flow laws for olivine to focus flow in the upper mantle without the Weenergy–can experimentallydrivenadjustmotionthe speedofof particles9. For moviesonof theliquidmotion of the tation,islands in Figs.we1 must18. F. D.understandDos Santos, T. Ondarcuhu,how Phys.olivineRev. Lett.LPOs75, rheological study, because combined com- and 2, see Science Online (www.sciencemag.org/cgi/ 2972 (1995). constant angular displacement rates corre- on March 31, 2010 the islands over several orders of magnitude content/full/290/5496/1561/DC1). aggregates determined in axial compression experimentsneed(12to).inSteppingvoke a chtestsange iperformedn deformatiatonhighmech- surfaces (or liquids on solid surfaces) is a develop and19.evolveC. D. Bain, G.withD. Burnett-Hall,strainR. inR. Montgomerie,particular pressional and simple shear components con- by changing the temperature slightly around 10. For convenience, our Cu crystals were grown as films Nature 372, 414 (1994). roomspondingtemperature.familiar Inphenomenonan Arrheniusto constantplot of(18the –24);on asheartheRu substrate.observa-Thestrainfilms weredeformationmoreratesthan 20 20.ofsettings.M. K. Chaudhury,strainsFlowG. M. inWhitesides,theyieldedmantleScience 256, in-stresstributeexponentsto the deformationn (n andϭ because3.2 atthe␥ ϳanism1.2 and(25n). ϭIn 3.3subductionat ␥ ϳ zones,3.2) typicalhigh shearof atomic layers thick, and the spacing between surface 1539 (1992). velocity (Fig. 5B), the linearity suggests that atomic steps was often several micrometers. Precise tion of the similarϪmotion5 Ϫ1of camphor parti- Ϫ4volvesϪ1 high21.strainF. Brochard,andLangmuirnon-coaxial5, 432 (1989). deforma- state of stress is poorly known at high strains the processes responsible for the island mo- x-ray crystallography measurements show that the deformation by dislocation creep. Shaded area (left) shows magnified view of the low strain interval either 6 ϫ 10 s or 1.2 ϫ 10 s at the22. P. G. de Gennes, Physics A 249, 196 (1998). strains may lead to strain localization where tion areclesactivatedacrosswith anwaterenergy ofdates900 meV.to 1686lattice(23,spacing24of )theseandcrystals istion,indistinguishablebut most experimental studies on poly- (15). Here we present results on the experi- from that of bulk Cu [H. Zajonz, D. Gibbs, A. P. 23. L. E. Scriven, C.atV. Sternling,rightNature 187(␥, 186Յ(1960).0.2), up to equivalent strains (ϳ10%) where rheological data are typically obtained Fromouterthewasabovethesurfaceanalysis,subjectthis shouldofofmuchbe thehalf discussionsample.Baddorf, V. Jahns,inD.ForM.theZehner, Surf.bothcrystallineSci. 447, L141defor-olivine24. A. W. Adamson,have PhysicalbeenChemistryperformedof Surfacesun- mental deformation of olivine aggregates in hot mantle convects past the upper side of of the activation energy associated with the (2000)]. (Wiley, New York, ed. 4, 1982), pp. 97–116. product19thD ␳ ␤century.. Because diffusionIt allowedbarriers, Lord11. EvidenceRayleighfor Sn-Cu exchangeto withinderislandsuniaxialrather 25.compression,Lord Rayleigh,inPhilos.compressionMag.in 30which, 386 (1890).deforma-experiments.torsion yieldingAtbulktheseshear strainslow strains,up to ␥ ϭ deformation often appears to be steady-state mationd d series, a peakthan stressat island edges comesoccurredfrom experiments in at26. Supporteda by the Office of Basic Energy Sciences, cold slabs. In regions of extensional deforma- defect estimateformation energies,molecularand exchangedimensionsen- whichandwemotivatedcompletely covered the tionsurface withis coaxialthe Divisionand oftheMaterialsbecausetotalSciences,amountU.S. Departmenttheof strainonsetof of5. Tweakeninghe aim of this stisudyrarelywas to areached.chieve non- ergies are typically on the order of hundreds (2ϫ2) Sn phase. Despite the absence of island edges, Energy, under contract DE-AC04-94AL85000. of meV,shearhimthe activationtostrainperformenergy␥weanmeasureϳaccurate0.1,is determinationfollowedthe rate of creation ofofbythe (͌15%3(equivalentϫ͌3)R30° alloyweak-strains Ͻ0.5) is limited (9–12). coaxial deformation and to investigate the de- tion, strain weakening may focus on major phase was identical to the observed rate for islands. 10 July 2000; accepted 26 September 2000 reasonablethewithinsurfacethis model.tensionFrom ourofmea-water (25), forFexam-ig. 3. MOlivineicrostsamplesructuredeformedof olivtoinsheare agstrainsgregaupte detafiolerdmmeicdrotstoruc␥turϳal, t5extsuhraol,wanind grheroiblobgiocanl surementseningat 270 K,upwe findtothat ␥the Sn-Cuϳ 0.5 (Fig. 1). At higher detachment zones near the crust-mantle bound- exchangeple,rate isandroughlynowone atomreappearsper 4000 s. in the contextHighgraiofnsShearwithinStrainrecrystaofllizeOlivined matrix. (A and B) eHvoigluhtio-nreosfoploulytciroynstaollirnieeonltivainteioonverima laargg-- Assumingstrains,solidsa valueonofthethe! on thenanoscale.floworder of 100stress(iii) Controlwasof thenearly steady- Fig. 2. Optical micro-er strain range than previously accessible. ary, for example during the development of meV, typical diffusion prefactors on the order ing mapsGeologischeswith colInstitut,or keyETH-Zentrum,according8092to iZun¨vrich,ers on March 31, 2010 e pole figures of shear direction and of 1012islandsϪ1, and themotionexperimentallypresentsmeasuredthe possibilityAggregates:of ma- Switzerland.Rheologicalgraphsandin cross-polarizedOlivine aggregates were hot-pressed from passive margins (26, 27). Seismic disconti- state, leading to2 a total weakeningshear plane ofnorthemal, respectively. Thick lines between pixels indicate grain activation energy, we find that (Dd␳d␻ !␤/ 1 nipulation of surface alloy formation to create San Carlos olivine powders (16). Deforma- k T) ⁄2 yields island velocities of the ob- *To whom correspondencelightshould[frombe addressed.thinE- sections B Seismic Consequences nuities recently identified in the upper mantle www.sciencemag.org servedaggregateusefulorder of andmagnitude.novelofSoaboutnanoscalethe simple 20%structures.frombouthendarpeakmail:ies ([email protected] neighbor misorientation ationnglexperimentse ␻ Ͼ 15°were) andcarriedthinoutlineinsamhigh-ark picture of Fig. 4 seems to be consistent with M. Bystricky,* K. Kunze, L. Burlini, J.-P.cutBurgas in (18)] of olivine subgrain boundaries (2° Ͻ ␻ Ͻ 15°). Recrystallized grains in the matrix have sizes of about 3 to 6 ␮m (28, 29) may correspond to such shear zones. on March 31, 2010 everythingto thewe knowflowat presentstressabout thisat ␥ ϳ 5. Stress exponents aggregates deformed at 1564system. Given theasimplicitynd haofvthee drivingvarious oHigh-pressurerien24taNOVEMBERtiandonhigh-temperatures with2000[1torsion00VOL]experimentspr290eferonaSCIENCEbolivinely aggregatestowwww.sciencemag.orgard the shear direction. A highly elongated forceoffor nthe motionϭ 3.3of the reactivedeterminedSn is- in dislocationafter creeptheshowweakeningabout 15 to 20% strain weakening6 ϫ before10Ϫsteady-5 sϪ1 to different For these regions of localized shear strain, the lands, this mechanismribbofonsurfaceshoalloyingws similstatearlbehavior,y sizecharacterizedd subgbyrsubgrain-rotationains and recrystallizationis in apandpraostrongximate owww.sciencemag.org rientation (031)[100]. (C) Scanning mightsuggestbe expected todislocationbe common when sur- creeplatticeaspreferredtheorientation.rate-limitingSuch weakening may providesheara way to strains.focus flow Shear sense application of rheologies derived from low face diffusion is fasterelethanctrexchangeon mintoictheroscopein theorupperientmantleatiowithoutn coanchangetrastin deformationimage amechanism.t sameFlowmlawsagnification of a different region with similar substrate. derived from low strain data may not be appropriate forisusedextral.in modeling high(A and B) Start- strain data would be inappropriate. mechanism,We make threefecommentsatureaboutevens. the sur-at highstrainstrains.regions. In such areas, seismic wave propagation will be anisotropic with face bronze formation: (i) The trajectories an axis of approximate rotational symmetry about theingshearmaterial.direction. In Olivine ag- shown in Figs.M1icandro2 dostnotruconveycturtheefulls werecontrastanatolycurrentzedthinking,withthe anisotropyopticawilll not indicate the orientation of animation of the Sn island motion (9); the Sn the shear plane in highly strained, recrystallized olivine-richgregaterocks. hot-pressed for

Downloaded from References and Notes islandsandofteneappearleFig.cttororeact4.n Poletomtheiricrsur-figuresoscopyand. Thin sections were roundings in a complex way (pausing when Seismic anisotropy in oceanic and continental to 1.5 using a diagonal-cut12 hours,assembly (13, 14showing) an 1. H. H. Hess, Nature 203, 629 (1964). theircubestt pathpeisrseismicnotpeobvious,ndicforulpropertiesexample)ar to lithospheresthe ofcisycommonlylindeattributedr radtoidefor-us ayieldednd useful microstructural and textural

and are efficient in finding new unalloyed mation-induced lattice preferred orientations observations. equigranularHowever, these experimentsfabric with 2. N. I. Christensen, Geophys. J. R. Astron. Soc. 76, 89 Downloaded from regions. It is interestingolivinethat suchaggregatescomplexity (LPOs) ofde-olivine and pyroxene (1–6) and is may not have produced steady-state deforma- (1984). canwariseitfromhinsuch20a seemingly0 to 3simple00situ-␮musedoftotdeterminehe saflowmppatternsle oinutheteupperr edgtione.textures. Inanaddition,averagesuch an experimen-grain size of ation as surface falloying.orme(ii)dSurfacein defreextramantlel sh(e7,a8).rTo. give weight to this interpre- tal arrangement does not allow a detailed 3. A. Nicolas, N. I. Christensen, in Composition, Struc- energy–In drivensucmotionh poflaparticlesnes,on liquiddefortation,mawetimustonunderstandis nhoweaolivinerly LPOssimprheologicalle study,20because␮m.combinedMostcom-grains have surfaces (or liquidsAlonl solidfigsurfaces)ures isaaredevelopuppandeevolver with strain in particular pressional and simple shear components con- ture and Dynamics of the Lithosphere-Asthenosphere

euhedral shapes with low www.sciencemag.org familiarshephenomenonar (h1e8m()18.–24isWp);htheiethobserva-reinecqruedeformationaalsianregasettings.straFlowin,intthehemantleavin-eratributege to the deformation and because the System, K. Fuchs, C. Froidevaux, Eds. (American Geo- tion of the similar motion of camphor parti- volves high strain and non-coaxial deforma- state of stress isapoorlyspecknownt raatthighiosstrains. Grain bound- clesgacrossrainwatersipdateszreotojre1686ecdtiu(o23,cne24sd.) andbPyotion,ldeybutnfiamostgm-experimentalic recrstudiesystaonllpoly-izati(o15n). Here we present results on the experi- physical Union, Washington, DC, 1987), vol. 16, pp. was the subject of much discussion in the crystalline olivine have been performed un- mental deformationarieofsolivineareaggregatesstraiginht to curved. 111–123. 19th(Fcentury.ig. It2uallowed).reAs LordatresRayleighhsecaarletodstrderinauniaxialinmsul␥compression,- ϳ 0in.5which, adeforma-typictorsional yielding bulk shear strains up to ␥ ϭ estimate molecular dimensions and motivated tion is coaxial and the total amount of strain 5. The aim of Fewthis studydeformationwas to achieve non- features 4. V. Babuska, M. Cara, Seismic Anisotropy in the Earth himdtoeperformformantaiaccurateptiloensdeterminationomf riacnrodofsotmr(equivalentucdtiustrrestrainsi- diϽs0.5)plaisylimiteds e(v9–12id).encocaxeial deformation and to investigate the de- the surface tension of water (25), for exam- Olivine samples deformed to shear strains up tailed microstrusuchctural, textuasral, anlamellaed rheological and sub- (Kluwer, Dordrecht, 1991). ple, and now reappearsbutioinnthe(mcontext.r.dof). Texture evolution of polycrystalline olivine over a larg- 5. D. Mainprice, P. G. Silver, Phys. Earth Planet. Inter. 78, solidsofonithencnanoscale.ipien(iii)t rControlecryofsthetalliGeologischeszationInstitut,(FETH-Zentrum,ig. 2, C8092 aZu¨nrich,d Der s)tr.ain range grainsthan previously areaccessible.visible within island motion presentsindtheicpossibilityes arofema-indSwitzerland.icated Olivine aggregates were hot-pressed from 257 (1993). nipulationAt of␥surfaceϳ alloy2,formationcoreto-createand-m*To awhomntcorrespondencele strucshouldturbeeaddressed.s suE-ggeSanstCarlos olivinethepowdersgrains.(16). Deforma-The micro- useful and novel nanoscalefor thstructures.e overall tmail:[email protected] tion experiments were carried out in a high- 6. W. Ben Ismail, D. Mainprice, Tectonophysics 296, 145 recrysta(lJl)izanatdioenacbhyposulebfiggrauirne rotation (Fig. 2, E graph in (A) was taken us- (1998). 1564 24 NOVEMBER 2000 VOL 290 SCIENCE www.sciencemag.org ing a gypsum plate. (C 7. M. K. Savage, Rev. Geophys. 37, 65 (1999). and F)s.eTpahreatmelyatr(iJxa, wJbr,apJcs). around porphyro- clasts aMnidnimreasemanbdlesmathxeimma osaic texture de- and D) Deformed proto- 8. K. M. Fischer, E. M. Parmentier, A. R. Stine, E. R. Wolf, Downloaded from lith. Deformation micro- J. Geophys. Res. 105, 16181 (2000). scribedairne innadtiucarateldpteoritdhoetilteefst (21). With further and right of measured structure at shear strain 9. H. G. Ave´ Lallemant, N. L. Carter, Geol. Soc. Am. Bull. 81, 2203 (1970). strain, ptheole porphyroclastsfigures, respectivebecome- more elon- ␥ ϳ 0.5. Evidence of dis- gated, with an oblique shape fabric consistent location creep and recov- 10. P. N. Chopra, M. S. Paterson, in The Effect of Defor- ly. Seismic properties mation on Rocks, G. S. Lister, H. J. Behr, K. Weber, with thewesensere derandived amountfrom theof shear. At ␥ ϳ 5, ery in the form of defor- J. J. Zwart, Eds. (Elsevier, Amsterdam, Netherlands, recrystallizationLPO data isunearlysing thcompletee and a flu- mation lamellae and 3 to 1981), vol. 78, pp. 453–473. idal mosaicVoigt microstructurevolume averag-(21) with a strong 4 ␮m subgrains. Grain 11. P. N. Chopra, M. S. Paterson, J. Geophys. Res. 89, ing scheme [programs boundaries are curved to 7861 (1984). foliationfromsub-parallel(32, 33)]. to the shear plane has lobate, with bulges indi- 12. S. I. Karato, M. S. Paterson, J. D. FitzGerald, J. Geo- developed (Fig. 2, G and H). The ϳ5% of cating the onset of recrys- phys. Res. 91, 8151 (1986). remaining porphyroclasts consist of ribbons tallization. A weak oblique 13. S. Q. Zhang, S. Karato, Nature 375, 774 (1995). 14. ࿜࿜࿜࿜ , J. FitzGerald, U. H. Faul, Y. Zhou, Tectono- with highly stretched tails (blue grains) and shape preferred orientation physics 316, 133 (2000). asymmetric porphyroclasts showing sub- is consistent with the sense of shear. (E and F) Protomylonite. Partially recrystallized microstructure grains and deformation features (white at ␥ ϳ 2. Recrystallized grain size is 3 to 4 ␮m. Elongated porphyroclasts with deformation lamellae 1566 24 NOVEMBERhave aspect2000ratiosVOLR fro290m 2 toSCIENCE6 (R ϭ 5.8www.sciencemag.orgfor the finite strain ellipse at ␥ ϭ 2). Core-and-mantle grains). High-resolution orientation imaging structures and a subgrain size similar to the recrystallized grain size provide evidence for maps are consistent with the observed micro- recrystallization by subgrain rotation. (G and H) Ultramylonite. Fluidal mosaic microstructure at structures (Fig. 3). The spatial distribution of ␥ ϳ 5. The recrystallized matrix (ϳ95% volume) is very homogeneous with ϳ3 ␮m equant grains. small-angle misorientations within the clasts Two types of porphyroclasts with distinct crystallographic orientations remain. Highly elongated confirms the formation of subgrains with a ribbon grains (blue) are in an orientation for easy slip on (010)[100] and track the bulk strain. Their size similar to the recrystallized grains, pro- aspect ratios (R ϭ 25 to 40) are consistent with the finite strain ellipse at ␥ ϭ 5 (R ϭ 27). A second type of porphyroclasts (white) with lower aspect ratios (R Ͻ 10) are full of subgrains and viding further evidence for recrystallization deformation features, are strongly asymmetric in shape, and have an oblique lattice orientation. by subgrain rotation. Both types have average grain areas equal to those of the starting grains (equivalent diameter, ϳ20 LPOs were measured using electron back- ␮m), suggesting that their shapes are due entirely to strain without major coalescence.

www.sciencemag.org SCIENCE VOL 290 24 NOVEMBER 2000 1565 The principles of electron back-scatter diffraction (EBSD) pattern analysis and lattice preferred orientations (LPO)

b

a c ρ(σ) σ2 (14) ∝ ρ(σ) σ2 (14) ∝ ε˙ σ3+ (15) ∝ ε˙ σ3+ (15) ∝ 5. Grain Growth2 Grain size evolution 2 Grain size evolution small grains get consumed faster. static grain growth is driv2en. 1by the Grain growth reduction of surface energy of 2.1 Grain growth grain boundaries (proportional to the curvature, that locally drives diffusion. dn dn = kt (16) − 0 dn dn = kt (16) − 0 n = 2+ n = 2+ 3 Piezometers 3 Piezometers

in reality, muck and second phases slow grain growth...

3

3 dislocation production

motion: glide climb

recrystallization: grain sub-grain rotation GB - migration growth

low T high T high stress low stress 1. REVISIT Intro: Observations Experimentally deformed: Naturally deformed in, nss) gime 1 w stra e o R (l gime 2 e R gime 3 e R

from: “Atlas” Dislocation creep regimes in quartz aggregates 147

1400 regimes determined by relative rates of

1. dislocation production 2. dislocation climb i R.,ml j : 3. grain boundary I- / J Regime 1 migration 10 -8 1; "7 1; -6 1 ; - s 1 .4 .8 1 ; - 7 1; .8 1 ; - s 1, -4 Strain Rate (l/s) Strain rate (l/s) these rates are a function Fig. 2. Plots of temperature vs strain rate showing the location of the dislocation creep regimes for quartz aggregates deformed (a) 'as-is' and (b) with 0.17 wt% water added. The boundaries between the regimes are gradational. Open circles of stress, temperature and represent regime 1, plus symbolsrepresent regime 2, open squares represent regime 3, and a plus inside a square represents water content gradational between regimes 2 and 3.

depends on the development of a steady state micro- 1000 structure. Since the axial shortening geometry of the experiments limits the amount of strain and thus the a 0 0 0 0 aoo. ¢P o W-377 (quartzite) amount of recrystallization possible, we also describe microstructures from samples of the fine-grained nova- 600- .oO o 0 R1 (high stress, lowc uT):lite t o illustrate tR2he p r(loocewsseers w hstressich we b,e higherlieve woul d R3 (low stress, high T): occur in quartzite at higher strains. 400- production fast, recovery T): recovery by climb recover• y by climbCQ- 8fast,3 (novaculite) by climb and GBM slow moderate, but GBM but Grain200. O Boundary Microstructural observations and interpretations slow, so recrystallization Migi . ration is also fast, so 0 04 l I I I I Regime 1. At looccurwer temsp ebraytu subgres andrain faste r strain complete0 10 20 30 40 50 60 rates microstructurrotational observat i(requiringons suggest th at dislo- recr1ystallization000. occurs cation climb is diffclimb)icult and that recovery is accommo- by both bsubg rain rotation dated by grain boundary migration recrystallization. 800. Quartzite samples deformed in this regime show the and GBM. (see relatively highest strengths, and they are the only ones to undergo unifor| . . m grain size) appreciable strain weakening (Fig. 3a). However, nova- CQ- 94 (novaculite) ,o0 culite samples deformed in regime 1 show little strain

weakening (Fig. 3a). e + Microstructures from quartzite samples shortened w-339 (quartzite) D 20% show that dislocation climb is not able to accommo- 0 date recovery in regime 1. T E M observations from the o . cores of original grains show high densities of tangled dislocations (up to -1016 m -2) which commonly have straight segments and show no evidence for the develop- | c ment of subgrain boundaries (Fig. 4a). At the optical :1 scale these grains exhibit irregular and patchy undula- tory extinction (Fig. 4b). The high densities of tangled dislocations and the absence of subgrain boundaries 400- suggest that within this regime the rate of dislocation i =,=: . . . CQ-84 (novaculite) production is greater than the rate of recovery by dislo- 2 0 0 - ° B ° B o B o o [] cation climb. J C0-82 (quartzite) a 0~ , , , , , The onset of strain weakening shown by the quartzite 0 10 20 30 40 50 60 at - 2 0 % strain (Fig. 3a) corresponds to the point where Axial Strain (%) recrystallization initiates at the grain boundaries. At the Fig. 3. Differential stress vs axial strain curves for quartzite and optical scale the recrystallized grains are too small to be novaculite samples deformed in (a) regime 1, (b) regime 2 and (c) resolved, and the grain boundaries appear diffuse (see regime 3. Curves are only shown out to 45% shortening. (a) W-377, arrow in Fig. 4b). TEM observations show that the open circles: Heavitree quartzite (700°C, 10 6 s-l, 1.5 GPa, 'as-is'). CQ-83, filled circles: novaculite (700°C, 10 5 s-l, 1.5 GPa, 0.17 wt% mechanism of recrystallization in this regime is strain- water added). (b) W-339, pluses: Heavitree quartzite (800°C, induced grain boundary migration (Fig. 4c). We inter- 10 -6 S- 1, 1.5 GPa, 'as-is'). CQ-94, filled circles: novaculite (700°C, pret the microstructure shown in Fig. 4(c) to indicate 10- 6 s- l, 1.5 GPa, 0.17 wt% water added). (c) CQ-82, open squares: Black Hills quartzite (900°C, 10-6s 1, 1.5GPa, 0.17wt% water that the grains with a low dislocation density (marked added). CQ-84, filled squares: novaculite (900°C, 10 -6 S- l , 1.5 GPa, with asterisks) were migrating into the work-hardened 0.17 wt% water added). dislocation production

motion: glide climb

recovery+ recrystallization: sub-grain rotation GB - migration

low T high T high stress low stress from: “Atlas” Summary

1. dislocations move through crystals to reduce elastic strain energy and cause deformation

2. each one causes only very small displacement... so many must be produced and move through the crystal to cause strain. The production is caused by stress acting on a “Frank-Read Source”.

3. The motion of dislocations depends on the anisotropy of the lattice, and the orientation of the lattice relative to the stress.

4. The strain rate is a non-linear function of the stress, due to the rate of production, the density and the velocity of dislocations. The velocity depends on the ease of climb, which is a thermally activated diffusive process, so dislocation creep has the exp(-Q/RT) dependence.

5. Large strains require recovery, which can be effected by sub-grain rotation, grain boundary migration recrystallization.

6. Regimes defined by microstructures depend on the relative rates of these processes of dislocation production and recovery 6. PIEZOMETERS

1. Empirical Piezometers (dislocations, subgrain and grain size), for olivine and quartz. Independent of T, water content, melt.

2. Theory(?): Why should these be independent of temperature and water?

3. The WATTMETER

Where this is going: Stress across shear zones (Kohlstedt and Weathers) Stress in the lithosphere from xenoliths (Mercier) ρ(σ) σ2 (14) ∝ ε˙ σ3+ (15) ∝ 2 Grain size evolution

2.1 Grain growth

dn dn = kt (16) − 0 n = 2+ and k exp(pieQ/RzometerT ). (stress-meter) is meaningful for ∝ stead− y state deformation (i.e. time scale longer than the transient time). 3 Piezometers in principle and empirical observation, the steady state lengthscale is proportional to the stress This is the general form(ratethat of productionall the empiri = ratec alof piezometersannihilation andfollo self-w. stress = applied stress ) Dislocation density: (17)

+n1 2 σ ρ = A b− (18) 1 µ ! " or Dislocation spacing: (19) σ nd δ = A b − (20) d d µ ! " Subgrain size: (21) σ nsg δ = A b − (22) sg sg µ ! " Grain size: (23) σ ng δ = A b − (24) g g µ ! " There are many complex models for the higher order piezometers, and none are generally ac- cepted. However, a common observation is that the piezometers are independent or only weakly sensitive to Temperature, water content, melt fraction, i.e. all the parameters to which creep is very sensitive. This is a mystery, not well understood.

However, one possible clue for the grain size piezometer comes from a model of Derby & Ashby (1987), which says that the recrystallized grain size has this form (derivation not shown here): v δ = Cε gbm (25) g c ε˙ ! " where C is a slightly temperature sensitive constant, εc is the critical strain for nucleation of re- crystallization, vgbm is the velocity for grain boundary migration, and varepsil˙ on is the strain rate. The implication is that, if the two factors in the ratio are affected similarly by temperature, water

3 Van der Wal et al.: RecrystallizedOlivine Grain Size 1481

10oo i i i i i +0.0004 03)' 1.33+0.09 R =0.96 (1) Dg= 0.015=0.0003

wherethe recrystallized grain size (Dg) is in metersand the stress(0) in MPa. R denotesthe correlationcoefficient. Note thatwe haveexcluded the "dry"data from theregression, in- cluding them would slightly flatten the regressionline. The ß Jd•eimdtmite significanceof this, however, would be questionable,be- ß AnitaBay dtmite causethe "dry"data are poorly constrained due to the limited 0 Cooledunder load samples recrystallizationthat occurred during the experiment. Figure 3 neverthelessshows that the "dry" Anita Bay and ]theim io ' • I I [ I I I ]o 100 dunitedata do not systematicallydeviate from the aboverela- recrystallizedgrain size (gm) tionship. Discussion Fig. 2. Log-Logplot of recrystallizedgrain size against deviatoricstress for/•heim dunite and Anita Bay dunite de- The combineddata of this studyand Karato et al. [1980] formedunder "wet" conditions. Also plotted "Cooled under indicatethat the dynamicrecrystallized grain size in olivineis load"tests which showno significantdeviation from the trend definedby the "wet"data, which suggests that, during final controlledmainly by flow stressand is largelyindependent of water content and temperaturewith a small dependenceon cooling,no significantstatic growth of therecrystallized flow law in the stressrange 30-300 MPa. The data do not grainsoccurred. Errors in stressmeasurement are around1 MPa. extend to even lower stresses, so it can not be evaluated if there is a lower stress-limitfor recrystallization[Twiss and and strain-ratestep tests show a well definedrelationship be- Sellers,1978]. The high temperaturesin the Earth'smantle, tween recrystallizedgrain size and stress.The data show a however, would favour recrystallizationeven at low stress levels. 2 tendencyfor the grain sizein Anita Bay duniteto be slightly ρ(σ) σ (14) smallerthan in ]theim at the samestress level. It is clearthat The∝stress - grain size relationship measured for/•heim and the cooledunder load sampleshave similar grain sizesto ε˙Anitaσ3+ Bay dunitesapplies not only to sampleswith a constant (15) thosefrom the othertests deformed at similarstress (Figure strain-rate∝ history but also to sampleswith a more complex 2). This suggeststhat static recrystallization has no significant multiple strain-ratehistory. Most of our strain-ratestep tests involved a decrease of strain-rate. Similar results were ob- effecton the2 grainGrsize aiin thesen siexperiments. ze evol ution Figure3 showsthe resultsfrom the "wet"samples plotted tainedby Rosset al. [1980] for stepswith both increasing as a singlegroup, the "dry"single crystal data of Karatoet al. anddecreasing strain-rate. This suggeststhat the dynamic re- [ 1980] which2.1were Gr obtainedain atg r 1650owthø C on dry recrystallized crystallizedgrain size can in manycases adjust to a changeof GEOPHYSICALsingleRESEARCH crystals,and LETTERS, the limited VOL. 20, data NO. obtained14, PAGESfrom 1479-1482, our JULYdry 23, 1993flow stressrather rapidly, i.e. on laboratorytime-scales, pro- samples.It is clear that all of the availabledata for recrystal- vided that enoughstrain has been achieved (>3% shortening lized grain sizes from this study, and from other studies in ourdn experiments).dn = kt (16) RELATIONSHIPS BETWEEN DYNAMICALLY RECRYSTAI•IZED GRAIN SIZE AND All of the0 experimentalolivine materials [Zeuch and Green, DEFORMATIONwherethe CONDITIONSstresses are INaccuraiely EXPERIMENTALLYknown, DEFORMEDfall along OLIVINEthe same ROCKS − trendlargely independent of temperatureand water content. 1984; Chopra and Paterson, 1984; Bus•od and Christie, WeDirkhave Van der combinedn Wal = 1,2, 2+Prame theand "wet"Chopra k Anita3, Martyn exp(Bay Drury andQ/R 2 and]theim TJohn ).Fitz data Gerald with 2 1991], includingthe "dry" singlecrystals [S-I. Karato,pets. comm.], contain a very small to small volume fraction of GEOPHYSICALthat of KaratoRESEARCH et LETTERS,al. [ 1980] VOL. and∝ 20, findNO. 14, that− PAGES the1479-1482, combinedJULY data 23, 1993 Abstract. New microstructuralcanbe statisticallydata on experimentallydescribed by where theterriperaturesfollowing arestress lower -and grainwater is probably melt.present As farin as we are aware there are no data available on re- deformed"wet" and "dry" natural olivine rocks (Anita Bay the olivine grainsor in a fluid phase[Bell crystallizedand Rossman,grain sizesin melt free olivine rocks. In conse- and/[heimdunitc), together sizewithrelationship: the other reliable experimen- 1992]. The resultsof Rosset al. [1980] suggestthat the re- RELATIONSHIPS3 BETWEENPi ezomDYNAMICALLY eterRECRYSTAI•IZEDs GRAIN SIZE AND quence,it is possiblethat stress- grainsize relationships in tal data,indicate that the DEFORMATIONexperimental stress-CONDITIONS recrystallized IN EXPERIMENTALLY crystallized DEFORMEDgrain size OLIVINEmay be ROCKS sensitive to watercontent and grainsize relationship in olivine-rocks is largely independent recenttheoretical models of dynamicrecrystallization melt-free[Derby materials could be different to those in materials of watercontent and temperature, DirkandVan isder onlyWal slighfiy1,2, Prame depen- Chopra 3, Martyn andAshby, Drury 2 and1987; John Drury, Fitz Gerald 1992]2 imply thatthe containingrecrystallized melt. denton the flow properties of thematerial. ThisThe isexperimental the general grainformsize could thatalso be alltemperature the empiridependent. cal piezometers follow. The melt contentand the watercontent of the polycrystal datacover Abstract.a stressrange Newof microstructural 30-300MPa, data water on contents experimentallyfrom whereWeterriperatures have studiedarelower the microstructuresand water is probably in apresent suiteinof experi- <30 ppmdeformedto 300"wet" ppm, and and "dry" temperatures natural olivine in the rocks range (Anita 1100- Bay thementally olivine deformedgrains or in"wet" a fluid and phase "dry"[Bell olivine and Rossman,rocks, mightto be deter- expected to influencethe stress- grainsize relation- 1650øC. and/[heimIxx:al melt dunitc),contents together of withupto the 10other volume% reliable experimen-cannot be 1992]. mineß Theif there resultsis anyof Rossdependence et al. [1980]of thesuggest dynamicthat ship therecrystallized re-because the presenceand amount of bothmelt andwater demonstratedtal data,toindicate have a that significant the experimental effect on stress-the stress recrystallized - grain crystallizedgrainsize grainon temperaturesize may be sensitiveand water to watercontent. Dislocontent These and cation experi- density: (17) grainsize relationship in olivine-rocks is largely independent recenttheoretical models of dynamicrecrystallization will[Derby affect the kineticsof the basicprocesses involved in re- sizerelationship. mentsextend the availablequantitative data to a muchwider +n1 of watercontent and •temperature, lOOand is onlyslighfiy depen- andrangeAshby, of conditions1987; ßDrury, than 1992]has imply previously thatthe beenrecrystallized available. crystallization. 2 σ Recent theoretical models [Derby and Ashby, denton the flow properties Introductionof the material. The experimental grainsize could also be temperature dependent. ρ = A1b− (18) datacover a stressrange of 30-300MPa, watercontents from We have studiedthe microstructuresin a suiteof experi-1987] suggestthat the stress- grain size relationshiparises ExperimentalMethods µ At high<30 temperatures ppmto 300 ppm,and- and pressurestemperatures in thein Earth, the range minerals1100- mentallydeformed "wet" and "dry"olivine rocks, to because deter- of ! a dynamic" balance between grain size reduction 1650øC. Ixx:al melt contents of upto 10volume% cannot be mineif thereis anydependence of the dynamicrecrystallized and rocksbecome ductile •and can flow in responseO Thisto devia-study, "wet" or Dislocation spacing: (19) demonstratedto have a significanteffect on the stress - grain grainTwosize on natural temperatureolivine and rocks water from/l•heimcontent. These inprocessesexperi- Norway and and graingrowth processes. Following Derby and toric stress. The mechanisms• involved are theß same Karato as in duc-et al. [1980] ("dry") • sizerelationship. . This study, "dry"mentsAnita extend Bay inthe New available Zealandquantitative have been data tostudied. a muchBothwider rocks are nd tile metals and ceramics where deformation is accommodated rangeof conditionsthan has previously been available. Ashby [ 1987]σ and− Drury [ 1992], the dynamicrecrystallized Introduction hydrateddunites with initial olivine grain sizes of 900gm i i! i i i i δ = A b (20) by the motionof crystaldefects such 10 as vacancies, i and •, dislo- • i• I (Aheim)i ' and' ' 100 gm'1 (Anita Bay). Bothrocks d graincontain sized 5-10% (Dg) canbe described by thefollowing relationship cations[Poirier, 1985]. When large permanentstrains are ExperimentalMethods µ At high temperaturesand pressures1 in the Earth, minerals10 orthopyroxeneand 100 5-10% hydratedphases such as chlorite,! " producedandby rocks thebecome motionductile of dislocationsand can flow(dislocation in responseto creep), devia- C E Vgbm recrystallizedtalcTwo andnaturalgrain phlogopite. size olivine (gm) Therocks deformationfrom/l•heim experimentsin NorwaySubgrainand were con-size: (21) microstructuraltoric stress.processes The mechanismssuch as involved dynamic are therecrystallization same as in duc- Anitaducted Bayin in a New Paterson Zealandgas have apparatus been studied. [Paterson, Both rocks1970] are at confin- Dg - (2) oftenaccompany tile metals anddeformation ceramics where[Pokier, deformation1985]. is accommodated nsg Fig. 3. As Figure2, includingthehydrateding recrystallized pressures dunites ofwith 300 initial MPa,grain olivine temperaturessize grain sizes of of1100-1400 900gm øC and σ − Studiesby the on motion metalsof crystaland rocks,defects deformedsuch as vacancies by dislocationand dislo- (Aheim)strain-rates and 100of gm 10 -4(Anita to 10 Bay).-6 s-1 Both (Table rocks 1).contain Steady5-10% state creep cations[Poirier, 1985]. When large permanentstrains are δsgwhere= A Csg isb a slightlytemperature dependent constant, E is a (22) Van der Walcreep, et al.:have Recrystallized found that thedata grainOlivineof size Karato formed Grainet by al. Size dynamic [ 1980] re- and orthopyroxenethe was"dry" achieveddata and during 5-10%of this the hydrated experimentsstudy. 1481phases such reported as chlorite,here. Stress- µ crystallizationproducedis by apparently the motion ofcontrolled dislocationsby (dislocation the flow creep), stress talc and phlogopite.The deformationexperiments were con- microstructuralprocesses Notethatsuch as a dynamic continuousrecrystallization band of datastraincan curves be recognized for tests4265 and 4267 (Table critical1) are shownstrain in! for " nucleationof recrystallization,Vgbm is the [Twiss, 1977]. This dependencehas provideda methodof ductedin a Patersongas apparatus [Paterson, 1970] at confin- oftenaccompany deformation [Pokier, 1985]. ingFigure pressures3 ofof Chopra 300 MPa,and temperaturesPaterson [1981of 1100-1400]. GraingrainøC andboundarysize: migration rate and k is the strain-rate.Models (23) 10oo i suggestiveof onesingle relationship between recrystallized i i i i estimatingStudies the onflow metals stresses and rocks, responsible deformed by for dislocation natural "Wet" experimentswere conductedon the as-received +0.0004 strain-ratesof 10-4 to 10-6 s-1 (Table 1). Steadystate creep ng deformationscreep,fromhave found microstructuralgrain that the grainsizeinformation. size and formed stress.Recrystallizedby dynamic re- of thistype σcan account for theindependence of recrystallized Dg= 0.015 03)' 1.33wasrocks.+0.09 achievedBreakdown R during =0.96 theof experimentshydrous (1)minerals reported occurs here. Stress-on heating so − stress- crystallizationgrain sizerelationships is apparentlyhave controlled been obtained by =0.0003 the flowfrom stress ex- strainthe wetcurves experiments for tests4265 were and 4267conducted (Table 1)under areδg shownwater-saturated = A in gb (24) [Twiss, 1977]. This dependencehas provideda methodof perimentaldeformation studies on dry recrystallizedsingle Figureconditions,3 of Choprai.e., inand thePaterson presence[1981 of ]. a free fluid. Infraredspec- µ crystalsestimatingof olivine inthe the flow range stressesof 1550- responsible 1650 øC for [Karato naturalet "Wet" experimentswere conductedon the as-received ! " deformationsfrom microstructuralinformation. Recrystallized troscopyshows that the "wet" samplescontain about 200-300 al., 1980] andon a varietyof olivinerocks at lower tempera- rocks.Breakdown of hydrousminerals occurs on heatingso stress- grain sizewhererelationships the haverecrystallized been obtained from grain ex- size ppm(Dg) (by is weight) in meters of waterand whichthe is partitionedbetween (i) a turesof 1100-1300 øC with a rangeof waterThere contentsare[Post, manthe wety experimentscomplexwere moconducteddelsunder forwater-saturated the higher order piezometers, and none are generally ac- perimentaldeformation stressstudies (0) inon MPa.dry recrystallized R denotessingle the correlationconditions,water-richi.e., fluid coefficient. in the and presence (ii) a ofrestricted Note a free fluid.intergranular Infraredspec- melt, and 1977; Mercier et al., 1977; Ross et al., 1980; Zeuch and crystalsof olivinethat in thewe range haveof 1550-excluded 1650 øC [Karatothe "dry"et data troscopy(iii) from withinshows the olivinethat regression, the "wet"and orthopyroxene samplesin- contain about grains 200-300 [Chopra and Green, 1984].al., 1980] The and workon a varietyon olivineof olivine rockscepted.rocks has beenat lowerconducted Ho tempera-wev er,ppmPaterson,a (by commonweight)1984]. of water"Dry" which observexperiments is partitionedationwere between conductedis (i) that a on sam-the piezometers are independent or only weakly usingolder tures solid-mediumof 1100-1300cluding øCdeformation with a them rangeof apparatus would water contents slightlywhich, [Post, as flatten water-richples thepredried regression fluid atand 1200 (ii) aøC line.restricted for 60 The hoursintergranular at an oxygenmelt, andfugacity of recently1977;suggested Mercier[Green etsignificance al., and 1977;Borch, Ross et1989], of al., this,1980;may however,Zeuch have sys- and would(iii) within be olivine questionable, and orthopyroxenebe- grains [Chopra and ß Jd•eimdtmite Green, 1984]. The work on olivinesensitiv rockshas beeneconducted to Temp 10erature,-5 Pa. Water contentsw aterincon the dried tensamples t, meltare less fraction,than 30 i.e. all the parameters to which creep is very tematicallyoverestimated the stresslevels by a factor of Paterson,1984]. "Dry" experimentswere conductedon sam- usingolder solid-mediumcausethedeformation "dry"apparatusdata are which, poorly as constrained ppm (by weight)due to [Chopra the limitedand Paterson, 1984]. Estimatesof ß AnitaBay dtmite somewherebetween 3 and7. Thus,the quantitative aspects of plespredried at 1200øC for 60 hoursat an oxygenfugacity of recentlysuggested [Green and Borch, 1989], may havesys- 10the-5 Pa. waterWater content contents in in the the upperdried samples mantle arerange less than from 30 50-1000 0 Cooledunder load samples the stress- grain size relationshipsrecrystallizationobtainedsensitiv thatfrom occurred e. theThis olderduring is a the mysteryexperiment. , notFigure well3 understood. tematicallyoverestimated neverthelessthe stressshowslevels by thata factorthe of "dry" ppmppm (byAnita [Bell weight) andBay [ChopraRossman, and and ]theim1992]. Paterson, 1984]. Estimatesof io ' • I I [ I I I solid-mediumsomewhereexperiments between 3 must and 7. be Thus, treatedthe quantitative with someaspectscaution. of the waterRheologicalcontent inresults the upperfor somemantle of range thesamples from 50-1000have been re- The stressthe stress- grain- grainsize sizerelationship relationshipsobtained obtained atfrom higher the oldertem- ]o 100 dunitedata do not systematicallydeviateppmported [Bell by andfrom ChopraRossman, theand above1992]. Paterson rela- [ 1981, 1984]. Additional ex- solid-mediumexperiments must be treatedwith somecaution. recrystallizedgrain size (gm) peraturesfor dry recrystallizedtionship.olivine single crystals [Karato perimentsRheologicalreportedresults for here some(Table of thesamples 1) includehave (i)been are- number of et al., 1980]The is stressmore- reliable, grainsize butrelationship thereis someobtained doubt at higherabout tem- its portedby Chopraand Paterson [ 1981, 1984]. Additional ex- peraturesfor dry recrystallizedolivine single Hocrystals wev[KaratoDiscussioner, one Anita possibleBay samplescluewhich forwere thecooled grainunder load sizein orderpiezometerto comes from a model of Derby & Ashby Fig. 2. Log-Logplot of recrystallizedgrain size againstapplicability to natural deformationsin the upper mantle perimentsassessthe reported effect ofhere static(Table recovery 1) includeand (i)recrystallization a numberof on the et al., 1980] is more reliable, but thereis somedoubt about its AnitaBay sampleswhich were cooled under load in orderto deviatoricstress for/•heim dunite and Anita Bay dunite de-applicability to natural deformations(1987),in the upperwhic mantleh saysassessdeformationthatthe effect themicrostincturesof static recrystallizedrecovery andand recrystallization (ii) a numbergrainonof/l•heim the sizeand has this form (derivation not shown here): The combineddata of this studyand Karato et al. [1980] formedunder "wet" conditions. Also plotted "Cooled 1Faculty under of EarthSciences, Utrecht University deformationAnita Baymicrostinctures samples with and either(ii) aa number simpleof/l•heimconstant and strain-rate indicatethat the dynamicrecrystallized Anitahistory Baygrain or samples a multiplesize with in either strain-rateolivine a simpleis "step" constant deformation strain-rate history. load"tests which showno significantdeviation from 2 RSES,the trend1AustralianFaculty of EarthNational Sciences, University Utrecht University 3Australian 2 RSES,GeologicalAustralian controlledSurveyNational OrganizationUniversity mainly by flow stressand historyAfteris largely ormost a multipleexperiments independent strain-rate the "step" specimensof deformation were history. unloaded and v gbm definedby the "wet"data, which suggests that, during final 3Australian Geological Survey Organization Afterthen mostcooled experiments at a rateof the 1-2 specimens øCper second.were unloadedδThe g = samplesand C εwerec (25) Copyright1993 by thewaterAmerican contentGeophysical and temperatureUnion. with thenat temperatures cooled a smallat a rate dependenceaboveof 1-21000 øC per øC second.on for onlyThe samples 1-3 minuteswere after the ε˙ cooling,no significantstatic growth of therecrystallized at temperaturesabove 1000 øC for only 1-3 minutesafter the Copyright1993 by flow theAmerican law inGeophysical the stressUnion. range 30-300load MPa.was removed The datathus limiting do notthe extent of staticannealing ! " grainsoccurred. Errors in stressmeasurement are around1 loadwas removed thus limiting the extent of staticannealing Paper numberPaper number93GL01382 93GL01382 extend to even lower stresses, soof ofit the thecan microstructures. microstructures. not be evaluatedSamples Samples cooled if cooled under load under were load were MPa. where C is a slightly temperature sensitive constant, εc is the critical strain for nucleation of re- 0094-8534/93/930094-8534/93/93 GL-01382503.00 thereGL-01382503.00 is a lower stress-limitfor recrystallizationcooledcooledat at a similar a similar rate [Twissrate to the to others the and othersbut werebut subjected were subjectedto a to a 14791479 and strain-ratestep tests show a well definedrelationship be- Sellers,1978]. crystallization, The high temperaturesvgbm inis thethe Earth'svelomantle,c ity for grain boundary migration, and varepsil˙ on is the strain rate. tween recrystallizedgrain size and stress.The data show a however, would favour recrystallizationeven at low stress levels. tendencyfor the grain sizein Anita Bay duniteto be slightly The implication is that, if the two factors in the ratio are affected similarly by temperature, water smallerthan in ]theim at the samestress level. It is clearthat Thestress - grain size relationship measured for/•heim and the cooledunder load sampleshave similar grain sizesto Anita Bay dunitesapplies not only to sampleswith a constant thosefrom the othertests deformed at similarstress (Figure strain-ratehistory but also to sampleswith a more complex 3 2). This suggeststhat static recrystallization has no significant multiple strain-ratehistory. Most of our strain-ratestep tests involved a decrease of strain-rate. Similar results were ob- effecton thegrain size in theseexperiments. Figure3 showsthe resultsfrom the "wet"samples plotted tainedby Rosset al. [1980] for stepswith both increasing as a singlegroup, the "dry"single crystal data of Karatoet al. anddecreasing strain-rate. This suggeststhat the dynamic re- [ 1980] whichwere obtainedat 1650ø C on dry recrystallized crystallizedgrain size can in manycases adjust to a changeof single crystals,and the limited data obtainedfrom our dry flow stressrather rapidly, i.e. on laboratorytime-scales, pro- samples.It is clear that all of the availabledata for recrystal- vided that enoughstrain has been achieved (>3% shortening lized grain sizes from this study, and from other studies in our experiments). wherethe stresses are accuraiely known, fall alongthe same All of the experimentalolivine materials [Zeuch and Green, trendlargely independent of temperatureand water content. 1984; Chopra and Paterson, 1984; Bus•od and Christie, Wehave combined the "wet" Anita Bay and ]theim data with 1991], includingthe "dry" singlecrystals [S-I. Karato,pets. that of Karato et al. [ 1980] and find that the combineddata comm.], contain a very small to small volume fraction of canbe statisticallydescribed by thefollowing stress - grain melt. As far as we are aware there are no data available on re- sizerelationship: crystallizedgrain sizesin melt free olivine rocks. In conse- quence,it is possiblethat stress- grainsize relationships in melt-free materials could be different to those in materials containingmelt. The melt contentand the watercontent of the polycrystal mightbe expectedto influencethe stress- grainsize relation- ß shipbecause the presenceand amount of bothmelt andwater will affect the kineticsof the basicprocesses involved in re- • lOO ß crystallization.Recent theoretical models [Derby and Ashby, 1987] suggestthat the stress- grain size relationshiparises -• O Thisstudy, "wet" becauseof a dynamicbalance between grain size reduction • ß Karatoetal. [1980] ("dry") • processesand graingrowth processes. Following Derby and . This study, "dry" Ashby[ 1987] andDrury [ 1992], the dynamicrecrystallized i i! i i i i 10 , i •, • i• I i ' ' ' '1 grainsize (Dg) canbe described by thefollowing relationship 1 10 100 C E Vgbm recrystallizedgrain size (gm) Dg - (2) Fig. 3. As Figure2, includingthe recrystallized grain size dataof Karatoet al. [ 1980]and the "dry" data of thisstudy. where C is a slightlytemperature dependent constant, E is a Notethat a continuousband of datacan be recognized criticalstrain for nucleationof recrystallization,Vgbm is the suggestiveof onesingle relationship between recrystallized grainboundary migration rate and k is the strain-rate.Models grain size and stress. of thistype can account for theindependence of recrystallized the transient time is related to the lengthscale of the microstructural property, ie dislocation density < subgrain size < grain size.

an example of transients:

for dislocation density: olivine: 1% strain quartz: 3% strain

for grain size, closer to 100% strain

Karato, p 249 (idea from Goetze, 1975; Weathers et al., 1979) low T (~1000 C): piezometer is high T (~1000 C): piezometer is near the field boundary far into dislocation creep field (at geologic strain rates) (at geologic strain rates)

Olivine, P=300 MPa, T=1015 C Olivine, P=300 MPa, T=1308 C !8 !8 !8 !8 !4 !4 !4 !4

2 2 !10 !10 !10 !6 !6 !6 !6 !10

8 1.5 ! 1.5 !12 !12 !12 !8 !8 !8 8 6 ! 1 1 ! !14 !10 !10 !10 10 14 !14 ! 12 ! , stress (MPa) 0.5 ! , stress (MPa) 0.5 ! ! !16 !12

log !12 log 12 0 !16 0 !

8 ! 10 ! !0.5 !0.5 !14 !14 16 ! 14 !18 ! 18 12 ! ! !1 !1 1 1.5 2 2.5 3 3.5 4 1 1.5 2 2.5 3 3.5 4 log d, grain size (microns) log d, grain size (microns) ρ(σ) σ2 (14) ∝ ε˙ σ3+ (15) ∝ 2 Grain size evolution

2.1 Grain growth ρ(σ) σ2 (14) ∝ 2 n n 2 ρ(σ) σ d d = kt ρ(σ) σε˙ σ3+(14) (16) (14) (15) ∝ 0 ∝ 2 − ∝ ρ(σ) σ ε˙ σ3+ (14)ε˙ σ3+ (15) (15) ∝ ∝ ∝ n = 2+ andε˙ kσ3+ exp( Q/RT ).2 Grain size evoluti(15)on ∝ ∝ − 2 Grain size evolution 2 Gr2ai.1n siGrzeainevgolroutiwthon 2 Grain size2.1evolGrutiain3ongroPiwthezometers 2.1 Grain growth n n 2.1 Grain growth d d0 = kt (16) This is the general form that all the empirical piezometers follow. − dn dn = kt dn dn = (16)kt (16) − 0 − 0 n n n = 2+ and k exp( Q/RT ). d d0 = kt ∝ − (16) n = 2+ and k exp( Q/RT ).− n = 2+ and k exp(DisloQ/RcationT ). density: (17) ∝ − ∝ − +n1 n = 2+ and k exp( Q/RT ). 3 Piezometer2s σ ρ = A b− (18) ∝ − 1 µ 3 Piezometers 3 Piezometers ! " This is the general form that all the empirical piezometers follow. 3 Piezometers or Dislocation spacing: (19) n This is the general form that all the empiriThiscalis piezometersthe general formfollow.that allσthe−empirid cal piezometers follow. δd = Adb Dislocation density: (20) (17) This is the general form that all the empirical piezometers follow. µ +n1 Dislocation density: ! " Dislocation densit(17)y:2 σ (17) Subgrain size: ρ = A1b− (21) (18) σ +n1 σ +n1 µ Dislocation density: 2 (17)nsg 2 ! " ρ = A1b− σ − ρ = A1b−or Dislo(18)cation spacing: (18) (19) +n1 µ δsg = Asgb µ (22) 2 σ ! " ! " nd ρ = A1b− or Dislocation spacing: µ (18)or Dislocation spacing:(19)σ − (19) µ ! " δd = Adb (20) ! " nd Grain size: nd µ (23) or Dislocation spacing: σ − (19) σ − ! " δd = Adb ng δd = AdbSubgrain(20)size: (20) (21) nd µ σ − µ σ − ! " ! " n δ = A b δg = Agb (20) σ − sg (24) Paleowattmeter: Literature σ σ d d Subgrain size: Subgrain size: (21) (21) ε˙ Calcite ε˙ Quartz µ δ = A b (22) 2 Paleowattmeter: Inversion 8 =10 8 =10 µ sg sg 10 -20 Protomylonites Mylonites and ultramylonites:! " nsg ! " nsg µ Field boundary -1 σ − σ − Schmid et al., 1980 Mylonites 2 SubgrainTonale fault zonesize: (21) ! " σε˙ σ δsg = Asgb δsg = Asgb (22) (22) Rutter, 1995 6 =10 Ultramylonites 6 ε˙ Grain size: (23) -8 =10 There are many complexnsg modelsµ for the higher order piezometers,µ and none are generally ac- Mylonites and -8 σ − ! " ! " n Diffusion m) σε˙ g

Dislocation creep µ =10 σ ultramylonites: σ δsg = Asgb Grain size: (22)Grain size: (23) − (23) creep 4 -4 Helvetic nappes 4 cepted.ε˙ However, aµ common observation is that the piezometersδgare= Aindepgb endent or only weakly (24) 1 σ =10 10 ε˙ ! " ng ng µ =10 σ -4 σ − σ − ! " 523 K 0 ε˙ Grain size: (23) ln (d) ( =10 σ K 2 -16 2 δg = Agb δg = Agb (24) (24) Stress (MPa) σ sensitivε˙ e to Temperature, water content, melt fraction, i.e. all the parameters to which creep is very ε˙ =10 σ σ K =10 σ ε˙ ˙ ng µ µ 573 4 ε˙ ε There are many complex models for the higher order piezometers, and none are generally ac- =10 0 =10 σ =10− ! " ! " 623 -12 δg = Agb -16 -20 (24) A) Calcite 0 0 sensitive. This is aµmystery, not well understood. 0 673 K Α There areΒ many complex models for theTherehighercepted.areordmaneHorypiezometers,wcomplexever, a commonmoanddels noneforobservtheareationhighergenerallyisordthaterac-piezometers,the piezometersandarenoneindepareendengenerallyt or onlyac- weakly 10 ! " 0 1 2 3 10 10 10 10 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 There are mancepted.y complex1/T (10Ho-3wmoKev-1dels)er, aforcommonthe higherobservordationercepted.piezometers,is thatsensitivHothewevandpiezometerser,e toanoneTcommonempareerature,aregenerallyobservindepwaterationendenac-conisttenorthatt,onlymeltthewpiezometersfraction,eakly i.e. areall theindepparametersendent ortoonlywhicwheaklycreep is very 573 K Paleowattmeter: Literature 2 Field boundary Why the independence from temperature, water content, melt fraction ? 10 Stipp and Tullis, 2003 Figure 4. Grain-size versus temperaturesensitiv relationshipse to Temp forHo sampleserature,wev er,fromwaterone naturalconp shearossibleten zones.t,sensitivmeltclue fraction,eforsensitivto Ttheempi.e.e.grainerature,Thisall thesizeis waparametersatermpiezometerysteryconten, nottot, meltwhicwcomesell hfraction,understocreepfromisoi.e.vd.aerymoall thedel parametersof Derby &to Ashwhicbhycreep is very 673 K cepted. However, a common observation is that the piezometers are independent or only weakly A: Calcite (cf. Herwegh et al., 2005; de Bresser et al., 2002, and references therein). B: Quartz Dislocation creep 773 K (Stipp et al.sensitiv 2002). Contourse to Temp are forerature,sensitiv constante.w σσεaterε˙ Thiscalculated(1987),conis tena usingmt,whicysterymelt (A) Equationh,fraction,notsays w8 thatewithlli.e.understo param-sensitivtheall therecrystallizedoe.d.parametersThis is a mtograinysterywhic,hsizenotcreepwhaseisll understovthisery formod.(derivation not shown here): eters from inversion (Fig. 2B), and (B) Equation 9 using same parameters as in Figure 2C, 873 K assuming deformation is completely accommodated by dislocation creep (β = 1). However, one possible clue for the grain size piezometer comes from a model of Derby & Ashby 1 sensitive. This is a mystery, not well understood. 10 973 K 1073 K vgbm However, one possible clue for16thep grain–1 Hosizewev(1987),piezometerer, onewhicpossibleδcomesh sa=ysCcluefromthatε fortheathemorecrystallizeddelgrainof sizeDerbpiezometerygrain&DeAshrbsizeyb &ycomes hasAshbthisyfrom, 1987forma mo(derivdel ationof(25)Derbnoty &shoAshwnbhere):y Stress (MPa) 1273 K stant (Fig. 2 ). The inverted value for p (4.9) is sig- our paleowattmeter (Kg = 7.0 × 10 µm s ) and g c Diffusion creep ε˙ 1473 K nifi cantly larger than that observed in experiments the paleopiezometric calibration of Van der Wal ! " 3 K However, one p(1987),ossible whicclue hforsatheys thatgrainthesizerecrystallizedpiezometer(1987),comesgrainwhicsizefromhhassaa ysmothisthatdelformoftheDerb(derivrecrystallizedy ation& Ashnotbygrainshownsizehere):has this formv (derivation not shown here): B) Quartz 167 (Evans et al., 2001). However, it is striking that this et al. (1993). The temperature dependence for gbm 0 δg = Cεc (25) 10 0 1 2 3 three-param(1987),eter fi t recwhiconcilesh labsaorysatorythat data ththeat recrystallizedthiswhere material iCs smisgrainall abecslighausizese actlyhastivatiotempthisn enerform- erature(derivsensitivation note shoconstanwn here):t, εc is the critical strainε˙ for nucleation of re- 10 10 10 10 vgbm vgbm ! " are discrepant with any of the empirical piezome- gies for dislocation creep (530 kJ/mol; Hiδrtgh = Cεc δg = Cεc (25) (25) ters (i.e., Equation 1). The elevated p value might and Kohlstedt, 2003) and for grain growth ε˙ ε˙ ˙ 2 C) Olivine crystallization, vgbmv is the velocwhereity forC grainis a slighboundarytly temperaturemigration,sensitivande constanvarepsilt, εonc isisthethecriticalstrainstrainrate.for nucleation of re- K gbm ! " ! " 10 Dislocation creep indicate that β includes a grain-size dependence, (520 kJ/mol; Karatδog, 1=989C) aεrec similar. (25) 873 which perhaps refl ects an increased cowheremponent oCf is aTheslighimplicationtly temperatureis that,εsensitiv˙ whereif etheconstanCtcrystallization,wisoat,factorsslighε istlytheintempvcriticalgthebmeratureisratiothestrainsensitivvareelocforiatyffenectedforucleationconstangrainsimilarlyt,bofoundaryε re-is thebymigration,criticaltemperature,strainand forvarwnepsilaterucleation˙ on is theof re-strain rate. K ! " c c 973 diffusion creep for smaller grain sizes. DATA FROM NATURAL SHEAR ZONES ˙ ˙ 3 K where C is a slighcrystallization,tly temperatureWhenv sgtsensitivbmudyinisg ntheateuraconstanvl selohearc zitoynt,esfor, tεwcograin crystallization,oisf ththee boundarycriticalThe implicationstrainvmigration,gbm isforthenisucleationandvthat,elocviartyifepsilfortheof grainre-ontwoisfactorsbtheoundarystrainin themigration,rate.ratio areandaffectedvarepsilsimilarlyon is thebystemptrainerature,rate. water 1 107 Austin & Evans, 2007 3 K 10 7 EXTRAPOLATION TO CONDITIONS OF most readily available pieces of information are 1 1 Diffusion creep K crystallization, v Theisimplicationthe velocityisforthat,grainif theboundarytwo factorsmigration,Theinimplicationthe ratioand areε˙isisthat,atheffectedstrainif thesimilarly3trate.wo factorsbThey tempin theerature,ratiowareateraffected similarly by temperature, water gbm 73 K NATURAL DEFORMATION grain size and temperature. Average recrystallized 73 Stress (MPa) 12 13 Extrapolation from laboratory conditions to grain size often correlates with inverse tempera- 3

Paleowattmeter: Literature K Field boundary 3 natural deformation of rocks requires knowl- ture, as shown in Figure 4 for mylonites hosted 73 K Van der Wal et al., 1993 15 0 147 3 3 10 edge of the effects of stress, temperature, and within calcite and quartz rocks. For a given strain if those parameters affect 0 1 2 3 10 10 10 10 strain rate. The empirical piezometers (Equa- rate, both power and fl ow str3ess will correlate the factors in the ratio Grain size (µm) tion 1) are temperature and strain rate indepen- with inverse temperature. Recently, de Bresser approximately equally, the Figure 3. Deformation mechanism maps at dent, whereas kinetic balance models, such as the et al. (2002) found consistent discrepancies ε˙ = 10−12 s–1 for (A) calcite, (B) quartz, and (C) paleo wattmeter or the fi eld boundary model, may between paleopiezometers and stresses inferred ratio will not change ! olivine, with stress–grain-size relationships contain temperature dependence. Extrapola tions from currently available fl ow laws. In Figure 4, from paleowattmeter, empirical paleopiezo- to natural conditions are shown in Figure 3. we compare the predicted power for natural metric calibrations (see respective legends), For calcite, our paleowattmeter predicts shear zones as a function of temperature. Using and fi eld boundary hypothesis (de Bresser stresses lower than the piezometer of Rutter the parameters shown in Figure 2 (B and C), we et al., 1998, 2001). For olivine, Kg was taken as 7.0 × 1016 µmp s–1. Dislocation and diffu- (1995), and different from that of Schmid et al. see that both laboratory and natural data are par- sion creep laws used are, respectively, (A) (1980). For example, for a grain size of 10 µm allel to lines of constant σε˙ (Figs. 1 and 4), and calcite: Renner et al. (2002) and Herwegh (common in both natural shear zones and experi- the natural data are shifted to coarser grain sizes. et al. (2003); (B) quartz: Hirth et al. (2001) and Rutter and Brodie (2004); and (C) olivine: ments), the paleostresses predicted by Equation 1 For calcite, the natural samples span an immense Hirth and Kohlstedt (2003). are 105 MPa (Rutter, 1995) or 46 MPa (Schmid range of σε˙, where most proto mylonites lie on et al., 1980), regardless of temperature; the paleo- contours that are essentially static (e.g., σε˙ << tions in grain-boundary mobility owing to differ- wattmeter predicts 75 MPa at 573 K, a tempera- 10−14 MPa s–1). However, virtually all the calcite ences in water fugacity, porosity, or solute impuri- ture consistent with many calcite-hosted shear ultramylonites are consistent with power levels ties. For example, for olivine rocks (Fig. 2D), zones (A. Ebert, 2005, personal commun.). In necessary for strain rates of 10−12 < ε˙ < 10−6 s–1. Karato (1989) used hot-pressed powders to study a recent study of the Helvetic nappes, Herwegh The grain-size–temperature relationship for grain-growth kinetics. This technique invari- et al. (2005) noted that stresses predicted by quartz rocks of Stipp et al. (2002) is consistent ably results in small amounts of porosity, which Rutter’s (1995) piezometer are much greater than with σε˙ < 10−12 MPa s–1. reduces grain-boundary mobility. Conversely, the expected lithostatic pressure, perhaps sug- The paleowattmeter may also explain the pro- Karato et al. (1980) and Van der Wal et al. (1993) gesting that the actual shear stresses along these gressive reduction in recrystallized grain size used olivine single crystals and natural samples, faults were lower. For quartz rocks, the paleowatt- from shear-zone margin to core (i.e., from proto- respectively. These probably have lower porosity meter also predicts consistently lower stresses for mylonite to ultramylonite). If the average grain and, thus, higher grain-boundary mobility. If the lower-temperature deformation. For example, for size at a given position in the shear zone is a func- pre-exponential in the grain-growth equation is a grain size of 10 µm, the Stipp and Tullis (2003) tion of stress only, then the stress within the shear adjusted to account for experimental differences, piezometer predicts a stress of 110 MPa, while zone is inferred to increase toward the core. If the resulting agreement is striking (Fig. 2D). our paleowattmeter predicts stresses of 130 MPa grain size refl ects the local power dissipated, then, For calcite, we also inverted the piezometric and 44 MPa at 1373 K and 773 K, respectively. for constant stress conditions, those regions with data using Equation 8, assuming that β was con- For olivine, there is very little difference between higher strain rates will have smaller grain sizes.

GEOLOGY, April 2007 345 Int J Earth Sciences .Geol Rundsch) .2001) 90 : 28±45 DOI 10.1007/s005310000149

ORIGINAL PAPER Field boundary hypothesis (de Bresser et al., 2001)

J.H.P. De Bresser ´ J.H. Ter Heege ´ C.J. Spiers Grain size reduction by dynamic recrystallization: can it result in major rheological weakening? Int J Earth Sciences .Geol Rundsch) .2001) 90 : 28±45 DOI 10.1007/s005310000149 I. Shimizu / Journal of Structural Geology 30 (2008) 899–917 quartz: 909

TReceived:ableOR3 I22GSeptemberINA1999L / APccepted:APE9 OctoberR 2000 / Published online: 20 December 2000 10 50 100 200 500 1000  Springer-Verlag 2000 Olivine, P=300 MPa, T=1015 C Parameters used for calculation on quartz 100 100 !8 !8 !8 !8 Parameter Value Remarks Source β-quartz Abstract It is widely believed that grain size reduc- a 3 Olivine, quartz IntroductionKohlstedt and N-H 1050°C tion by 2dynamic recrystallization can lead to major 50 S&T 50 brheological 3weakening and associated strain !local-10 W!10eathers (1980) creep 10 !10 ization by bringing 4about a! switch from grain size The localization of strain in deforming rocks is mJ.H.P. De4.2 Bresser10 MPa ´ J.H. Ter Heegebelieved´ C.J.toSpiersTwissbe of prime(1977)importance in controlling the insensitive dislocation8  creep to grain size sensitive dif- n 1.5 0.! 15 dynamics ofTwissthe (1lithosphere.977) Commonly, localized fusion creep. Recently,4 however, we advanced the !12 b 5 10À mm !12 deformationTwisszones(1show977) a grain size that is reduced dislocation hypothesis that, rather than a switch, dynamic recrys- !12 KGraintallization0 leads7.8sizeto a balancereductionbetween grainCalculatedsize reduc-bbyy Eq.with(A.23)dynamicrespectPennockto that inettheal.recrystallization:protolith .e.g. White et al. creep qtion and1grain2 growth processes set upHalitin thee neigh- 1980; Tullis(20and05)Yund 1985; Jin et al. 1998; Shige-  matsu 1999). Associated with this reduction in grain borhood of the boundary between the dislocation !14 qcanc it12 result in major rheologicasize, a general rheologicall weakeningweakeningis inferred .Poi- ? creep field and the diffusion5 3creep field.1 In this paper, V 2.370 1010À m molÀ b-quartz 14 Berman (1988)!14 10 c 10 ! 12 ! rier 1980; Hobbs et al. 1990). Meaningful modeling[µm] of c = we, stress (MPa) compare0.5 the predictions implied! by our hypothesis = 0.5

d 2 with! those of other models for dynamic recrystalliza- deformation behavior of rocks must therefore include Dislocation creep of b-quartz the rheological changes linked with grain refinement tion. We also evaluate4 the fulln range1 of models against !16

A log 1.1 10À MPaÀ sÀ Black Hills quartzite,and localization,Gleasonas emphasizedand Tullis in work by, amongst experimental0 data on a variety of materials. We con- !16 5 5 nclude that a4temperature dependence ofwithoutthe relation-melt others, Furlong(1995).1993), Govers and Wortel .1995) and 1 Braun et al. .1999). Coble creep Journal of Structural Geology 30 (2008) 899–917 Qshipc between223recrystallizedkJ molÀ grain size and flow stress Several different mechanisms can result in grain cannot!0.5be neglected a priori. This should be taken 16 size reduction during deformation. These include cata- Dislocationinto accountcrewhenep of bestimatin-quartzg natural flow stresses! Contents lists available at ScienceDirect 4.93 n 1 14 clasis .e.g., Hippler and !Knipe18 1990), syntectonic, met- ! 18 using experimentally calibrated12 recrystallized grain A 300 10À MPaÀ sÀ Brazilian quartz ! Rutter and Brodie R&B Received: 22ÂSeptember! 1999 / Accepted: amorphic9 Octoberreaction2000.e.g.,/ PublishedNewman etonline:al. 1999),20andDecember 2000 nsize piezometers. We also demonstrate experimental 2.97 (hot-pressed aggregatdynamices) recrystallization(2004a) during recovery-controlled Journal of Structural Geology supportSpringer-Verlag!1for the field boundary12000 hypothesis. This sup- Q 242 kJ molÀ portc implies1 that significant1.5 weakening2 2.5by grain size3 creep3.5.e.g., Tullis4 et al. 1990; Hirth and Tullis 1992). The last of these mechanisms is of particular interest reduction in localizeda shearlog d,zones grainis possible size (microns)only if journal homepage: www.elsevier.com/locate/jsg Diffusion creep of b-quartz because experimental work on numerous metals, 1 caused by a process other2 than1 dynamic1 recrystalliza- A À À tion .such as0.4/d(syntectonicmm) MPareactions or cataclasis)Brazilian orquartzif ceramics, andRuttrockser andhasBrodiedemonstrated that the mean 50 100 200 500 1000 nAbstractgrain growth1is inhibited.It is widely believed(hot-pressedthataggregatrecrystallizedgraines) (20sizegrain04b)diameterreduc-D .usually referred to as σ [MPa] 1 the ªgrain sizeº) can be related to the steady state Qc gs reduction220 kJ molÀ -> diff creep field -> Introduction tionKeywordsbyDeformationdynamic´ Microstructuresrecrystallization´ canflow stressleads by antoequationmajorof the typeTheories and applicability of grain size piezometers: The role of Oxygen diffusion in b-quartz Fig. 7. Comparison of the deformation mechanism map for b-quartz at 1050 C and the rheologicalPaleopiezometry ´ Rheologyweakening´ Recrystallizationand associatedD = Ks±m strain local- dynamic r.1)ecrystallization mechanisms 0grain gro11wth2 1 -> disl. field -> gs reduction recrystallized grain size at 1000–1100 C (solid circles) after Stipp and Tullis (2003). Dv 4 10À m sÀ Brazilian quartz, c-axis Giletti and Yund The localization of strain in deforming rocks is ization by bringing1 about a switchwherejj fromK and mgrain.typicallysize0.7±1.6) areS&Tmaterial-: empirical* andd–s relations within the temperature range of 700–1100 C after Stipp Qv 142 kJ molÀ (1984) I. Shimizu 0 17 3 1 mechanism-specific constants .Takeuchibelievedand Tullisand Argon(20to03)be. Theofboundariesprimebetweenimportancethe deformationin mechanismscontrollingof dislocationthe wDinsensitivegb 2.6 dislocation10À m sÀ creepArkansatos novgrainaculitesize sensitiveFarver and Yunddif- Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  1 1976; Twiss 1977; Drury et al. 1985; creepRutterJournaland1995).diffusionof StrIf ucturalcreep (thickGeologysolid line)30 (20and 08)the iso-899–9c line1(short7 dotted lines), and Qgb 113 kJ molÀ calibrated experimentally,(1991b) this typedynamicsof D± relationof the lithosphere. Commonly, localized fusion creep. Recently, however, we advanced the the boundarys between Nabarro–Herring (N–H) creep and Coble creep (thin lines) are J.H.P. De Bresser . ) ´ J.H. Ter Heege ´ C.J. Spiers ) can be used to estimate paleo-stressesdeformationafromr t thei c lmicro-e i n fzoneso showa bas graint r a c t size that is reduced OxyhypothesisHPT-Laboratory,gen diffusionFacultythat,in aof-qEarthuartzratherSciences, UtrechtthanUniversity,a switch,structuredynamicof naturallyrecrys-deformed rockscalculated.White 1979),from the dislocation creep flow law of ‘as-received’ quartzites after Gleason P.O.0 Box 80.021, 3508 TA5Utrecht,2 1The Netherlands Dv 2.9 10À m sÀ Brazilian quartz, c-axis Farver and Yund withandArticleTullishistoryrespect: (1995) andtothethatoxygenindiffusionthe protolithcoefficients in.e.g.‘wet’ quWhiteartz (Gilettietandal.Yund, E-mail: [email protected] providing a useful tool in quantifying lithosphere The average grain size (d) arising from dynamic recrystallization (DRX) is often used as an indicator of tallization Âleads1to a balance betweenjj grain size reduc- Received 10 January 2007 QPhone:v +31-30-2534973243 kJ molÀ dynamics. In(1addition,991a) D±s relations Contentsobtained1984; Farvforergeo-andlistsYund,available1991b). R&B:flowboundaryatstressScienceDirect(s); howbetweenever, a theoreticalthe dislocationbasis for theandscaldiffusioning relation between d and s has yet to be well 0 12 3 1 1980;Received inTullisrevised form 3andMarch 2008Yund establis1985;hed. InJinthis papeetr, theoriesal.for1998;the developmentShige-of recrystallized grain size are reviewed and their wDtionFax: +31-30-2537725and1.7grain10À mgrowthsÀ processesCalculated in thissetlogicalupmaterialsin thesuchneigh-as olivine and calciteAcceptedhave12 Marchbeen2008 gb creep fields calculated from the flowappliclawabilityparametis examiersned.ofSp‘dry’ecial qattentionuartz, afteris paidRutterto the depandendence of the d–s relation on DRX  1 matsuAvailable online1999).31 March 20Associated08 with this reduction in grain Qgb 193.4 kJ molÀ paper (see text) Brodie (2004a,b). The parameters usedmechainnisms.the Scalculationteady-state DRXareis listedclassifiedinintoTabledisconti3. nuous DRX with bulging (BLG) nucleation borhood of the boundary between the dislocation þ grain boundary migration (GBM) and continuous DRX with subgrain rotation (SGR) nucleation GBM. a size,Keywords:a general rheological weakening is inferred .Poi- þ The nucleation-and-growth model derived from Derby–Ashby theory describes the former case, whereas creepThe flowfieldlaw isandwrittenthein diffusionthe form of Eq.creep(8). field. In this paper, Dynamic recrystallization that derived from Shimizu theory applies to the latter. A static energy-balance model derived from Twiss JournalrierGrain 1980size ; ofHobbsStructuralet al. 1990). MeaningfulGeologmodelingy of we compare the predictions implied by our hypothesis theory is applicable to subgrain size, but not to recrystallized grain size. The lower limit of grain size is coefficientStress wD in fine-grained quartz aggregates measured under deformationDislocation creep behaviorgb ofpossiblyrocksconstrainedmustby a chanthereforege in deformationincludemechanism from dislocation creep to diffusion creep, with those of other models for dynamic recrystalliza- Quartz because deformation-induced grain size reduction ceases in the diffusion creep field. Scaling relations can be compared with the grain size data of Stipp and Tullis (2003) ‘wet’ conditions (Farver anddeterminedYund,in1the99laboratory1b) arsuppoe employrt the Shimized.u modelBecausein the case of SGR GBM. The theoretical pi- the rheological changes linked with grain refinement þ tion. We also evaluate the full range of models against ezometer calibrated for quartz suggests significant temperature effects under low-temperature meta- obtained at 1.5 GPa. The water fugacity fH O in the experiments of oxygen diffusion in quartz is faster than silicon diffusion by a factor experimental data on a variety of2 materials. Wejocon-urnalandhomlocalization,epage: waswemphasizedw.emorphlseic condivitions.einr.cworkom/lby,ocaamongstte/jsg Rutter and Brodie (2004a,b) would have been very low because of others,of 2 or Furlong3 (Farver and.1993),Yund,Govers2000), theandstrainWortelrate of.1995)diffusionandcreepÓ 2008 Elsevier Ltd. All rights reserved. scludemall P thatand thae temperatureoccurrence of a dependenceredox reaction bofetwtheeenrelation-water and estimated here is considered to represent an upper limit. Oxygen c Braun et al. .1999). ishipron jacbetweenkets; this rerecrystallizedaction consumes Hgrain2O andsizeprodanduces Hflow2-richstressvapors diffusion and recovery-controlled creep of quartz are both dependent1.SeveralIntroduction ondifferentf (Farvermechanismsand Yund, 199can1a; PateresultTherson,stress1in989exponentgrain). Hence,p0 of subgrain size is found to be close to icannotn pore spbeacesneglected. In addition,athpriori.e water cThisontentshoulds of the tbewo takensets of H2O unity, but p of recrystallized grain size is slightly larger than 1 in sintotartingaccountmaterials whendiffered estimatinby two ordgersnaturalof magnitflowudes: 1stresses0–20 H/ sizeit isDynamicreductiondesirarecryblestallizationtoduringuse(DRX)diffusionisdeformation.consideredcoefficientsto be one of theTheseandmanayincludecases,dislocationas detailedcata-belowcre.epEmpirical relations in the form of Eq. 6 clasislakeywmechanismsobtained.e.g., ofHipplergrinainsimilarsize reductionandchemicalinKnipeshear zonesenvi1990),(Turonmellis and syntectonic,nts.(1) arThee wideldiffusiony used tmet-o estimatdatae paleostress in crustal shear zones 1using0 Si in thexperimentalle Brazilian quartyz ucalibratedsed by Rutter arecrystallizednd Brodie (2004a,grainb) and Yund, 1985). It has been demonstrated that during DRX the average (White, 1979b; Christie and Ord, 1980; Kohlstedt and Weathers, several tTheorieshousand H/106Si in andthe BlackapplicabilHills quartzite (Gleasityon andofamorphicwegragrinresizeainobtainedd is primarilreactionsizeyunderdependenth.e.g.,pieonydrfloothermaw strzometNewmaness s andl conditionsthat nor-eters:1980;al.withEtheridg1999),Thea ewandateWilkie,andr pres-r1981;oleOrd and Christie,of 1984; Sto¨ckhert size piezometers. We also demonstrate experimental malised values of d and s for various materials obey the following et al., 1999), orogenic belts (Dunlap et al., 1997; Zulauf, 2001) and sure of 100 MPa, whereas the deformation experiments of Gleason Tsupportullis, 1995for) usedtheby Sfieldtipp aboundarynd Tullis (200hypothesis.3). In sample pThisreparasup-tion, dynamicuniversal relation:recrystallization during recovery-cthe upper mantleontrolled(Mercier, 1980), because recrystallized grain size p is easily measured under the optical microscope and is believed to water addynamicsorbed on the powdrerecryed Brazilistallizationan quartz was incorporatemechanismsd andd Tus llisÀ (1995) were conducted at much higher confining pres- creepK .e.g.,; Tullis et al. 1990; Hirth(1) andbe stableTullisduring the1992).post-deformation history of the rock mass port implies that significant weakening by grain size b ¼ m into the grains that displayed rapid coarsening during hot pressing; sures (about 1.5 GPa). In the latter case, waterrelativfugacitye to otherwstressas poorlindicatorsy such as dislocation density and The last of these mechanisms is of particularsubgrain size (White,interest1979a; Ross et al., 1980). reduction in localized shear zones is possible only if where b is the length of the Burgers vector, m is the shear modulus, however, the total water contents after deformation remained in the controlled, and their ‘as-received’ quartzite samplesThe theoryweprreesentenotd bysat-Twiss (1977) has long been quoted as * becauseand K is a non-dimensionaexperimentall constant, the orderworkof which isontypicallnumerousy metals, I. Shimizu6 a theoretical basis for grain size piezometry. In this theory, ‘stable’ rcausedange of 1by00–2a00processH/10 Si (otherRutter athannd Brodynamicdie, 2004a)recrystalliza-. Therefore, the ur10 ate(Twiss,d with1977; Poirierwate, 1r985(Hirth). A similaret relatioal., 20n, although01). Althoughwith water pressure and grain size or subgrain size is derived based on a comparison be- ceramics,a different scalingandfactorrocksK , is knownhasfor thedemonstratedsubgrain size d of that the mean wtionater-w.sucheakeniasng esyntectonicffects would hareactionve been morinorcataclasis)in the experimorenifts the chemical envir0 onments in the oxygen0 diffusiontween free dislocationexperimentsenergy and (sub)grain boundary energy in various materials (Takeuchi and Argon, 1976; Twiss, 1977; Derby, Department of Earth and Planetary Science, University of Tokyo,recrystallized7-3-1 Hongo, grainBunkyo-ku,diameterTokyo 1D13-0.usually033, Japanthereferredsame volume. ThetoTwissastheory predicts that the coefficients K cgrainonductgrowthed by Rutisterinhibited.and Brodie (2004a,b). In fact, the diffusion creep (1Giletti991): and Yund, 1984; Farver and Yund, 1991b) were not the same the ªgrain sizeº) can be related to theand K0steadyare temperature-stateindependent, meaning that recrystallized 0 as0 those inp the deformation experiments (Gleasongrain sizeandandTulsubgrlis,ain1size995are),solely dependent on flow stress; flow law calibrated in their experiments is consistent with a theoret- d s À K 0 : (2) ical diffusion creep flow law with the D value of ‘dry’ oxygen diffusion flowtheb ¼ presstressm entsstudyby andoesequationnot makeofanthey corrtypeectionhowevforer, Twissf (1.977) presented only an outline of his model, and Keywords Deformation ´ Microstructuresv ´   its validity hasH2yOet to be closely scrutinized. A full description of his in quartz (Fig. 10 of Rutter and Brodie, 2004b). The deformation mechanism map of b-quartztheoryatappear1050ed inCa latinerFig.confer7ence volume (Twiss,1980), and it is Paleopiezometrya r t i c´ Rheologyl e i n ´fRecrystallizationo a b s t r ±am c t this full account that is summarized in the present paper. A deformation mechanism map is now considered under H O- Dw=aKs constructs ed based on the flow law parameters and diffusion.1) 2 * Corresponding author. Tel.: 81 2 5841 4513; fax: 81 3 5841 45469. While the piezometer of Twiss (1977) is still used for paleostress þ þ rich conditions. For dislocation creep of b-quartz, the flow law dataE-maillistedaddress: [email protected] Tableokyo.ac.jp3. The grain size data of Stippanalysis,andseveralTullisdifferent(20models03) have since been proposed with where K and m .typically 0.7±1.6) are material- and parametersArticleof historyGleason: and Tullis (1995) are employed, whoseThe averappear0191-81age41/$ –graintosee frontbematterconcorsizeÓ 2008(ddantElsevier) arisingLtd.withAll rightsthefromreserved.fielddynamicboundaryrecrybetwstallieen zatiodislo-n (DRX) is often used as an indicator of doi:10.1016/j.jsg.2008.03.004 sample Rassembleceivedy1and0 Januaryexperimental2007 conditions were similar flowto mechanism-specificstresscation(scre);ephowandeverdiffusion, aconstantstheoreticalcreep; how.Takeuchibasisever,forif silicontheandscaldiffusionArgoning relationin between d and s has yet to be well those ofReceivStippedandin revisedTullis form(2003)3 .MGleasonarch 2008and Tullis (1995)establis1976;qhed.uartzitTwissIne ithiss slow1977;papeer thanrDrury, theoriesoxygenetdiffusion,foral. the1985;developmenttheRutterdislocation1995).creof eprecryIffieldstallized grain size are reviewed and their Accepted 12 March 2008 calibrated experimentally, this type of D± D relationD presented flow laws for samples ‘with melt’ and ‘without melt’.applicabilitywould eisxtendexamifurtherned.doSpwnecialwardattention. For example,is paidif s v toandthegb depof endence of the d–s relation on DRX AlthoughJ.H.P. DeAvailabletheBresseractionline.v)ation) ´ 3J.H.1energyMarchTer Heege20determined08 ´ C.J. Spiersfor the former cansiliconbe weusedre lessto thanestimatethe oxpaleo-stressesygen diffusion coefficientsfrom thebymicro-a factor of mechanisms. Steady-state DRX is classified into discontinuous DRX with bulging (BLG) nucleation (HPT-Laboratory,Qc 137 34 kJ/mol)Facultyis muchof EarthsmallerSciences,thanUtrechtthat forUniversity,the latter structure2, the fieldofboundarynaturallywoulddeformedshift to the positionrocks .Whiteof the c 1979),0.5 line in ¼ Æ ¼ þ (P.O.Q 223Box 80.021,56 kJ/mol),3508 TAthe differUtrecht,enceThein strainNetherlandsrate is not largegrainat boundaFig. 7. It shouldry migrationbe noted(GBMthat each) anddatacontipointnuousin Fig. 7DRXrepresentswith subgrainthe rotation (SGR) nucleation GBM. c ¼ Keywords:Æ providing a useful tool in quantifying lithosphere þ highE-mail:[email protected] above 1000 C. In the present analysis, the flowThe nucleatigeometricon-andmean-groof wththe diametmodelersderiveof recdrystallizefrom Derby–Ashbyd grains; accord-theory describes the former case, whereas Phone: +31-30-2534973Dynamic recrystallization dynamics. In addition, D±s relations obtained for geo- law for the melt-free samples is employed. that derivinglyed, thefromaverShimizage grainu theorysize in termsappliesof tovoltheumelatterfraction. Awostaticuld beenergy-balance model derived from Twiss Fax: +31-30-2537725Grain size logical materials such as olivine and calcite have been For diffusion creep, the self-diffusion coefficient D of oxygen in shifted toward the dislocation creep field. Moreover, most of the Stress v theory is applicable to subgrain size, but not to recrystallized grain size. The lower limit of grain size is b-quartz (Giletti and Yund, 1984) and the effective grain boundary run products described by Stipp and Tullis (2003) contained Dislocation creep possibly constrained by a change in deformation mechanism from dislocation creep to diffusion creep, Quartz because deformation-induced grain size reduction ceases in the diffusion creep field. Scaling relations determined in the laboratory support the Shimizu model in the case of SGR GBM. The theoretical pi- þ ezometer calibrated for quartz suggests significant temperature effects under low-temperature meta- morphic conditions. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The stress exponent p0 of subgrain size is found to be close to unity, but p of recrystallized grain size is slightly larger than 1 in Dynamic recrystallization (DRX) is considered to be one of the many cases, as detailed below. Empirical relations in the form of Eq. key mechanisms of grain size reduction in shear zones (Tullis and (1) are widely used to estimate paleostress in crustal shear zones Yund, 1985). It has been demonstrated that during DRX the average (White, 1979b; Christie and Ord, 1980; Kohlstedt and Weathers, grain size d is primarily dependent on flow stress s and that nor- 1980; Etheridge and Wilkie, 1981; Ord and Christie, 1984; Sto¨ckhert malised values of d and s for various materials obey the following et al., 1999), orogenic belts (Dunlap et al., 1997; Zulauf, 2001) and universal relation: the upper mantle (Mercier, 1980), because recrystallized grain size

p is easily measured under the optical microscope and is believed to d s À K ; (1) be stable during the post-deformation history of the rock mass b ¼ m   relative to other stress indicators such as dislocation density and subgrain size (White, 1979a; Ross et al., 1980). where b is the length of the Burgers vector, m is the shear modulus, The theory presented by Twiss (1977) has long been quoted as and K is a non-dimensional constant, the order of which is typically a theoretical basis for grain size piezometry. In this theory, ‘stable’ 10 (Twiss, 1977; Poirier, 1985). A similar relation, although with grain size or subgrain size is derived based on a comparison be- a different scaling factor K , is known for the subgrain size d of 0 0 tween free dislocation energy and (sub)grain boundary energy in various materials (Takeuchi and Argon, 1976; Twiss, 1977; Derby, the same volume. The Twiss theory predicts that the coefficients K 1991): and K0 are temperature-independent, meaning that recrystallized 0 0 p grain size and subgrain size are solely dependent on flow stress; d s À K 0 : (2) b ¼ m however, Twiss (1977) presented only an outline of his model, and   its validity has yet to be closely scrutinized. A full description of his theory appeared in a later conference volume (Twiss,1980), and it is this full account that is summarized in the present paper. * Corresponding author. Tel.: 81 2 5841 4513; fax: 81 3 5841 45469. While the piezometer of Twiss (1977) is still used for paleostress þ þ E-mail address: [email protected] analysis, several different models have since been proposed with

0191-8141/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsg.2008.03.004 olivine:

322 J. Precigout et al. / Tectonophysics 445 (2007) 318–336 diff-disl creep only + disl. accommodated GBS field:

Available online at www.sciencedirect.com

Tectonophysics 445 (2007) 318–336 Available online at www.sciencedirect.comwww.elsevier.com/locate/tecto Fig. 2. Olivine deformation maps (shear stress in MPa as a function of the grain size in μm) with strain rate isolines at a constant temperature (T=850 °C). a) Dislocation/Diffusion-related rheology: OlivStrainine defolocalisationrmation map winiththetwo subcontinentaldeformation mechamantlenisms, na—melay dislocation creep (Eq. (1)) and diffusion creep (Eq. (2)). Creeping parameters are given in Tableductile1. The blalternativeack arrow reptorestheents brittlethe grainmantlesize reduction by dynamic recrystallisation at a constant strain rate. The recrystallised grain size could be defined as the boundary between the two deformation mechanisms (Balance hypothesis, curve ① De a,⁎ a a b c Bresser et al., 1998; 2001) or as the experimeJ.ntalPrecigoutly calibrated, F.VGueydanDW's pie,zDom. Gapaiseter (cur,vC.J.e ②GarridoVan der,WAa.lEssaifiet al., 1993). b) GBS-related rheology: Olivine deformation map with three deformaa tGéoscienceion mes cRenneshan(UMRismsCNRS:Tdectonophysis61lo18),caUniversitétion crdeeeRennes1p,icsdi, Bâtimenf445fusiot 15,n(2007)cCampusreepde,anBeaulieu,d318dryCS-G–74205,BS336F-35042creepRennes(Eq.Cedex,(4)).FranceCreeping parameters are given in b DP. Mineralogia y Petrologia, Universidad de Granada, 18002, Granada, Spain Table 1. During dynamic recrystallisation, grain sizceDépartemes reduntcedeanGéologie,d tenBPd t2390,owarFacultéds thdeseSciencesrecrySemlalia,stallise40000d gMarrakecrains bh,alMoranoccoce that is here defined as the boundarywwwbetwee.elseviern .com/locate/tecto dry-GBS creep and diffusion creep at high stressReceivedand 14beFebruarytween2007;disreceivedlocatiinorevisedn creeformp 3aSeptembernd diff2007;usioacceptedn cre13epSeptemberat low2007stress (modified balance hypothesis). Available online 20 September 2007

grain size is defined by the VAbstractan der Wal's piezometer, an called dry-GBS creep mechanism, respectively. The important stress decrease is pItrisenodwicadmteditted tduhat thrie hngigh stregrngthaofinthe suibzcoentinental upperstmorast mainntleracontetrols thise firthst orusder straengtlhoofwthe-plithoowspherere. Anfunction of stress (non- incipient narrow continental rift therefore requires an important weakening in the subcontinental mantle to promote lithosphere-scale reduction that has to be achistraevin loedcalisaintion anthd esubsedquifenfut cosnitionenntalcrbreaeek-up. Based on theNclaesswicaltronheoliaogicnal layverinsgcofothseictoynti)nentaalnlitdhospheofre, thegrain size (GSS). The Strain localisationorigin of a lithospheric mantle shear/fault zonine has been athettributed to the exsubcontinentalistence of a brittle uppermost mantle. However, the lack of mantle — a domain. Braun et al. (1999)mantcole eanrtshqiuasketes nantdlythe amobsencedeof llfieledd occurthreences in the mdiantlfeffaeurltezonneceled tionthethideaeofga rdaucitinle-relsiatedzeweakexeninpgonent between this law mechanism, instead of brittle-related, for the incipient mantle strain localisation. In order to provide evidence for this mechanism, we dynamic recrystallisation reinlavetsteigdated thtoe mictrhositrsucturehs anedolalttogice pyre,ferreid.eori.e,ntations of ma(nmtle GBrocksSin=a kil2o)meatren-scadlethducetile sdtraiifnfugradsieinot inthecrRoeendap law (mdiff =3; Eq. (2)) ductilePeridotites (Betics cordillealternativera, Spain). Two main features were shown: 1) graiton size reducttheion by dynamic rebrittlecrystallisation is found to be mantle based on the piezometric relthaetionloynrelsevhanipt w,eaankenidng methcheanyismorebsptoansiibnele fodr strain localismatioan arnkd 2)s, wdithifinfecreraseinng sttraityn, grpeainssizeoredfucdtioinfisfucoesvialon creep; namely coble with both the scattering of orthopyroxene neoblasts and the decrease of the olivine fabric strength (LPO). These features allow us to strain localisation. However,ptrohepose asthatsugrainmbopuntdaiorynslidofing (GgrBSai) panrtlysaiczcoemmodates dynacrmieec recprys(taelli.sagti.onsuandrsufbasecqueentdgirafinfusizsie reonduct)ion.and Nabarro–Herring creep A new GBS-related experimental deformation mechanism, called dry-GBS creep, has been shown to accommodate grain size reduction within the GSS diffureductiona,s⁎ionduringcreedynamicp isrecrystallisationnot suppandortoteinduceda significant(e.gweakening. volatulowmetemperaturesdaiffus(Tibo800n;°C).KThenippresente, 1989; Wbheeler, 1992), c J. Precigout microstructural, Fstudy. demonstratesGueydanthe occurrence of the,grainDsize.sensitiveGapaisdry-GBS creep in natural, continentalC.J.peridotitesGarridoand , A. Essaifi by any published experimentallowsal reussuto proposelts (Da newerbrheologicaly andmodelAsforhbtheysubcontinental, remantle.specDuringtivedynamicly. Arecrystallisation,s a resutheltaccommodation, the combination of the three of grain size reduction by three competing deformation mechanisms, i.e., dislocation, diffusion and dry-GBS creeps, involves a a 1987; Drury et al., 1991). grain size reduction controlled by the sole dislocation creep at highdetemperaturesformat(Nio800n°C),mwhereasechanidislocationsms,creepi.e.and, ddry-islocation creep, diffusion Géosciences Rennes (UMR CNRSGBS61creep,18),areUniversitéthe accommodating mechanismsde Rennes1at low temperatures, Bâtimencree(b800p °C).atndConsequently15,drCampusy-G, weakeningBS crde,iseveryepBeaulieu,limitedyielifdsthe theCSfoll74205,owing deF-35042finition Rennes Cedex, France b grain size reduction occurs at temperatures higher than 800 °C, whereas a large weakening is expected in lower temperatures. This 2.2. GBS-related rheologyDPlar. geMineralogiaweakening related to GBSycreepPetrwouldologia,occur at depthsUniversidadlowerofthanth60ekmoandvedethereforeralGranada,l sprovidestrainanraexplanationte18002,: for ductileGranada, Spain c strain localisation in the uppermost continental mantle, thus providing an alternative to the brittle mantle. Départeme©nt2007deElsevierGéologie,B.V. All rights reserved.BP 2390, Faculté des Sciences Semlalia, 40000 Marrakech, Morocco

Keywords: Grain size; Dynamic recrystallisation; Grain boundary sliding; Ronda: peridoti: tes : : Recently, a new deformation mechanism was pro- e edisl ediff eGBS 6 Received 14 February 2007; received in revised form¼ 3þSeptemberþ 2007; accepted 13 September ð2007Þ posed to account for grain size reduction in a grain size Available online 20 September 2007 sensitive creep (Hirth and K⁎oCorrespondinhlstedt,g author19. 95; 2003). This The deformation map of Fig. 2b shows different slopes E-mail addresses: [email protected] (J. Precigout), [email protected] (F. Gueydan), mechanism has been called drdenis.gapay-GBis@univ-renneS cres1.frep(D. byGapais),Hcarlosg@ugrirth a.esnd(C.J. Garrido). of the strain rate isolines in the dry-GBS creep and Kohlstedt (2003), or GSS-p0040-1951/$ower - seelafrontw mattercre©e2007p Elsevierby DrB.V. Allurrightsy reserved. diffusion creep fields due to their different grain size doi:10.1016/j.tecto.2007.09.002 Abstract(2005), and consists of grain boundary sliding (GBS) exponents, and thus displays three deformation mecha- creep accommodated by dislocation and diffusion creeps nism domains. Similar to the deformation map provided (Hirth and Kohlstedt, 2003; Drury, 2005): in Fig. 2a, far away from the transition line between each It is now admitted that the high strength of the subcontinentalmeupcphearnmismostdmaomnaitln,e cothnetroplarstithale fstirrastinorderater stfroenr gtthheof the lithosphere. An Q : : : incipien:t narrow continental riGBfSt thernGefBSore mrGeBqS uires an importancot wensidaerkeedninmeg cihnanthisemsu(bcedisonl, tiednifef nortalemaGBSn)tlceontotribpruomtes ote lithosphere-scale eGBS AGBS exp À s dÀ ; 5 : strain localisa¼tion anÁ d subseRqu:TentÁcontiÁnental break-upð. BÞ asedasona thwehoclelatsso icthaelorvheeraolllostgraicinalralateyeer.inIngcoofnttrhaset,cinonthtienental lithosphere, the   neighbourhood of the transition line, all partial strain rates origin of a lith:ospheric mantle shear/fault zone has been attributed to the existence of a brittle uppermost mantle. However, the lack of mantle weahretrheqeuaGBkeS,sAGaBnSd, QthGBe Sa,bnsGBenSceandofmfiGBelSdaroecctheurrestrncaines in cothnte rimbuatentltoe thfaeuoltvezraollnestraleind ratote t(hEeq.id(6ea)). Aotfthea dexucactitle-related weakening rate, the pre-exponential constant, the activation energy, transition line between dislocation creep and dry-GBS mechantihsems,trinessts eeaxpdonoefnbt rainttdleth-reeglarateind,sifzeorexthpoeniennctipofietnhet maso- ntle stcrraeeinp (loGcSaI/lGisBatiSotnra.nsInitioornd),erditosloprcatovionidane devGidBeSncsterafionr this mechanism, we investigated the microstructures and lattice preferred orientations of mantle rocks in a kilometre-scale ductile strain gradient in the Ronda Peridotites (Betics cordillera, Spain). Two main features were shown: 1) grain size reduction by dynamic recrystallisation is found to be the only relevant weakening mechanism responsible for strain localisation and 2), with increasing strain, grain size reduction is coeval with both the scattering of orthopyroxene neoblasts and the decrease of the olivine fabric strength (LPO). These features allow us to propose that grain boundary sliding (GBS) partly accommodates dynamic recrystallisation and subsequent grain size reduction. A new GBS-related experimental deformation mechanism, called dry-GBS creep, has been shown to accommodate grain size reduction during dynamic recrystallisation and to induce significant weakening at low temperatures (Tb800 °C). The present microstructural study demonstrates the occurrence of the grain size sensitive dry-GBS creep in natural continental peridotites and allows us to propose a new rheological model for the subcontinental mantle. During dynamic recrystallisation, the accommodation of grain size reduction by three competing deformation mechanisms, i.e., dislocation, diffusion and dry-GBS creeps, involves a grain size reduction controlled by the sole dislocation creep at high temperatures (N800 °C), whereas dislocation creep and dry- GBS creep, are the accommodating mechanisms at low temperatures (b800 °C). Consequently, weakening is very limited if the grain size reduction occurs at temperatures higher than 800 °C, whereas a large weakening is expected in lower temperatures. This large weakening related to GBS creep would occur at depths lower than 60 km and therefore provides an explanation for ductile strain localisation in the uppermost continental mantle, thus providing an alternative to the brittle mantle. © 2007 Elsevier B.V. All rights reserved.

Keywords: Grain size; Dynamic recrystallisation; Grain boundary sliding; Ronda peridotites

⁎ Corresponding author. E-mail addresses: [email protected] (J. Precigout), [email protected] (F. Gueydan), [email protected] (D. Gapais), [email protected] (C.J. Garrido).

0040-1951/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.tecto.2007.09.002 Equations 8–9 suggest that the stable grain size and temperature can be seen for rocks COMPARISON WITH size during dynamic recrystallization scales deformed at constant σε˙ (Fig. 1). The expo- EXPERIMENTAL DATA with the rate of mechanical work done by dis- nents, m′, predicted by particular models dif- In Figure 2, we compare the wattmeter to location creep. Thus, recrystallized grain size, fer in details, but are similar in magnitude, e.g., stress–grain-size data reported from deforma- rather than being a paleopiezometer, is likely a m′ is predicted to be 1.5 (Derby); 1.25–1.33 tion experiments on calcite (Schmid et al., 1980; paleowattmeter. (Shimizu); 1.0 (fi eld boundary), and 1.33 (watt- Rutter, 1995; Barnhoorn et al., 2004), quartz meter, taking n = 3, p = 2). Additional data and (Stipp and Tullis, 2003), and olivine (Karato et al., COMPARISON TO OTHER MODELS OF observations will be necessary for validation 1980; Van der Wal et al., 1993) rocks. Inserting DYNAMIC RECRYSTALLIZATION of one model or another. However, note that, material parameters from published creep and Because this treatment includes neither the unlike the de Bresser fi eld boundary model, the grain-growth laws without fi tting, the wattmeter exact physics of the grain-size reduction process wattmeter does not constrain the grain size to predicts recrystallized grain sizes in remarkably (e.g., Derby, 1991, 1992), nor the kinetics of lie at the boundary between creep mechanisms good agreement with the data (Figs. 2A, 2C, and nucleation (e.g., Shimizu, 1998), this wattmeter (Figs. 2 and 3). 2D). Slight misfi ts could be explained by varia- is properly called a scaling relation, rather than a kinetic law. The model contains several explicit assumptions, including: −5 −4 −3 −2 −1 0 −5 −4 −3 −2 1. Other elements of microstructure have 5 5 Barnhoorn et al., Stipp and Tullis, 2003 achieved steady state or do not infl uence the parti- 2004 Gray Scale = log (σε˙) tioning of work into the stored boundary energy. 4 Schmid et al., 1980 4 2. The rate of production of new boundary Rutter, 1995 m) 3 3 area is proportional to the rate of work done on µ Gray Scale = log (σε˙) the material. 2 2 3. Grain growth and reduction processes ln (d) ( operate independently and, at steady state, bal- 1 1 ance each other. Α) Calcite Β) Quartz 4. The growth mechanism is governed by an 0 0 equation similar to that for normal grain growth. 0.8 1.0 1.2 1.4 0.7 0.8 0.9 1.0 1/T (10-3 K-1) 1/T (10-3 K-1) 5. During diffusion creep, the grain-size reduction rates are small compared with grain Figure 1. Relationship between temperature and grain size for (A) calcite and (B) quartz. See growth rates. legend for data sources. Gray scale and symbol size both scale with log of power in each 6. Diffusion and dislocation creep operate as experiment. linearly independent processes. 2 2 These assumptions arise either because 10 10 Barnhoorn et al., 2004 Parameters from inversion of Eq. 8: of lack of contrary evidence (3, 4, and 6), or A p = 4.9 B Rutter, 1995 Qg = 179 kJ/mol because of positive experimental evidence, Schmid et al., 1980 Kgcγ/βλ = 5.9e13 albeit fragmentary (1 and 5). The constant Grain growth parameters of proportionality in the second assertion (λ) (Covey-Crump, 1997): 1 p = 3 1 might conceivably depend on the dislocation 10 Qg = 174 kJ/mol 10 K = 2.5e9 density and/or the recrystallization mecha- g Dislocation creep flow law: Renner

nism. For example, Hirth and Tullis (1992) m)

µ et al., 2002 suggested that three distinct recrystallization Barnhoorn et al., 2005 Rutter, 1995 regimes exist for quartz rocks. If so, λ might Schmid et al., 1980 0 Calcite 0 Calcite 10 10 not simply be a numerical constant over all 0 1 2 0 1 2 conditions; it would, of course, be bounded 10 10 10 10 10 10 2 by 0 < λ < 1. Assertion 4 is based on experi- 10 Stipp and Tullis, 2003 C Van der Wal, 1993; D mental observations for olivine (Karato, 1989) Kg = 2.37e15 Grain growth parameters 2 Karato et al., 1980; and calcite rocks (Walker et al., 1990; Austin (Wightman et al., 2006; refit 10

Predicted grain size ( Kg = 2.37e15 from Tullis and Yund, 1981): Paleowattmeter: Literature σ σ and Evans, 2005), but may not be true for all ε˙ Calcite ε˙ Quartz 2 Paleowattmeter: Inversion Van der Wal,8 1993; =10 8 =10 p = 3.2 10 -20 Protomylonites Mylonites and ultramylonites: Field boundary -1 materials. If not, Equations 5–7 could be modi- Qg = 215 kJ/mol Schmid et al., 1980 Kg = 7.0e16 MylonitesGrain 2 Tonale fault zone 1 σε˙ σ 10 and/or, an alternativRuttere ,idea: 1995 the “Wattmeter”6 =10 Ultramylonites 6 ε˙ fi ed to include changes to the grain-size evolu- Kg = 1.3e8 Karato et al., 1980; -8 growth =10 Mylonites and -8 grain sizDiffusione is proportional to1 theK dissipated= 7.0e16m) σε˙ energy, not justparameters the stress . Dislocation creep µ =10 ultramylonites: 10 g σ tion rate. Likewise, if solute drag or additional creep 4 -4 4 ε˙ Dislocation1 creep flow Helvetic nappes =10 10 σε˙ (Karato, 1989 dry, law: Hirth et al. (2001): =10 σ -4 phases infl uence either the grain-size evolution based523 K on ideas from non-equilibrium thermod0 ynamics:300 MPa): the recrε˙ ystallized 2.5 ln (d) ( =10 σ f(H O) =K 10 MPa 2 -16 2 Stress (MPa) σ ε˙ 2 ε˙ σ K p = 2 =10 σ kinetics or the power dissipation, the appropri- =10 σ ε˙ ˙ grain siz573e represents the budget of energy being4 stored in grainε˙ ε =10 0 =10 =10 623 Qg = 520 kJ/mol -12 boundaries and energy dissipated by flo0 w. 0 -16 -20 ate equations may be modifi ed accordingly. 0 A) Calcite Quartz 0 Olivine 0 673 K Α Β 10 10 10 0 01 1 2 2 03 1 2 Previous treatments of dynamic recrystal- 10 1010 10 10 10 1010 0.5 10 1 1.5 210 2.5 3 0.5 1 1.5 2 2.5 3 total work/dt = energy stored + energy dissipated 1/T (10-3 K-1) lization also predict relations similar to Equa- 573 K Paleowattmeter: Literature 2 MeasuredField boundary grain size (µm) 10 Stipp and Tullis, 2003 Figure 4. Grain-size versus temperature relationships for samples from natural shear zones. tion 10 (de Bresser et al., 1998; Derby, 1992; 673 K A: Calcite (cf. Herwegh et al., 2005; de Bresser et al., 2002, and references therein). B: Quartz Dislocation creep fraction of Derby and Ashby, 1987; Drury, 2005; Shimizu, Figure 2. Grain size predicted773 K from paleowattmeterin creation (Equations (Stipp et al. 2002). 8 or Contours 9) comparedall are diffor constantfusion to thatσσεε˙ calculated using (A) Equation 8 with param- eters from inversion (Fig.+ 2B), and (B) Equation 9 using same parameters as in Figure 2C, 1998). In common with those models, the measured during dynamic recrystallization873 K ofof ne(A)w calcite, GB usingdislocation dislocation cree creepp fl ow law assuming deformation is completely accommodated by dislocation creep (β = 1). 1 of Renner et al. (2002) and10 grain growth973 K parameters from Covey-Crumpcreep (1997); (B) calcite, paleo wattmeter suggests a modest temperature 1073 K using inversion of Equation 8; (C) quartz using dislocation creep fl ow law of Hirth et al. 16 p –1

Stress (MPa) stant (Fig. 2 ). The inverted value for p (4.9) is sig- our paleowattmeter (K = 7.0 × 10 µm s ) and dependence of the ds-σ relation, involving the (2001) and grain growth parameters from W1273ightman K et al. (2006); and (D) olivine, using dislo- g Diffusion creep 1473 K nifi cantly larger than that observed in experiments the paleopiezometric calibration of Van der Wal difference between Q and Q (cf. Eqn. 10). net grain 3g Krowth/dt = growth rate + reduction rate g disl cation creep fl ow law of HirthB) Quartz and Kohlstedt (2003)167 and grain(Evans egrowtht al., 2001) .parameters However, it is strik froming that Karatothis et a l. (1993). The temperature dependence for 0 10 Such a correlation between recrystallized grain (1989). Solid lines indicate0 where the1 predicted2 grain3 sizethr eequalse-paramet ether fi t r emeasuredconciles labora tgrainory data tsize.hat this material is small because activation ener- 10 10 10 10 aPrea dliescorewpaantt wtmith eantye orfs t:h eA e mspciraiclailn pgiez roemlea- tiogines ffoor disyloncatimon iccraeelpl y( 530 kJ/mol; Hirth 2 C) Olivine ters (i.e., Equation 1). The elevated p value might and Kohlstedt, 2003) and for grain growth K Paleowattmeters: A scaling relation for dynamically 10 Dislocation creep irnediccartye sthtaat lβl iiznceludde sg ar agrianin -ssiizze edependence, (520 kJ/mol; Karato, 1989) are similar. 873 recrystallized grain size which perhaps refl ects an increased component of K Nicholas J. Austin* J. Austin* Department Depar of Earth,tment Atmospheric, of Ear andth, Planetar Atmospheric,y Sciences, Massachusetts and Planetar Institutey ofSciences, Technology, Massachusetts Institute of Technology, 344 Brian Evans* 77 Massachusetts GEOLOGYAve., Cambridge, Massachusetts, April 02139, USA2007 973 dBrianiffus iEvans*on cree p for sm77a lMassachusettsler grain size sA.ve., Cambridge, MassachusettsDATA FRO M02139, NA USATURAL SHEAR ZONES 3 K ABSTRACT also diWssipahtede, bnut sa tfruacdtioyn, iλn, isg r enspoanstiubler al shear zones, two of the 107 1 During dislocation creep, mineral grains often evolve to a stable size, dictated by the deforma- for increases in internal energy owing to grain- 10 3 K 7 EAXBSTTRAACPTOLATION TO CONDITIONS OF most readily available pailesoc ediss soipfa teidn,f bourtm a afrtaicotinon a, rλe, is responsible 1 tion conditions. We suggest that grain-size evolution during deformation is determined by the boundary area. If the total strain rate can be par- 1 K Diffusion creep rateD ofu mreicnhgan diciasl lwoocrka.t Pioronvi dcerde tehapt ,o mtheirn eelermaenl tgs rofa minicsro ostfrtuectnu ree vhaovlev aec htioev aed ssteaabdyl e stitzioen,e d biecttwaeetne di sbloyca ttiohne a ndde dfifofursmiona c-reep, for increases in internal energy owing to grain- 73 K NATURAL DEFORMATION grain˙ s˙ize and temperature. Average recrystallized 73 state and that the dissipation rate is roughly constant, then changes in internal energy will be then β = Wdisl/Wtot, and the dissipation rate is Stress (MPa) 12 tpiroopno rctioonadl tiot cihoanngse.s Win gera sinu-bgogunedsatr yt haraeat. Igf nroarimna-l sgirzaien -egrvoowlthu atniod ndy ndaumrici gnrgai nd-seizfeo rmation is determined by the boundary area. If the total strain rate can be par- 13 θ˙ = (1 – β)σε˙ + (1 – λ)(β)σε˙. (4) reduEctixont orcacupr soimlualttanieoounsl y, tfhreno tmhe s tealday-bstaoter garatino srizye i s dcetoermnidneidt biyo thne bsa lantcoe of grainirr size often correlates with inverse tempera- Paleowattmeter: Literature rate of mechanical work. Provided that other elements of microstructure have achieved steady titioned between dislocation and diffusion creep, K those rates. A scaling model using these assumptions and published grain-growth and mechani- Incorporating Equation 4 into Equation 3 and ˙ ˙ Field boundary 3 nstaattue raanld tdheatf othrem daistsiiopant ioon fr atreo ics krosu grhelyq ucoinrestsa ntk, nthoewn cl-hangetsu irne i,n taesrn sahl eonwerngy iwni lFl bige ureth e4n βf o=r W mdisl/yWltoot, nanitde tshe h doissitpeadtio n rate is 73 K cal relations matches stress–grain-size relations for quartz and olivine rocks with no fi tting. For rearranging yields 15 marbles, the model also explains scatter not rationalized by assuming that recrystallized grain Van der Wal et al., 1993 147 proportional to changes in grain-boundary area. If normal grain-growth and dynamic grain-size 0 −cγ ˙ ˙ ˙ esidzeg is ea fuonctfio nt ohf setr esse aflofnee. cWthsen eoxtfra psolattreed tso sco,n dtiteionms typpiecarl afotr unartuera,l maylnondite s, within cσaε =lcited read . nd qua(r5t) z rocksθ. irFr =o r(1 a – g βi)vσeε n+ (s1t r–a λi)n(β )σε. (4) 10 d 2 0 1 2 3 rtheed muocdetli ois nco nosciscteuntr w sitihm fi ueldlt caonsetroauintssl oyn, stthreessnes tahnde sstrtaeina rdayte-s.state grain size is determined by βλthe balance of 10 10 10 10 stthroasien r atreas.t eA. s cTalihneg meomdepl uirsiincga lt hepseie azssoummpettieornss an(Ed qpuubali-shed rgTarhate eisnta,-b glebr goroawitn htshiz e a pisn odftwe mn peeorsci theda tnno iod-cc urfl owInc osrtproersatsin gw Eiqllu actionr r4e ilnatot eE quation 3 and Grain size (µm) cKaeyl wroerldas:t ireocnrysst amllizaaticohn, edsef osrtmraetisosn–, rgheroaloigny,- csailzciete ,r qeulaarttzi, olnivsin ef,o grra iqn usizaer, ptize zaomnedte ros,l ivwinhen rgoracink gsro wthit ahnd n reod ufic ttiotnin ragte.s Fareo erq ual rearranging yields tlioocanliz at1io)n, daissripeat iotne. mperature and strain rate indepen- (dwe Birteshse r ietn avl., e19r9s8e; H atlle amnd Pparemrenatiteru, re. Recently, de Bresser marbles, the model also explains scatter not rationalized by assum2i0n03g) . tWhhaetn rnoe cmreychsatnaiclalli zweodrk igs rdaonine, the −cγ Figure 3. Deformation mechanism maps at dsINiezTneR tiOs,D awU CfhuTIneOcrNteioans okf isntreetsisc a lbonael.a Wn19ch90ee).n mC elexoartlrdy,a eap ltohsel,ao rtesetiucdac l tuohn d ceaorssnta dntdhiitniego onf s ttyoetpatli c garaali nfl-os.iz re n(eva2otl0utir0oan2 lr am)te ymlfuoosnt uietqneuasdl, t he c onsistent dσiεsc=repandcie. s (5) Reduced grain size is a common and striking recrystallization is necessary to interpret natural static growth rate. Lacking any specifi c infor- βλd 2 red ε˙ = 10−12 s–1 for (A) calcite, (B) quartz, and (C) ptfehaaetlu erme ooof w dmeyall otintsim tecso. enCtrseeiesprt eoxnprte r witmhietnehts fiin deeil-dld cm obicnrosstturuanctiundretass. royn mstroesdseesl a, nmd astyra in rmbaattieoenst .won ehoewn (o rp ifa) lgerowothp ained zreodumctioen ters and stresses inferred cate that average grain size evolves to a stable processes couple, we assume they are linearly ˙ ˙ ˙ The stable grain size is often posited to occur olivine, with stress–grain-size relationships cvoalune (tdas) idnete rmteinmed bpy teher daeftourmrateio n dcoendpi- enSCdAeLnINcGe B.E TEWxEEtNr aPOpWoElRa tions infdrepoenmden t: cdtuot =r drred =n dtgrl. y available fl ow laws. In Figure 4, Ktionesy. Ewarolyr wdosrk: sruegcgersytesdt tahlalt idzs awtaiso innv,e rdseelyf ormANaDt iGoRnA, IrNh eSIoZlEogy, calcite, quartz, olivTihnues,, grain size, piezometers, when grain growth and reduction rates are equal from paleowattmeter, empirical paleopiezo- related to stress (σ) (Twiss, 1977), and, thus, During deformation, external forces do work tlooc nalaiztautiroanl, dciossnipdaittioion.ns are shown in Figure 3. we comp−caγ re the predi(cdtee Bdr espsoerw ete ra l.,f o19r9 8n; aHtaullr aanl d Parmentier, mylonite grain sizes have been used to infer (per unit volume) at a rate of W˙ = σε˙ + σ˙ε, σε = (d − d ). (6) metric calibrations (see respective legends), tot βλd 2 tot gr paleFostoresrse s (ec.g.a, Slticppi taned, T ulliso, 2u00r3 ): palwehoerew a daot dtemnotes tae timr e derpivartieved, aincd σts an d shear zones as a functio2n00 o3)f. Wtehmenp neor amtuecrhea.n iUcasl iwnogrk is done, the and fi eld boundary hypothesis (de Bresser ε are differential stress and incremental strain, Assuming that deformation does not modify I NTRODUCdT =I AOσ –Nm, (1) 1990). Clearly, a theoretical understanding of total grain-size evolution rate must equal the stresses loswer than the rpespieectziveolym. Fore sitmeprlic ityo, wfe coRnsiduert utneiarxi al notrhmael gpraian rgarowmth ekitnetircs, thsehn tohew ratne ofi n Figure 2 (B and C), we et al., 1998, 2001). For olivine, K was taken ˙ –1 1–p g wheRre eAd aundc emd a rge reamipnir icsailzlye d eiste ram icneodm. Tmhe ons traenssd a nsdt croikaxiinalg st rain.r Ief cσr vyarsietas lelsisz raatpiidolny isg rnoewcthe iss sdagrr =y K tgo e xipn[–teQrgp/RrTe]pt ndat,u wrhaelr e p static growth rate. Lacking any specifi c infor- 16 p –1 application of this idea was clouded by evidence than grain size, then, on some time scale, σ˙ = 0 is the exponent in the normal grain growth law. as 7.0 × 10 µm s . Dislocation and diffu- (f1ea9tu9r5e )o, f amnydlo ndiitefsf.e rCerenetp ferxopmeri mthe˙ natts oinfd i-Schmmicirdos terutc taulr.e s. see that both laboratory maantido nn aontu hroawl d(oart aif )a rgero pwathr -and reduction of independent infl uence of metamorphic condi- and Wtot = σε˙. The rate that work is done equals The total grain-size evolution rate is cate that average grain size evolves to a stable ˙ processes couple, we assume they are linearly sion creep laws used are, respectively, (A) tions (Etheridge and Wilkie, 1981). More recent the rate of increase in internal energy, Eint, plus 2 Q ˙ (1980). For example, for a grain size of 10 µm allel βλtoσε dlines o−f g c o−1n1−sp tant σε (Figs. 1 and 4), and workers have suggested that d is also a function the rate of energy dissipation, θ˙ , including heat d = + K exp p d . (7) value (d ) determineds by the deformation condi- SCAirLr ING BETWEEtotN PcOWERg  RT  independent: d˙ = d˙ = d˙ . calcite: Renner et al. (2002) and Herwegh of temperatus re or strain rate (Poirier and Guil- production and any other irreversible increases γ   tot red gr (tcioonms. Emarolyn w ionrk b sougtghe snteadt uthraat ld s whaesa inrv zerosenlye s aAnNdD e GxpReArIiN- SIZtEhe natural data are shiftedT htuos ,coarser grain sizes. et al. (2003); (B) quartz: Hirth et al. (2001) and lopé, 1979; de Bresser et al., 1998, 2001; Drury, s in entropy (Poliak and Jonas, 1996; Bercovici The steady-state grain size occurs when the r2e00la5)t.e Adt p rteose nst, tthrerse sis n(oσ u)n iv(eTrsawllyi sacsc,e p1te9d 77a)n,d aRnicdar,d , t2h00u5s)., If the stoDreud reinnerggy d aessfoocir-matottiaol nev,o leuxtiotne rantea ils zfeoror:ces do work mtheeoryn tot sex)p,la itnh thee r eplataiolneshoip sbettwreens ds eansd parteed dwiithc ftreee dis lbocaytio Ens rqapuidlay tatitaoinns s t1ead y For calcite, the natural samples span an immense Rutter and Brodie (2004); and (C) olivine: s −Q −cγ ˙  g  −˙1 ˙ mthey dloefnorimteat iogn rcaoinnd itisoinzs,e bsu t hfacvtoers btoe eben ustsaeted, t hteon thien rfaeter o f cha(npgee ro f uinnteirtn avl oenleurgmy e) at a rate oKfg eWxptot = σpε cγ+ σε, σε = 2 (dtot − dgr ). (6) ˙RT  Hirth and Kohlstedt (2003). acronesi de1re0d a5re tMransiPtioans in( Rthe urectrtyestarll,i za1tio9n 95is )p rooporr tio4na6l toM theP raate o(fS chcanhgem of igdra in rangde1+ p of σε, whe,r e m(8)ost proto mylonβλitdes lie on paleostresses (e.g., Stipp and Tullis, 2003): where a dot denotes a times d=erivative, and σ and emte cahaln.i,sm 1 (P9oi8rie0r a)n,d rGeuiglloapér, d19l79e; sRsut teor, f tbeoumndapry earreaa: ture; the paleo- contours thβγaσtε are essentially static (e.g., ˙ << 1995), stabilization of the grain size at va –lumes bal- ε are differential sinttroe wshsi cha nβ dm aiyn bce rseubmstietuntetda, ly iseltdriangi:n, Assuming that deformσatiεon does not modify d = A , (1) −cγ ancing diffusion and dislocatiosn creep σ(de Bresser E d , (2) int = r2esrped ectively. For simpli−c1i4ty, we cons−–iQd1er uniaxial normal grain growth kinetics, then the rate of tions in grain-boundary mobility owing to differ- wattmeter predicts 75 MPa at 573 K, da tempera- 10 MPa s )g . H−1 owever, virtually all the calcite et al., 1998, 2001), subgrain nucleation and grain- K g exp p cγ  RT  –1 1–p where A and m are empirically determined. The stress and coaxial strain. I1+fp varies les s rapidly growth is d˙ = K exp[–Q /RT]p d , where p growth rates (Shimizu, 1998), and nucleation-site where c is a geometric constant, d is the grain ds σ= . (9) gr g g ences in water fugacity, porosity, or solute impuri- ture consistent with many calcite-hosted shear ultramylonλσiteεdisl are consistent with power levels adepnpsiltyic (Saatkiaoi nan do Jfo nthasi,s 1 9i8d4e).a was cloudeds izbey a t etivmied te, anncd eγ is thet havaenra gge rsapeincifi sci gzreai,n then, on some time scale, σ˙ = 0 is the exponent in the normal grain growth law. ties. For example, for olivine rocks (Fig. 2D), zoofF nienled soebp se(rnvAadtieo.n st E oifn nbflo meuirenatnl,lcy e s2 inog0fl e0m-p5heats,ae mpobeorurpnshdaoircyn ecnaoenrlg dy.ic T-hoenm, amnd uW˙n. )=. I˙n. TheF rinnaaetlleyc , tiefh tashets rwoacrko ydrek fo firsmo sd rbo y nspoetw rearq-liuanwa lcsrre eapt, eTsh eo tfo ta1l 0gr−a1i2n <-si z˙e e

© 2007 The Geological Society of America. For permission to copy, contact Copyright Permissions, GSA, or [email protected]. GEOLOGYGeology, April, April 2007; 2007 v. 35; no. 4; p. 343–346; doi: 10.1130/G23244A.1; 4 fi gures. 343