Diffusion and Diffusion Creep

1. Introduction: Observations of textures associated with pressure solution and diffusion creep

2. Theoretical description of Diffusion: a. Random walk b. continuum theory. -driven by concentration gradients -driven by stress gradients

3. Diffusion Pathways: Diffusivity and Diffusion paths and a word on Grain boundaries.

4. Models for diffusion creep in rocks at high temperature: Lattice and grain boundary diffusion creep...

5. Experimental data: Rheological data for olivine

6. Effects of Water

7. Effects of Melt -Coble Creep, Cooper-Kohlstedt model, Contiguity model 1. Intro: Observations

from: “Microtectonics” from: “Microtectonics” Calcite mylonite straight boundaries, triple and 4 grain jcns weakly elongated grains uniform grain size

from: “Fault-related Rocks: A Photographic Atlas” Mylonite with Ultramylonite (um) derived from granite (S. California):

from: “Fault-related Rocks: A Photographic Atlas” 2. Theoretical descriptions of diffusion

1. Random walk: diffusion is a random walk process (when, in the presence of a gradient, gives a ﬂux down that gradient)

2. Can be described by a continuum behavior: Fick’s ﬁrst law (the delta spike decay).

3. Fick’s second law (1st Law + continuity eqn)

4. Generalized to any gradient ? Stress gradient (or electromagnetic, eg)? How does stress drive diffusion ? on a lattice, these jumps are limited to “sublattices” from:Shewmon

http://en.wikipedia.org/wiki/Atomic_diffusion http://www.youtube.com/watch?v=QfQolcXrV1M&NR=1 http://www.youtube.com/watch?v=xDIyAOBa_yU 1 Observations: Textures associated with diﬀusion creep

A general problem in the study of the rheology of earth materials, it is very difficult to observe the processes in action (at the conditions in the Earth or approximated in experiment). We can observe only indirectly and/or frozen states, so how do we infer/interpret the predominant mechanisms act- 1 Observations: Textures associated with diffusion creep ing in a material? An important approach is to compare experimental microstructures with natural 1 Observations: Textures associated with diffusion creep microstructures,A general problemassumingin thethatstudythe ofmicrostructurethe rheology of earthrepresenmaterialts thes, itaccumis veryulateddifficultconsequenceto observe theof the A general problem in theprostudycessesproofatthecesseshandrheologyin(sactionteadyof earth(atstatethematerialconditionsmakess, itthisiningsvtheeryeasier).Earthdifficultor Ditoapproffobservusionximatede thecreepin expleaerimenves lesst). ofWeancanobobservviouse trace only indirectly and/or frozen states, so how do we infer/interpret the predominant mechanisms act- processes in action (atthanthe conditionsdislocationin thecreepEarth(exceptor approforximatedthe lacin kexpoferimendislot).cations!).We can observe only indirectly and/or frozen states,ing insoahomaterial?w do we Aninfer/inimpterpretortant approacthe predominanh is to comparet mechanismsexperimenact-tal microstructures with natural ing in a material? An importantmicrostructures,approach is to compareassumingexpthaterimenthetalmicrostructuremicrostructurrepresenes with naturalts the accumulated consequence of the microstructures, assumingAPPRthatprotheOcessesAmicrostructureCH:atSIMPLEhand (srepresenteadyto progressivstatets themakaccumeselythingsulatedCOMPLEXeasier).consequenceDi!ffusionof creepthe leaves less of an obvious trace processes at hand (steady statethanmakesdislothingscationeasier).creepDi(exceptffusionforcreepthelealacvkesoflessdisloofcations!).an obvious trace than dislocation creep (exceptTexturalfor theindicatorslack of disloofcations!).diffusion creep: Equiaxed grains and straight grain boundaries (some 4 grain junctionsAPPRindiOACH:cativSIMPLEe of grainto progressivswitching.ely COMPLEXWhy do w! e associate these textures with diffusion APPROACH: SIMPLEcreepto(asprogressivopposeelyd toCOMPLEXdislocation! creep) ? Grain switching ? Grain boundary sliding ? Textural indicators of diffusion creep: Equiaxed grains and straight grain boundaries (some 4 Textural indicators of diffusiongraincreep:junctionsEquiaxeindidcatigrainsve ofandgrainstraiswitcghthing.grain bWhoundariesy do we(someassociate4 these textures with diffusion grain junctions indicative Also,of grcreepainstrainswitc(aslohing.oppcalization,oseWhd toy dislodomwylonitescatione associatecreep)andthese?uGrainltramtexturesswitcylonites...withhing ?diffGrainusionboundary sliding ? creep (as opposed to dislocationPressurecreep)solution:? Grain Hicswitckmanhing ?&GrainEvansboundary, Karzsliding& Scholz.? Also, strain localization, mylonites and ultramylonites... Also, strain localization,2 AmylonitesmPressureathemand usolution:ltramatiylonites...calHicpikmanctur& Eveansof, Karzdiff&usiScholz.on Pressure solution: Hickman & Evans , Karz & Scholz. 2.1 Di2ffusiAomn athemas a randoaticalm piwcturalk pre oofcessdiffusion 2 A mathematical picture of diffusion Use Shewmon2.1 DiChapterffusion 2.asCopa randoy it...mDowalthek prcalcsocessif time... Otherwise, just scan them... 2.1 Diffusion as a random walk process Use Shewmon Chapter 2. Copy it... Do the calcs if time... Otherwise, just scan them... Use Shewmon Chapter 2. Copy it... Do the calcs if time... Otherwise, just scan them... random walk (Karato, Ch. 8): 1 D = a2Γ (1) 1 D =c.n. a2Γ (1) 1 Γ = X c.n.Γ (2) D = a2Γ d m (1) Γ = XdΓm (2) c.n. Gm Γ = XdΓm Γm = ν exp − Gm (2) (3) Γm = ν exp R−T (3) ! RT " Gm ! " Γm = ν exp − 1 Gm (3) RT 1 2 2 Gm ! " D =D = a aννexpexp −− (4) (4) 1 G c.n.c.n. RRT D = a2ν exp − m !! ""(4) c.n. RT ! " Thoughc.n.theseis thec.n.equationscoisordinationthe coordinationarenumgeneralbner.umbΓ,er.theyisΓaisprobabilitaareprobabilitdescribingy yofofmigrating.migrating.diffusionννisisthedrivthelatticelatticeen bvibrationy vibrationconcenfrequencytrationfrequency. gradi-. c.n. is the coordination number. Γ is a probability of migrating. ν is the lattice vibration frequency. ents.

J = J (5) http://staff.aist.go.jp/nomura-k/english/itscgallary-e.htm i − v XiDi = XvDv (6) 2 X = 1 X 1 (7) 2 i −2 v ≈ D X D (8) i ≈ v v

Get to the origin of δD = √Dt

2.2 Diﬀusion as a continuum process

Fick’s First Law: ∂c J = D i (9) i ∂x − ! " This is fine for steady state flows... if not steady state, need a second equation: first, the “continuity” or mass conservation equation ∂c = J (10) ∂t −∇ (J = D x) (11) − ∇ ∂c = ( D x) (12) ∂t −∇ − ∇ ∂c = D 2c (13) ∂t ∇ Or, in 1-D 2 ∂c ∂ ci = Di (14) ∂t # ∂x2 $

2.3 How to measure D in experiment?

See Demouchy refs... Mackwell paper.

3 Stress-driven diﬀusion

Lehner, Takei... Schmalzreid paper (ﬁnd it !) and section in shewmon?? Assume D is not stress dependent !

3.1 Pathways !

This is the critical concept in Diffusion creep: there are the effects on Diffusivity, but there are also the critical effects of the diffusion path. Here are some options (Figure): Lattice, Dislocations (Pipe Diffusion), Grain Boundaries, Fluid phase... draw them !

3 µµ bb ττmaxmaxµ b== (2.9)(2.9) τ = 22ππaa (2.9) max 2π a µµ µ b ττmax (2.10)(2.10) τmax = maxµ ≈ 22ππ (2.9) 2π aτmax ≈ (2.10) µ ≈ 2π TheThe slopslopee ofof thethe shearshear stressstressτ atat thethe zerozero crossingcrossing givgiveses thethe shearshear momodulusdulus µµ(2.10).. TheThe implicationimplication isis The slope of the shear stress at maxthe zero crossing gives the shear modulus µ. The implication is thatthat ifif yyouou decreasedecrease thethe atomicatomic spacingspacing≈ 2πbb,, thethe elasticelastic momoduduluslus willwill increase.increase. ThisThis predictspredicts aa relationrelationsshhipip that if you decrease the atomic spacing b, the elastic modulus will increase. This predicts a relationship The slope ofbbettheetwweeneensheardensitdensitstressyy andandat sheartheshearzeromomodulus:crossingdulus: aagivdenserdenseres thematerialmaterialshear mowillwilldulushahavveeµa.a higherhigherThe implicationshearshear momodulus.dulus.is TToo seesee thisthis between density and shear modulus: a denser material will have a higher shear modulus. To see this that if you decreaseexpexperimenerimenthe atomictallytally,,sespacingseeeKaratoKaratob, theFigFigelastic4.11.4.11. TheThemodurelationshiprelationshiplus will increase.bbetetwweeenenThiselasticelasticpredictswwaavveevaveloelorelationcitcityyVVship µ/µ/ρρ,,thereforetherefore ∝∝ between densitexperimenytheandtallyeffshearect, seofemoµKaratoondulus:veloFigcita densery4.11.is greaterThematerialrelationshipthanwillthathaofbveteρw.aehigheren elasticshearwavmoe vdulus.elocity VTo see µ/thisρ, therefore the effect of µ on velocity is greater than that of ρ. ∝ !! experimenthetallye,ffseecte Karatoof µ onFigvelo4.11.city isThegreaterrelationshipthan thatbetofweρen. elastic wave velocity V µ/ρ, therefore ∝ ! the effect of µ on veloHOHOcitWEVER,WEVER,y is greatertheretherethanisis NOTNOTthat ofaa relationshipρrelationship. bbetetwweeneen activactivationation energyenergy!andand shearshear momodulusdulus bbecauseecause thethe HOWEVER, there is NOT a relationship between activation energy and shear modulus because the elasticelastic strainsstrains areare vveryery small,small, andanddodo notnot feelfeel thethe heighheightt ofofthethe ppotenotentialtial wwellell !! HOWEVER,elastic strainsthere isareNOTverya small,relationshipand dobnotetweenfeelactivthe heighationtenergyof the andpotensheartial wmoelldulus! because the elastic strains are2.2.v22ery small,WhatWhatandiissdoaanotthertherfeelmmthealalheighllyy actiactit of thevvatedatedpotentialprproowcess?ellcess?! 2.2 What is a thermally activated process?

2.2 What is a thermally activated process? EEaa kk==AAEexpexp−− (2.11)(2.11) k = A exp − a RRTT (2.11) Ea RT EEa 11 k = A exp − ln(ln(kk))== −− a ++ln(ln(AA)) (2.11) (2.12)(2.12) RT Ea 1TT TT ln(k) = − "" +##ln(A) (2.12) E 1 T T ln(k) = − a "+ ln(#A)∂∂lnlnkk (2.12) EEa == RR (2.13)(2.13) T Ta ∂−ln k 11 "Ea =# R − $$ ∂∂ T %% (2.13) − ∂ 1 T PP ∂ ln k $ T %P R = 8.314 kJ/mol Ea = R (2.13) R = 8.314 kJ/mol − ∂ 1 R = 8.314 kJ/mol $ T %P R = 8.314 kJ/mol2.2.33 WhatWhat araree defects,defects, wwhathat dodo theythey do?do? 2.3 What are defects, what do they do? 2.3 WhatTTakarakei’sei’se defects,thermothermodynamicsdynamicswhat(Gibbs),(Gibbs),do theyKohlstedt’sKohlstedt’sdo? equilibriumequilibriumconcenconcentrationtration ofofvvacancies.acancies. Takei’s thermodynamics (Gibbs), Kohlstedt’s equilibrium concentration of vacancies. Takei’s thermodynamics (Gibbs), Kohlstedt’s equilibrium concentration of vacancies. The equilibrium SconceSconfconf ==ntrakkBBtionlnlnWW of vacancies (2.14)(2.14) S = k ln−−W (2.14) conf B Maxwell-Boltzmann $!! where,where, forfor aa ...... nono ininteractions,teractions, etc...etc... − where, for a ... no interactions, etc...Sconf = kB ln W Statistical Thermodynamics: (2.14) − ((NN++nnv)!)! #!! where, for a ... no interactions, etc... W = v (2.15) W(N=+ nv)! (2.15) increasing T W = NN!!nnvv!! (2.15) && '' (N + n )! N!nv! W = v& ' (2.15) ! N!n ! & v ' ff GG((TT,,PP,,NN,,nnvv))==GG00((TT,,PP))f++nnvvggvv kkBBTTlnlnWW (2.16)(2.16) G(T, P, N, nv) = G0(T, P ) + nvgv kBT−−ln W #!!(2.16) − ff ((NN++nnvv)!)! 1.13:/*+;<=708 GG((TT,,PP,,NN,,nnvv))==fGG00((TT,,PP))++nnvvggv kkBB(TTNlnln+ nv)! (2.17)(2.17) G(T, P, N, nv) = G0(T, P ) + nvg kBT lnf W v (2.16) G(T, P, N, nv) = G0(T,vP ) + nvgv kBT−−ln NN!!nnvv!! (2.17) − − N!&n&v! '' $!! f (N + nv)!& ' G(T, P, N, nv) = G0(T, P ) + nvgv kBT ln (2.17) − N!nv! ∂∂GG ff & ∂∂ ' ∂G ((TT,,PP))==gg ∂kkBTT lnlnWW %!! (2.18)(2.18) ∂n = gf k Tvv − Bln W∂n (2.18) ∂nvv v B − ∂nvv ! !"!!# !"!!$ !"!!% !"!!& !"!' ∂nv"" (T,P#)# − ∂nv ∂G " #∂f∂GG ∂ f ∂∂ ((NN++nnvv)!)! ( *+),-,.-/+-0.-1.23,240.+5613+70819+ (T, P ) = g kBT ln Wf (2.18) ) ∂G v ((TT,,PP))==ggvv ∂kkBBTT (N ln+lnnv)! (2.19)(2.19) ∂nv ∂∂nn−v ∂f nv −− ∂∂nnv NN!!nnv!! " # "" v ##= gv kBT ln v && v '' (2.19) ∂G ∂nv Figure(T,P ) 2.3: ∂LoT-hiT.jpg− (∂Nnv+ n )! N!nv! (T,"P ) =#gf k T Figureln 2.3: LoT-hiT.jpgv& ' (2.19) ∂n v − B ∂n 77N!n ! " v # v & v ' Figure 2.3: 7LoT-hiT.jpg with the solution : 7 with the solution : A ∂G f nv = g kBT ln (2.20) with the solution : ∂vG f nv ∂nv (T,P ) − = gN +knvT ln (2.20) ! " ∂n ! v − B " N + n ! v "(T,P ) ! v " ∂G∂G f f nv % = g=∂vGgv kBkTBlnT (lnXv) (2.21)(2.20) ∂n∂vn − − f N + n ! ! "v(T",P(T),P ) = gv! kBT vln"(Xv) (2.21) ∂nv (T,P ) − ∂G ! f" the equilibrium concentration occurs where ∂=G gv = 0kBT ln (Xv) (2.21) ∂n ∂nv − > the equilibrium conc!entrationv "(T,Po)ccurs where ∂G = 0 # $ ∂nv f ∂G g # $ the equilibrium concentration occurs whereeq =v 0 *+EF"+),-,.-/+-0.-"+5G=70819 X = exp∂n f (2.22)) v v gv kBeqT ( & # %$Xv &= exp (2.22) f %kBT &

eq gv ln Xv = exp (2.22) kBT > & ? '! '' '# 2.3.1 What is the equilibrium concentration%of po&int defects? '=B17613,2C31*+D $ 2.3.1 What is the equilibrium concentration of point defects? @+'! Karato 5.2 ; Entropy and Free Energy of point defects 2.3.1 What is the equilibrium concentration of point defects? configurationalKaratoentrop5.2y and; Envibrationaltropy and FenreetropEnergyy ; of point defects configurational entropy and vibrational entropy ; Karato 5.2 ; Entropy and Free Energy of point defects 2.configurational4 What iens tropa wyorandkivibrationalng definientitropony ;of High temperature? 2.4 What is a working definition of High temperature? The2.homologous4 Whattempis eraturea wor! kiThng= Tdefini/Tm. IttiisonwhereofthermallyHigh temactivatedperproaturcessee?s begin to influence the strength ofThethehomologousmaterial (toutempghnesserature, but! alsoTh =theT/Tpemak. Itstressis wherethatthermallya materialactivcanatedsuppproort).cesses begin to influence the strength of the material (toughness, but also the peak stress that a material can support). The homologous temperature ! Th = T/Tm. It is where thermally activated processes begin to influence the strength of the material (toughness, but also the peak stress that a material can support). 8 8

8 Though these equations are general, they are describing diﬀusion driven by concentration gradients.

J = J (5) i − v Though these equations are general, they are describing diffusion driven by concentration gradi-XiDi = XvDv (6) ents. Though these equations are general, they are describing diffusion driven by concentration gradi- Xi = 1 Xv 1 (7) ents. − ≈ Ji = Jv (5) − J = J Di XvDv (5) (8) i − v ≈ XiDi = XvDv (6) XiDi = XvDv (6) Get to the origin of δD = √Dt Xi = 1 Xv 1 X = 1 X 1 (7) (7) − ≈ i − v ≈ Though these equations are general, theyDare describingX D diffusion driven by concentration gradi-(8) Di XvDv i ≈ v v (8) ents. ≈ Get to the origin of δD = √Dt Get to the origin of δD = √Dt 2.2 Diffusion as a Jcoi =ntiJvnuum process (5) − XiDi = XvDv (6) 2.2 Diffusion as a continuum process Fick’s First Law: X = 1 X 1 (7) CONTINUUM VIEW: a tracei r −betwv ≈een two rods: 2.2 Diffusion as a continuumFicprk’soFircessst Law: ∂ci Di XvD∂vc J = Di (8) (9) J = ≈D i − ∂x (9) Fick’s First Law: − i ∂x ! " Get to the origin of δD = √Dt ! " or,Fick’ins 1stgeneral,∂ Laci w: J = D c. This is fine for steady state flows... if not steady state, need a second This isJfine=for Dsteady state flows... if not steady state, need a second equation:(9) first, the “continuity” − i ∂x − ∇ or massequation:conservation! efirst,qu"ation the “continuity” or mass conservation equation or, in general, J = D c. This is 2fi.ne2 forDiffsteadyusion asstatea coflontiws...nuumif prnotocess∂steadyc state, need a second − ∇ = J (10) equation: first, the “continuity” or mass “continconservationuity”equation eqn: ∂t −∇ ∂c Fick’s First Law: (J = D x) = J (11) (10) − ∇∂ci ∂t −∇ ∂c J∂=c D (9) = J =− i (∂xD x) (10) (12) ∂t −∇ ∂t −∇! − "∇ (J = D c) (11) This is fine for steady state flows... if not steady∂c state, need a second equation: first,− the∇“continuity” (J = D c) = D 2c ∂(11)c (13) or mass conserv−ation∇equation ∂t ∇ ∂c = ( D c) (12) Or, in 1-D = ( D c) ∂c ∂(12)t −∇ − ∇ = J 2 (10) ∂t −∇ − ∇ ∂∂ct −∇ ∂ ci = Di ∂c 2 (14) ∂c 2 ∂(Jt = D# ∂xx)2 $ = D c (11) (13) = D c − ∇ (13) ∂t ∇ ∂c ∂t ∇ = ( D x) (12) Or, in 1-D 2.3 How to measure D in experimen∂t t?−∇ − ∇ Fick’Or, sin 2nd1-D Law ∂c 2 = D 2c 2 (13) See Demouc∂c hy refs...∂ Macci kwell paper. ∂t ∇ ∂c ∂ ci = Di 2 (14) Or, in 1-D∂t # ∂x $ = Di 2 (14) 2 ∂t # ∂x $ 3 Stress-driven diffusion ∂c ∂ ci = Di (14) 2.3 How to measure D in experiment? ∂t # ∂x2 $ Lehner,2T.ak3ei... HoSchmwalzreidtopapmeasurer (find it e!) andD sectionin expin shewmon??erimenAssumet? D is not stress See Demouchy refs... Mackwell paper.2dep.3 endenHotw! to measure D in experiment? See Demouchy refs... Mackwell paper. 3.1 PSeeathwDemoucays ! hy refs... Mackwell paper. 3 Stress-driven diffusion 3ThisStris theess-drcriticalivconcepten diinffusiDiffusionon creep: there are the effects on Diffusivity, but there are also the critical3 effectsStrof theess-drdiffusion path.ivenHere diare someffusioptionson(Figure): Lattice, Dislocations (Pipe Lehner, Takei... Schmalzreid paperLehner,(find itTak!)ei...andSchsectionmalzreid inpapshewmon??er (find it !) andAssumesectionDin isshewmon??not stressAssume D is not stress Diffusion), Grain Boundaries, Fluid phase... draw them ! dependent ! dependent ! Lehner, Takei... Schmalzreid3 paper (find it !) and section in shewmon?? Assume D is not stress 3.1 Pathways ! 3.1 Pathways ! dependent ! This is the critical concept in Diffusion creep: there are the effects on Diffusivity, but there are also This is the critical concept in Diffusion creep: there are the effects on Diffusivity, but there are also the critical effects of the diffusion path. Here are some options (Figure): Lattice, Dislocations (Pipe the critical effects of the diffusion path.Diffusion),Here3.1Grainare somePBouathndaries,optionswaFluidys(Figure):!phase... draLattice,w themDislo! cations (Pipe Diffusion), Grain Boundaries, Fluid phase... draw them ! This is the critical concept in3 Diffusion creep: there are the effects on Diffusivity, but there are also the critical3 effects of the diffusion path. Here are some options (Figure): Lattice, Dislocations (Pipe Diffusion), Grain Boundaries, Fluid phase... draw them !

Contrib Mineral Petrol (2007) 154:279–289 283

cÃ Results 1 dx D~ cÃ zdc; 18 ð Þ ¼ À 2t dc c ð Þ Fe–Mg interdiffusivity Ã Z0

where t the duration of the diffusion experiment. The transmitted light micrograph in Fig. 1 shows the dif- The derivative term in Eq. 18 was determined by diffusion zone between MgO and (Mg,Fe)O, as well as the ferentiating Eq. 16, while the actual data are used to cal- initial interface in sample PI-1238. The two crystals of the culate the integral term, as described in Mackwell et al. diffusion couple bonded, during the high-temperature dif- (2005). In this way the entire data set was fit (a ‘‘global fusion anneals at temperatures over 1,100°C. The cracks fit’’) by varying D~0; Q; B and C in the following equation did not modify the concentration profiles, indicating that to define the interdiffusion coefficient (see Schmalzried they formed either during the quench or during the prep- 1981, p. 38; Mackwell et al. 2005): aration of the sample for microprobe analyses. Contrib Mineral Petrol (2007) 154:279–289 283 An interdiffusion profile for a sample annealed for 2 h at B Q C x =RT 2 1 D~ D~0 x expÀð þ Þ m sÀ ; 19 1,200°C as well as the fit to Eq. 16 is shown in Fig. 2. ¼ ð Þ Although the interdiffusion profile is clearly asymmetric, it cÃ Results 1 dx where D~0 is the pre-exponential term, which includes the is fit well by this equation. Using this fitted curve in ~ D cÃ zdc; 18 oxygen fugacity and water fugacity dependences, (Q + Cx) ð Þ ¼ À 2t dc c ð Þ Fe–Mg interdiffusivity Ã Z0 is the composition-dependent activation energy, B and C are fitting parameters, x is the mole fraction of FeO, T is the where t the duration of the diffusion experiment. Theabsoluttransmitte temperated lighture,micrograpand R ishthein gasFig.constant.1 shows Inthethisdif- The derivative term in Eq. 18 was determined by diffusion zone between MgO and (Mg,Fe)O, as well as the ContribdescripMineraltion,Petrolthe(2007)activati154:279–289on energy was expanded in a Taylor ferentiating Eq. 16, while the actual data are used to cal- initialDOIseries10.1007/s00410-007-0193-9interfto describeace in samits dependenceple PI-1238on. Tx.hOnlye twothecrysfirsttalstermofofthe culate the integral term, as described in Mackwell et al. diffusiontheORIGexpansionINAcoupleL PAPERbonded,is requiredduringto fitthethehigh-teexpermperatureimental datadif- (2005). In this way the entire data set was fit (a ‘‘global fusion(Schmalannealszried at1981temperat, p. 38ures; Zhaooveret 1,100al. 2004°C.; ThMacke crackswell fit’’) by varying D~0; Q; B and C in the following equation didetnotal. 2005modify). the concentration profiles, indicating that theyInfluenceAssumformedingofeitherthathydrogentheduringoxygentheon sublatticeFe–Mgquench orinterdiffusionis duringimmobilethecom-prep-in (Mg,Fe)O to define the interdiffusion coefficient (see Schmalzried and implications for Earth’s lower mantle 1981, p. 38; Mackwell et al. 2005): arationparedofto thethe samcationsple inforFemicroprobexMg1–xO (i.e.,analyses.the Matano interface is coincident with the original interface), the SylvieAn interdDemouchyiffusionÆ StephenprofilJ. Mackwele forl Æa sample annealed for 2 h at B Q C x =RT 2 1 DavidinterdL.iffusionKohlstedtcoefficient can be written as a function of the D~ D~0 x expÀð þ Þ m sÀ ; 19 1,200°C as well as the fit to Eq. 16 is shown in Fig. 2. self-diffusivities (D ) of the diffusing cations (i = Mg, Fe) ¼ ð Þ AlthoContribughMineralthePetrolinterd(2007) 154:279–289iffusioni profile is clearly asymmetric, it Fig. 1 Photomicrograph of a diffusion couple after a interdiffusion DOIaccordi10.1007/s00410-007-0193-9ng to the Nernst–Planck relation (Schmalzried experiment (PI-1238) at 1,200 C where D~0 is the pre-exponential term, which includes the is fit well by this equation. Using this fitted curve in ° 1981, p. 103): oxygen fugacity and water fugacity dependences, (Q + Cx) ORIGINAL PAPER Received: 28 September 2006 / Accepted: 27 February 2007 / Published online: 15 March 2007 is the composition-dependent activation energy, B and C Ó Springer-Verlag 2007 DFeDMg D~Fe Mg ; 20 are fitting parameters, x is the mole fraction of FeO, T is the InfluenceÀ ¼ xDofFehydrogenx 1 DonMgFe–Mg interdiffusionð inÞ (Mg,Fe)O absolute temperature, and R is the gas constant. In this andAbstracimplicationst Interdiffusionþ ofð FeforÀandÞEarth’sMg in (Mg,Fe)Olowerhas mantleIntroduction been investigated experimentally under hydrous condi- description, the activation energy was expanded in a Taylor tions.whereSinglex is thecrystalsmolofe fractionMgO in ofcontactFeO. withWe assumePericlase1thatand perovtheskite are the principal minerals of the Sylvie Demouchy Æ Stephen J. Mackwell Æ (Mgtherm0.73Feodynami0.27)O werecannealedfactorhydrothermis unitallyy,at as300 MPathe (Mg,lowerFe)Omantle.solidPerovskite is the most abundant mineral series to describe its dependence on x. Only the first term of David L. Kohlstedt betweesolution 1,000n is andnear1,250ly °idealC and forusingthea Ni–NiOcompositionbuffer. (and~80%conditionsby volume), followed by periclase (~15% by vol- the expansion is required to fit the experimental data After electron microprobe analyses, the dependence of the ume) with a composition of Mg1–xFexO with x between 0.1 (Schmalzried 1981, p. 38 ; Zhao et al. 2004; Mackwell interdinvestigatiffusivityedon Feinconcentrthis study.ation was determined using a and 0.2 (Ito and Takahashi 1987; Ringwood 1991; Guyot BoltzmSinceann–Mtheatanocationanalysis.self-difFor a fusivitieswater fugacityareofrelatetedal. 1988to the; Fiquetva-et al. 1998). Although periclase is un- et al. 2005). ~300 MPa, the Fe–Mg interdiffusion coefficient in likely to form an interconnected phase in the lower mantle, FecancyxMg1–xOselwithf-diffusivi0.01 £ x ty£ 0.25and canits beconcentratdescribed byion (Nakamit is still uraone ofandthe most abundant minerals inside the Assuming that the oxygen sublattice is immobile com- Received: B28 SeptemberQ Cx =2006RT / Accepted: 27 February 20074 / Published2 1 online: 15 March 2007 D~ D~0x expÀð þ Þ with D~0 5 1 10À m sÀ ; Earth, and its physical and chemical properties are of major ÓSchm¼Springer-Verlagalzried200719841 ) through¼ ð Æ Þ Â pared to the cations in FexMg1–xO (i.e., the Matano inter- Q 270 20 kJ molÀ ; B 0:8 0:1; and C = –80 ± interest for understanding the dynamics of the deep Earth ¼ Æ–1 ¼ Æ 10 kJ mol . For x = 0.1 and at 1,000°C, Fe–Mg interdif- (e.g., convection, chemical homogenization, rheological face is coincident with the original interface), the Abstract Interdiffusiontot of Fe and Mg in (Mg,Fe)O has Introduction fusionDi isDa VfactorVMe00of ~4 ;faster under hydrous than under properties). 21 interdiffusion coefficient can be written as a function of the beenanhydrinvestigatous conditio½ed ns.exper Thisimentalenhancedly undrateer ofhydrouinterdsiffusioncondi- Previous studiesð Þon Fe–Mg interdiffusion in periclase tions. Single crystals of MgO in contact with Periclase1 and perovskite are the principal minerals of the self-diffusivities (D ) of the diffusing cations (i = Mg, Fe) istheattributinterded iffusionto an increasedcoefficieconcentntrationfor theof presentmetal (andstudiroyn-containiis givenng periclase) examined the effects of i Fig.(Mgvacan10.73ciesPhotomicrograFe0.27resulting)O werefromannealedphtheofincohydrothermarporationdiffusionallyofathydrogcouple300 MPaen. afterlowertempera interdiffusionature,mantle.presPerovssure andkite oxygenis the mosfugacityt abunda(Yamazaknt mineri andal betweeby n 1,000 and 1,250°C and using a Ni–NiO buffer. (~80% by volume), followed by periclase (~15% by vol- according to the Nernst–Planck relation (Schmalzried experimentSuch water-i(PI-1238)nduced enhancat ement1,200of°Ckinetics may have Irifune 2003; Holzapfel et al. 2003; Mackwell et al. 2005). Afterimportantelectronimplicamicrotionsprobefor theanalysesrheological, the dependpropertenceies of the ume)Underwithdry aconditicomposions,tionFe–Mgof Mginterdiffusi1–xFexO wionth xratesbetweenincrease0.1 1981, p. 103): interdlower~ iffusivimantle.ty on Fe concentrtotation was determined using a andwith0.2increas(Ito anding Takahairon contentshi 1987;andRingwoodincreas1991ing ;oxygenGuyot DFe Mg DV VM00 e : 22 BoltzmÀann–M¼atano ½analysis. For a water fugacity of etfugacital. 1988y, but; Fiquetdecreaseð Þet al.with1998Fig.increas). 2AlthoughConcentrationing prespericlsureasebyisprofile~un-2.5 (x = Fe/(Fe + Mg)) of a sample (PI- D D ~Keywor300 MPa,ds Periclthe Fe–Mgase Fe–MginterdinterdiffusioniffusioncoefHydrogenficient in likelyorderstoofformmagnitudean interconnectedfrom onephaseatmospin theherelowerto 23mantle,GPa Fe Mg Fe Mg O with 0.01Á x 0.25 can be describÁ ed byÁ it is still one of the most1240),abundaannealednt mineralsatinside300 theMPa, 1,250°C for 2 h, as a function of the D~ ; 20 Earth’sTherex 1–manxfore,tle the depend£ £ ence of D~Fe Mg on x and(MackwfHellO etentersal. 2005). Fe Mg ~ ~ B Q Cx =RT ~ 4 2 1 2 position in the diffusion couple perpendicular to the original interface. À ¼ xDFe x 1 DMg ð Þ D D0x expÀð þ Þ with D0 5 1 10ÀÀ m sÀ ; Earth,Nomiandnallyits physicalanhydrousandminerchemicals ofal Earth’spropertiesmantleare of(olivinemajor through¼ the depend1 ence¼ofð ÆtheÞ Âtotal vacancy concentration þ ð À Þ Q 270 20 kJ molÀ ; B 0:8 0:1; and C = –80 ± interestand its highfor unpressurederstandingpolymorpInitialthe dynamichs, asinterfacewells oasf thepyroxenesisdeepset Earthatandz = 0. The solid line shows the non-linear ¼ Æ–1 ¼ Æ 10inkJthemolpres. Forencex = 0.1ofandhydrogat 1,000en°C,onFe–Mgtheseinterdif-quantitie(e.g.,garnet)s.convection,incorporatechemwatericalasleasthydrogenhomosquaregenizatin fitconjion,tounctionrheologithe datawithcal where x is the mole fraction of FeO. We assume that the fusionCommunicatedis a factorby T.L.ofGrove.~4 faster under hydrous than under propeintrinsirties).c point defects in their structure (Kohlstedt et al. anhydrous conditions. This enhanced rate of interdiffusion 1996Previous; KohlstedtstudiesandonMackFe–Mgwell interdiffusi1998; Ingronin andin periclaseSkogby thermodynamic factor is unity, as the (Mg,Fe)O solid S. Demouchy S. J. Mackwell is attributedÁ to an increased concentration of metal (and2000;iroBolfan-Cn-containiasanovang periclase2005). The) exampresenceined theof hydrogeffectens oinf vacanLunar andciesPlanetaryresultingInstitute,from 3600the incoBay ArearporationBlvd, of hydrogen. temperature, pressure and oxygen fugacity (Yamazaki and solution is nearly ideal for the composition and conditions Houston, TX 77058, USA nominally anhydrous silicate minerals, even at very low 123 investigated in this study. Such water-induced enhancement of kinetics may have Irifune 2003; Holzapfel et al. 2003; Mackwell et al. 2005). importantS. Demouchyimplica(&) tionsD. L. Kohlstedtfor the rheological properties of the Under1 Strictlydryspeaking,conditithisons,mineralFe–Mgpresentsinterdiffusiin the loweron ratesmantleincrshouldease Á Since the cation self-diffusivities are related to the va- lowerDepartmentmanoftle.Geology and Geophysics, withbe call ferropericlaseincreasing butironit iscontentnot a term andapprovedincreasby theingInternationaloxygen University of Minnesota, 310 Pillsbury Drive SE, fugacitMineralogicaly, butAssociationdecrease yet.withAlso,increasthe ingsamepresmineralsure (Mg,Fe)Oby ~2.5 Minneapolis, MN 55455, USA containing only a small amount of Fe, has been call magnesiowu¨stite cancy self-diffusivity and its concentration (Nakamura and Keywords Periclase Fe–Mg interdiffusion Hydrogen orders of magnitude from one atmosphere to 23 GPa e-mail: [email protected]Á Á Á for many years. Schmalzried 1984) through Earth’s mantle (Mackwell et al. 2005). Nominally anhydrous minerals of Earth’s mantle (olivine and its high pressure polymorphs, as well as pyroxenes and tot 123 Di DV VMe00 ; 21 garnet) incorporate water as hydrogen in conjunction with ½ ð Þ Communicated by T.L. Grove. intrinsic point defects in their structure (Kohlstedt et al. the interdiffusion coefficient for the present study is given 1996; Kohlstedt and Mackwell 1998; Ingrin and Skogby S. Demouchy S. J. Mackwell 2000; Bolfan-Casanova 2005). The presence of hydrogen in Lunar and PlanetaryÁ Institute, 3600 Bay Area Blvd, by Houston, TX 77058, USA nominally anhydrous silicate minerals, even at very low

1 tot S. Demouchy (&) D. L. Kohlstedt Strictly speaking, this mineral presents in the lower mantle should ~ Department of GeologyÁ and Geophysics, be call ferropericlase but it is not a term approved by the International DFe Mg DV VM00 e : 22 À University of Minnesota, 310 Pillsbury Drive SE, Mineralogical Association yet. Also, the same mineral (Mg,Fe)O ¼ ½ ð Þ Fig.Minneapolis,2 ConcentrationMN 55455, USAproﬁle (x = Fe/(Fe + Mg)) containingof a sampleonly a small(PI-amount of Fe, has been call magnesiowu¨stite ~ 1240),e-mail: [email protected] 300 MPa, 1,250°C for 2 h, as fora manyfunctionyears. of the Therefore, the dependence of DFe Mg on x and fH2O enters À position in the diffusion couple perpendicular to the original interface. through the dependence of the total vacancy concentration Initial interface is set at z = 0. The solid line shows the non-linear 123 in the presence of hydrogen on these quantities. least square ﬁt to the data

123

Though these equations are general, they are describing diﬀusion driven by concentration gradi- Though these equations are general, they are describing diﬀusion driven by concentration gradients. ents. J = J (5) Ji =− Jv (5) i − v XiDi = XvDv (6) XiDi = XvDv (6) X = 1 X 1 (7) Xi = 1− Xv ≈ 1 (7) i − v ≈ D X D (8) Di ≈ Xv Dv (8) i ≈ v v Get to the origin of δD = √Dt Get to the origin of δD = √Dt

2.2 Diffusion as a continuum process 2.2 Diffusion as a continuum process Fick’s First Law: Fick’s First Law: ∂ci J = D ∂ci (9) J =− Di ∂x (9) − !i ∂x" or, in general, J = D c. This is fine for steady state! flo" ws... if not steady state, need a second or, in general, J =− D∇ c. This is fine for steady state flows... if not steady state, need a second equation: first, the “con− tin∇uity” or mass conservation equation equation: first, the “continuity” or mass conservation equation ∂c ∂c= J (10) ∂t =−∇ J (10) ∂t −∇ (J = D c) (11) (J =− D∇ c) (11) ∂c − ∇ ∂c= ( D c) (12) ∂t =−∇ −( D∇ c) (12) ∂∂c t −∇ − ∇ ∂c= D 2c (13) ∂t = D∇ 2c (13) ∂t ∇ Or, in 1-D Or, in 1-D 2 ∂c ∂ c2i ∂c= Di ∂ 2ci (14) ∂t = D#i ∂x $ (14) ∂t # ∂x2 $ 2.3 How to measure D in experiment? 2.3 How to measure D in experiment? See Demouchy refs... Mackwell paper. See Demouchy refs... Mackwell paper. 3 Stress-driven diffusion 3 Stress-driven diffusion Stress-driven diffusion stress effect on chemical potential: 0 µi = µi (0T ) + Ωiσn (15) µi = µi (T ) + Ωiσn (15) vacancies µi = Ωi σn (16) consumed on ∇ µ = Ω∇ σ (16) ∇ i i∇ n face in compression σn (σ11 σ33)/l = σ!/l (17) ∇ σn≈ (σ11− σ33)/l = σ!/l (17) vacancies ∇ ≈ − so the flux then is so the flux then is created on face in tension Though these equations are general, theyJ =areD describingµ DΩi σ diffusion driven by(18)concentration gradi- J =− D∇ µ≈ − DΩ∇ σ (18) − ∇ ≈ − i∇ ents. atom flux in opposite3 direction as vacancy flux:3 J = J (5) i − v XiDi = XvDv (6) X = 1 X 1 (7) i − v ≈ D X D (8) i ≈ v v

Get to the origin of δD = √Dt

2.2 Diﬀusion as a continuum process

Fick’s First Law: ∂c J = D i (9) i ∂x − ! " or, in general, J = D c. This is fine for steady state flows... if not steady state, need a second − ∇ equation: first, the “continuity” or mass conservation equation ∂c = J (10) ∂t −∇ (J = D c) (11) − ∇ ∂c = ( D c) (12) ∂t −∇ − ∇ ∂c = D 2c (13) ∂t ∇ Or, in 1-D 2 ∂c ∂ ci = Di (14) ∂t # ∂x2 $

2.3 How to measure D in experiment?

See Demouchy refs... Mackwell paper.

3 Stress-driven diﬀusion

0 µi = µi (T ) + Ωiσn (15) µ = Ω σ (16) ∇ i i∇ n σ (σ σ )/l = σ!/l (17) ∇ n ≈ 11 − 33 so the ﬂux then is

J = D µ DΩ σ (18) − ∇ ≈ − i∇

3. Diffusion pathways

3.1g3rain.1P boundarathPwathyaysw!ays ! This is the critical concept in Diffusion creep: there are the effects on Diffusivity, but there are also This is the critical concept in Diffusion creep: there are the effects on Diffusivity, but there are also the critical effects of the diffusion path. Here are some options (Figure): Lattice, Dislocations (Pipe the critical effects of the diffusion path. Here are some options (Figure): Lattice, Dislocations (Pipe Diffusion), Grain Boundaries, Fluid phase... draw them ! dislocations grain Diffusion), Grain Boundaries, Fluid phase... draw them ! matrix Dgb >> Ddis > Dgm Dgb >> Ddis > Dgm Vgm >> Vgb > Vdis Vgm >> Vgb > Vdis

4 Rheological Data and Flow Laws 4 Rheological Data and Flow Laws Kohlstedt’s way of writing constitutive equations in the treatise... explain it, as alternative to the wayKohlstedt’syou see writtenwayevoferywhere.writing Mecconstitutivhanistic,etoequationsexplain theinphtheysics...treatise...Whereasexplainthis isit,theasempirialternativcal e to the way,wdefiningay you sethee wfactorsrittenthateverywhere.we can isolMecatehanistic,by curve tofitting.explain the physics... Whereas this is the empirical way, defining the factors that we can isolate by curve fitting. 0 p Qdf ε˙ = A σ d− exp − (19) df RT ! " 0 p Qdf ε˙ = A σ d− exp − (19) df RT 5 Effects of water, melt and GB crud ! " 5 Effects of water, melt and GB crud

620

Fig. 2a±d HREM images and diraction patterns of grain boundaries in the ultramylonite. a Boundary between two plagioclase grains. Lattice 618 fringes of the both grains are in direct contact at the boundary. in the north-east part of Japan (Otsuki and Ehiro 1992; boundary plane, the lattice fringes in two adjacent grains b Amorphous layer at a pla- Kubo et al. 1994). This zone has been estimated to ex- overlap in the HREM image. It is not easy to set the gioclase boundary formed by tend more than 200 km in a N-S trend and to have left incident beam exactly parallel to the grain boundary electron irradiation. c Boun- lateral displacements of about 60 km, that occurred plane, so HREM images were observed from thin re- dary between K-feldspar and plagioclase grains. d Boundary during the Cretaceous to Paleogene period (Otsuki and gions where the eect of the inclined grain boundaries is Phys Chem Minerals (1999) 26: 617±623 Ó Springer-Verlag 1999 between plagioclase and quartz Ehiro 1992). Some of the ultramylonite shows fracture minimized. The HREM images shown in this paper are grains ®lling in the original-granite rock at a macroscopic scale, one-dimensional lattice fringes, which can be obtained ORIGINAL PAPindicatingER that the mylonite originated from a brittle- under a wide range of conditions without exactly con- deformation fault, which later became a ductile-defor- trolling the diraction conditions and defocus values mation fault (Kubo and Takagi 1997). Plagioclase (Shindo and Hiraga 1998), so the grain boundaries re- porphyroclasts with asymmetric pressure shadows, ported in this paper are622not unusual. which are composed of ®ne-grained (£10 lm) plagio- Many minerals are sensitive to electron irradiation, T. Hiraga á T. Nagase á M. Akizuki clase, K-feldspar and quartz grains, are well developed and transform easily into the amorphous state at grain boundary (Ichinose 1992) where the segregation of hy- in this sample. As can be seen in a scanning electron boundaries (Carter and Kohlstedt 1981; Hay and Evans drogen results in expansion of one lattice spacing by The structuremicrosof graincope (SEM)boundariesimage (Fig. 1a),ineverygranite-originmineral pres- 1988;ultramyloniteBons et al. 1990). This irradiation damage is the 20% perpendicular to the boundary plane. studied by high-resolutionent exhibits a weak shapeelectron-preferredmicroscopyorientation. Most major impediment to the investigation of grain bound- Serrated and curved boundaries formed by grain grain boundaries are smoothly curved; a few are planar. aries in minerals by HREM. In this work, this diculty boundary migration are common in plastically deformed The present observations, in which the ultramylonite has been overcome by employing a low-dose technique Phys Chem Minerals (1999) 26: 617±623 Ó Springer-Verlag 1999 rocks like those observed here. In spite of the develop- contains phengite but no biotite (which is observed in used for HREM observations of zeolite (e.g., Alfredsson the host rock granite), indicate low-grade metamor- et al. 1995). A weak electron source of about 1 pA/cm2 ment of such boundaries, as can be seen in Fig. 1b, the ORIGINAL PAPphismER of the ®ne-grained layers. and an exposure time of about 5 seconds at a direct existence of the low-index plane grain boundaries even magni®cation 150,000 were used. In addition, to in the deformed rocks indicates that low-index plane minimize the total doseÂof electron irradiation, a highly- boundaries are stable compared with other high-angle grain boundaries. This type of boundary has been fre- Received: 15 September 1998Experimental/ Revised, accepted:procedures8 April 1999 sensitive TV camera was used in the HREM observa- T. Hiraga á T. Nagase á M. Akizuki tions. As for photographic ®lms, highly-sensitive ®lm quently observed in polycrystalline materials especially Thin specimens for TEM observations were prepared (Mitsubishi MEM), which is about four times as sensi- in covalent bonded materials, and its stability has been AbstracThe structuret The structurefromofof graipiecesgrainn bouof thinndariesboundariessectionsin athatgranite-wereinion-milleparticgranite-originular,d withtheanwidthtive asofultramylonitenormalthe graiFujin boundaryFG ®lm, wasis relatedused. Compared with interpreted as the lower grain boundary energy of the origin ultramylonite, acceleracomposedting mainlyvoltage ofof ®ne-3 kVgrainedat an angledirectofly15°to. Bograithn usualboundaryHREMdiobservausion, tionswhichwithplaya sdirectan magni®cation higher density plane of the matching atom (Ichinose of 400,000 , an exposure time of 2 seconds and normal et al. 1988). The results obtained indicate that low-index feldspastudiedr andbyquartz,high-resolutionsideswasofstudiedthe thinbysamphigh-resolutelectronles were coation edmicroscopyimportawith carbnt onroleto in metamorphicÂ reactions and in the electron microscopy (HREM)prevent charging. At mostduringof theTEMbound-observdefoations.rmationTEM,of rocks®lm, (e.g.,an incideJoestennt electro1983;nRubbeamie 1986)in the. low-dose tech- plane boundaries are abundant in most polycrystalline aries, not only betweenHREtheM,sameandmineralsanalyticalbutelecalsotronbe-micTheroscopystructure(AEM)of grainniqueboundariis approximates is,elytheref70 timore,es ofweaker.greatTaking account rocks. tween dierent minerals,observalatticetionsfringeswereincarrieadjacentd out grainson a 200interekV electrost to npemi-trologistsof theantimd structurale to set upgeologdiractioists,nyetconditionsfew and an op- Voids along quartz-quartz boundaries were reported timum defocus for HREM observations, the total elec- meet at the interface croscopwith noe othe(JEM-2010)r appreciablewith anphaseenes.rgy studiesdispersivehaveX-raybeen made of the major minerals, such as previously (e.g., White and White 1981; Behrmann 1985; tron dose is roughly estimated to be as little as a few In these boundaries, someanalyzerof the(EDstraiX) anghtd onsegmenta 300skVcor-(JEM-30feldspa10)relecandtronquartz in crustal lithologies. White and Bons et al. 1990). The origin of the voids has been in- respond to a low-indexmicrosplanecope.of HREMone of theobservconnectationsedof graiWhiten boundari(1981) inesan investigation of grain boundaries in a terpreted in terms of three mechanisms: grain boundary Received: 15 September 1998were/ Revised,made withaccepted:the incident8 April beam1999 parallel to the grain grains. Boundaries containing voids, with a spheroidal quartz mylonite withFig. 1transmissia, b Microstonructuelecres tronof the microsultramylocopynite. a SEM image of a sliding (Behrmann 1985), healing of ¯uid ®lled micro- shape elongated alonboundag the boury planendaries,to examinwereeobservthe possedible(TEM),existencereporteof thind preferential electron irradiation dam- cracks (Kronenberg et al. 1990), and transformation grain boundary phases and with diraction conditions ®ne-grained region etched by a HF solution. Darker grains with only between quartz grains. It is suggested that these age zones at grainsmbouoothndariescontrast arconte of qaininguartz andsomeother gvoidrainss.with scratches are of from continuous ¯uid ®lms that wetted grain boundaries appropriate for observing lattice fringes in both con- boundaAbstracriest Thewerestructureformedofbygraihealin boung ofndariesmicrocracks.in a granite-The Behrmparticular,ann (1985)the wiexaminfdtheldspaofrs.edtheb aTgrai®ne-graiEMn imboundaryagnede ofquathrtzeis urelatedmylltramo-ylonite. Qz quartz, (Urai 1983). We conclude that the boundaries observed nected grains. If the electron beam is inclined to the Pl plagioclase, K K-feldspar structuraorigin ultral widtmylonith ofe,majorcomposedboundariesmainly, ofdeduced®ne-grainedfrom nitedirectwithly tograigrain boun boundariesndary£100diusion,nm in whicwidth aplaynd withs an in this study formed by healing of microcracks, for the lattice-ffeldsparringeandimaging,quartz, iswaslessstudiedthan aboutby 0.5high-resolutnm. ion manimportay spheroidnt rolealinvoidmets; amorphiche interpretreaced tionsthe formatand ionin theof following reasons: (i) Grain boundaries with voids are electron microscopy (HREM). At most of the bound- thedefoboundarirmation esofasrockscreep(e.g.,damageJoestendue 1983;to grainRubboundaie 1986)ry. concentrated in local areas in the sample and tend to aries, not only between the same minerals but also be- The structure of grain boundaries is, therefore, of great should be noted that thisKeytypewordsof boundaryGrain boundaryis observá Lated tice fringe á sliding. In addition to TEM observations, Farver and extend for relatively long distances. Also, many are tween dierent minerals, lattice fringes in adjacent grains interest to petrologists and structural geologists, yet few only between quartz grains.High-resolutionThe voids tendelectronto accummicroscou- pyDiscussioná Ultramyloniand conclusionte Yund (1995) showed from diusion measurements of an straight, and only a few are smoothly curved. These meet at the interface with no other appreciable phases. studies have been made of the major minerals, such as late in speci®c boundaries and the boundaries commonly anorthite aggregate that the eective width contributing features resemble the form of cracks in rocks. (ii) Ob- connect to the same kindIn theseof boundarieboundaries,s assomeshownof thein straiTheghtpresentsegmentHREMs cor- observatofeldspagraitionsnr bouandwithndaryquartzthedilowinusioncrustaldoseis 3 tlitholo 5 nm.ogies.EvenWhitein theseand served voids have no tendency to develop at triple Fig. 5. Also, sub-grainIntroductionresponboundariesd to frequa low-entlyindexstartplanefromof onetechniquof thee yieconnectld valuabed lestudies,Whiteinforma(1981)howetionver,inatanlittainvestnanometle ofigationtheerrealofnagraituren boundaand characterries in a junctions, as can be seen in Fig. 5. Thus, the voids in the the boundaries with vograins.ids. SomeBoundaof theriesboundaricontaininges ap-voids,scalewithabouta spherothe oriidalginal structureofquartzgrainmylonitboundariof graine withboues wasndariestransmissidetermin ined.on electron microscopy present sample could not have resulted from grain pear to contain a secondaInshaperocks,ry elongatedphasethere. Figarealonuregrain6b,g theboundarian bouen-ndaries,aesgranite-betwwereeenorigintheobservsameultraedmylonit(TEM),Thee consipurposereportestingdofofpreferentiplagthe presentioclaseal ,elecstudtrony isirradiato examinetion dam-the boundary sliding or a continuous ¯uid ®lm because tri- larged image of the kindplaceonly ofbetwindicatedmineraleen quasbyanrtzdanphasegrains.arrowboundariItin is K-feldspsuges gestedbetwareenthatanddierentquartz.these structureInagethezonesmajoforityatgraingraiofnboundaribouboundariesndarieses,inconta granite-aining originsome voidultra-s. ple junctions would be the easiest locations to be opened Fig. 6a, shows a secondakindsboundary phaseofriesminerawereaboutls.formed1Innmthisinbypapewidthhealir, ngtheybothofaremicgrainallrocracks.referredboundaTheriesto andmylonitBehrmphaseanne forbou(1985)follndaries,owingexaminreasons:no edopena ®ne-grai(i) metamonedrphicquareactionrtz mylo-s by the grain boundary sliding or by a ¯uid ®lm (Watson along the boundary. Thisasstructurasecograinndarylbouwidtphasendaries.h ofdidmajorBecanot resultuseboutheyndariesspacarees,, deducedimpoand rtantno fromimpuriin typrogressnitephaseswithbygraiweremeansn bouobservendariesof dgraiatn£100abounmndaryin widtdiusih aonnd(e.g.with, and Brenan 1987; Xinggang et al. 1990). (iii) Many from beam damage, becgeologilattice-fause itcalringedidprocenotimaging,sses,changetheisitlesspropes widththanrtiesabnanomeofoutgrai0.5tern boundarinm.scale. Inesa fewBradmanboundariyy spheroid1983;es,Joestenvoidsal voidareands; cleheFisharlyinterpreter 1988),ed (ii)the grainformatbouionnd-of dislocations and sub-grain boundaries are observed near distinguishable. the boundaries as creep damage due toFiggrain. 6a, b boundaTEM anryd HREM images of grain boundaries with voids. a in images taken with dihaveerentbeeirradian studtionied;timfores.example, grain boundary diu- aries play important roles in the deformTEMationimageofof q®ne-uartz grain boundary with voids. b HREM image of the boundaries with voids, as shown in Fig. 5. This sion,Key wordsgrain boundaryGrain boundarymigration,á Latgraiticenfringeboundaá ry sliding graineslidingd. Irocksn additionby graitonTEMboundaryobservdiations,thusione regiFarveroandn indgrainicaandted by an arrow in a. Both quartz grains are observation can be explained by the rearrangement of andHigh-grairesolutionn boundaryelectronfracturemicroscoare impopy ártantUltramylonimechanismste boundaYund (1995ry sliding) show(e.g.,ed fromPatersondiusion1990),measurre(iii)presetheemenntedmylonitbtsy of(10an1)e lattice fringes. The boundary has a secondary dislocations that were induced at crack tip zones and mainlanorthyitecoaggregatensists of plagiocthat thelase,eectK-feldspive widtphaarshe,contandabouributingtqua1 nmrtz,width with no fringes, as indicated by the pair of in the formation of rocks (e.g., Boland and Tullis 1986; arrows wake zones of propagating cracks. In particular, sub- Joesten 1991; StuÈ nitz and Fitz Gerald 1993; Masuda whichto graiaren boumajorndarycomdiponentsusion isof3 crustalto 5 nm.lithEvenologies,in theseand grain boundaries appear at the turning points of the etIntroductional. 1997). The properties of materials strongly depend (iv)studies,a ®ne-graihowenedver,aggregalittle oftethein therealultramnatureyloniteand characterallows us grain boundary with voids, as can be seen in Fig. 5. This on the structure of their grain boundaries (e.g., Gleiter toofinvestgrainigateboundarimanyesgrainwas determboundariesined.inforonegrainspecimeboundaryn by diusion (Yan et al. 1977). Unfor- sub-grain boundary results from dislocations or a strain andIn rocks,Chalmersthere are1972;grainSuttonboundariandesBallubetween1995)the same. In TEMTheanpurposed high-resolutioof the npresentelectronstudmicroscoy tunateis topyexaminely,(HREM)lattice-frithe. nge images yield no information ®eld generated at the front of the re¯ected point of a kind of minerals and phase boundaries between dierent structure of grain boundaries in a granite-aboutoriginthe localultra-electron charge density. However, grain propagating crack. The reason why the microcracks kinds of minerals. In this paper, they are all referred to mylonite for following reasons: (i) metamoboundarphicriesreactionare specials places, distinct from the insides of developed mainly in quartz-quartz boundaries is not T. Hiraga (&) á T. Nagase á M. Akizuki Sampleprogressofbygranite-originmeans of graiultramylonitn boundarye diusion (e.g., Instituteas grainof Mineralogy,boundaries.PetrologyBecauseandtheyEconomicare Geology,important in grains and crystal surfaces, because the transformation known, but possibly the phenomenon can be compared FacultygeologiofcalScience,processes,TohokutheUniversity,propertiesAoba-yama,of grain boundaries Brady 1983; Joesten and Fisher 1988),to(ii)thegrainamobourphousnd- state occurs only at the grain with results of Carlson et al. (1990). They observed Theariesultraplaymylonitimportante usedrolinesthisin studythe deformwas collecationtedoffrom®ne- Aoba-ku,have beeSendai,n stud980-8578,ied; for Japanexample, grain boundary diu- boundaries and not at crystal surfaces or inside grains. preferential openings in quartz-quartz boundaries com- e-mail: [email protected] thegraineHatdagawarocks Fractureby grainZoneboundaryin Fukushidiusionma Prefecand grainture, sion, grain boundary migration, grain boundary sliding Also, the distortion of the lattice spacing at the boun- pared with quartz-feldspar and feldspar-feldspar boundary sliding (e.g., Paterson 1990), (iii) the mylonite and grain boundary fracture are important mechanisms dary, shown in Fig. 5, suggests that the segregated im- boundaries in granite and attributed this feature to a mainly consists of plagioclase, K-feldspar and quartz, in the formation of rocks (e.g., Boland and Tullis 1986; purities at the boundary may be, for example, water, a larger volumetric change of quartz against temperature which are major components of crustal lithologies, and Joesten 1991; StuÈ nitz and Fitz Gerald 1993; Masuda very common substance in the earth's crust and known compared with other minerals. The secondary phase (iv) a ®ne-grained aggregate in the ultramylonite allows us et al. 1997). The properties of materials strongly depend to interact with the surfaces of silicates (Parks 1984). A appears to have formed from an impurity in the mic- to investigate many grain boundaries in one specimen by on the structure of their grain boundaries (e.g., Gleiter similar distortion was observed in a metal-ceramic rocrack healing process. and Chalmers 1972; Sutton and Ballu 1995). In TEM and high-resolution electron microscopy (HREM).

T. Hiraga (&) á T. Nagase á M. Akizuki Sample of granite-origin ultramylonite Institute of Mineralogy, Petrology and Economic Geology, Faculty of Science, Tohoku University, Aoba-yama, Aoba-ku, Sendai, 980-8578, Japan The ultramylonite used in this study was collected from e-mail: [email protected] the Hatagawa Fracture Zone in Fukushima Prefecture, articles Grain boundaries as reservoirs of incompatible elemenarticts lesin the Ca Ca Ca represents the concentration if the solute were confined entirely to partition coefficient, DGM=GB XGM=XGB, decreases with decreas- Ca¼ the grain-boundary plane. This concentration may, in fact, rep- ing temperature, and (3) DGM=GB increases with increasing Ca Earth’s mantle resent the real grain-boundary concentration, as observed for concentration in the matrix. All of these features are predicted by segregation of Ca to within a single atomic plane at grain boundaries equation (1). in Ca-doped MgO (ref. 15). Strain energy minimization and space charge compensation are Takehiko Hiraga1 , Ian M. Anderson2 & David L. Kohlstedt1 * In Fig. 2, Ca monolayer coverage is plotted versus grain-matrix two important thermodynamic driving forces for segregation to Ca concentration (molar occupancy of octahedral M sites), XCa , grain boundaries16. A survey of all elements measured in the grain- 1Department of Geology and Geophysics, University of Minnesota, Minneapolis, Minnesota 55455, USA GM for the samples described above combined with the results from boundary profiles indicates an enrichment of Ca, K, Cr, Ti, Al and 2Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA samples of two natural peridotites, a fine-grained lherzolite and an occasionally Fe, depletion of Mg and occasionally Si, and no * Present address: Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan olivine phenocryst-bearing basaltic rock, as well as a synthetic detectable enrichment or depletion of Mn and Ni at the grain olivine diopside aggregate1. Individual data points represent boundaries1,11 (T.H., unpublished work). These observations ...... the coverageþ...... from a single grain boundary in the samples. The suggest that differences in ionic radius and valence state from that observed variation in Ca coverage among grain boundaries in a of host cations (that is, Mg and Si) control the enrichment of 2 The concentrations and locations of elements that strongly partitiongivintoen samplethe fluidcan phabe explainedse in rocksby theprovidevariationessentiof grain-boundaral constraintys oelementsn at grain boundaries. Divalent cations such as Ca þ, which geochemical and geodynamical processes in Earth’s interior. Asfundamtructureentalfor diquesfferentiont miremasorienins,tatiohowevns beter,weenasntoeighwherebourintheseg are enriched at grain boundaries, have much larger ionic radii articles grains and grain-boundary orientation. However, the plots demon- (0.100 nm) than those of the host isovalent cations incompatible elements reside before formation of the fluid phase. Here we show that partitioning of calcium between the grain 2 2 2 strate that (1) the concentration of Ca at the grain boundaries (Mg þ 0.072 nm and Fe þ 0.078 nm). In contrast, Mn þ and interiors and grain boundaries of olivine in natural and synthetic olivine-rich aggregates follows a thermodynamic model for 2 ¼ ¼ 2 increases with increasing concentration in the olivine grains, (2) the Ni þ, which do not segregate, have ionic radii (Mn þ 0:083nm equilibrium grain-boundary segregation. The model predicts that grain boundaries can be the primary storage sites for elements 2 ¼ Ca Ca Ca and Ni þ 0:069nm) close to those of the host cations. For these represents thewitconcentrationh large ionic radif theius—solutethat is,wereincompatconfinedible elementirelyents intothe parEarth’stitionmantle.coefficient,This observationDGM=GB provideXGM=sXaGBmechan, decreasesism forwtheith decreas-¼ Ca¼ isovalent cations, we conclude that segregation is controlled by the grain-boundarselectivey plane.extractionThisof ctheseoncentrelemationents andmaygives, in afact,framewrep-ork foringinterprtempeetingratgeoure,chemicaland (3signatu) DGMres=GBininmantcreasleesrockswit.h increasdifferencing esCain ionic radius. Similar partitioning behaviour takes resent the real grain-boundary concentration, as observed for concentration in the matrix. All of these features are predictedplace betweenby coexisting minerals (for example, olivine, pyroxenes and plagioclase) and between minerals and melt17,18. segregation ofSegregationCa to withinof incompatiblea single atomicelementsplaneresultsat grainin a markedboundariesdifference equatiosuccessionn (1).and visually inspected to ensure the absence of beam Aliovalent elements such as K, Cr, Al and Ti can segregate to grain in Ca-doped MgObetween(ref.the 15).composition of the interiors and the boundaries of damage.Strain energTheseyprofilesminimizationwere thenandsummedspacetochargeobtain ccompositeompensation are 1 boundaries to compensate space charge. Al commonly segregates to In Fig. 2, Cgaramonolayins withinerthcoe verocrageks thisat plottedcompriseversusthe Eagrain-matrixrth’s mantle . At twoprofilesimporwithtanta totalthermodynamicacquisition time ofdriv2.5 singper pixforcel.esCharacteristifor segregationolivc ine graintoboundaries and occasionally exceeds the amount of Ca, chemical equilibrium, the molar concentrations of soluCates in X-ray intensities 16were calculated by fitting reference spectra for Ca concentration (molar occupancy of octahedral M sites), XGM, grain boundaries . A survey of all elements measured inastheshowngrain-in Fig. 1a. Because Si and Fe concentrations are constant in for the samplesgraindescribedmatrices, XabovGM, ande combinedthe grain boundaries,with theXresuGB, canlts befromrelated boundarfamiliesyofprofileselementalindicateslines to thesean enrichmentcomposite spectra.of Ca, K, Cr, theTi,prAlofile,andAl must substitute for Mg at the grain boundaries. The through the grain-boundary segregation energy, g: A grain-boundary profile characteristic of the olivine solubility of Al in olivine grains is below the detection limit of samples of two natural peridotites, a fine-grained lherzolite and an occasionally Fe, depletion of Mg and occasionally Sþi, and no g anorthite sample is shown in Fig. 1a. All profiles exhibit segregationEPMA, indicating that it partitions more strongly to grain bound- XGB XGM 3 olivine phenocryst-bearing basaltic rock, as wexpell as a synthetic 1 detofectableCa andenrichmentconcomitant ordepdepletionletion of Mofg aMnt the andolivinNie gatraiariesnthethangraindoes Ca, even though the ionic radius of Al þ (0.054 nm) XGB01 2 XGB ¼ 1 2 XGM RT ð Þ 1,11 olivine diopside aggregate . Individual data points represent boboundaries.undaries In c(ontrast,T.H., theunpprominentublishedAlwsegregationork). Theshownse obinsisecloserrvatithanonsthat of Ca to the value of either the radii of host cations þ where XGB0 is the saturation level of XGB, R is the gas constant, and or the theoretical strain-free radius discussed later. Likewise the the coverage from a single grain2 boundary in the samples. The suggestFig. 1athatwas differobserveencesd onlyininionicthe olivradiusine andanorvalencethite specimen.state from that 3 T is the temperature . This thermodynamic model has not been Similar enrichments at olivine grain boundariesþ have been reportedionic radius of Cr þ (0.062 nm) is approximately equal to the observed variation in Ca coverage among grain boundaries in a of host cations (that is, Mg and Si) control the enrichmenttheoreticalofstrain-free radius for M2 sites in olivine, yet segregation applied to grain boundaries in Earth materials; grain boundaries 1,7,11–13 2 given sample can be explained by the variation of grain-boundary elementspreviouslaty grain. boundaries.The width ofDivalentthe peakscations(,5 nmsuchFWHasMCa) itsoþo,liwhichvine grain boundaries was detected1. These observations have simply been treated as possible sinksarticfor impuritieslesin rocks. characteristic of electron-beam spreading in the specimen; smaller structure forMandifyfestudiesrent mthatisohavrieeninferredtationsthebepresentweence ofnetracighebelementsouring at are enriched at grain boundaries, have much larger ionicindicateradiithat aliovalency is also a strong driving factor for partition- widths were observed for data acquired from thinner regions of the grains and grain-boundargrain boundariesy orientation.in rocks attributedHowevertheir obser, thevationsplots todemon-secondary (0.100 nm) than those of the host isovalentingctoatgrainionsboundaries. However, the combination of strain and specimen.2 Also, the change in2 composition at the grain boundarvyalen2ce factors complicates the development of a quantitative effects such as contamination in intergranular regions introduced þ þ þ Ca Ca Ca strate that (1) the concentration of Ca at therepr3–10grainesentsboundariesthe concentration(Mgdoes notifindicate0.072the solutenmthe presencandwereFee ofcaonfinedsecond0.078phase.nm).entirelyThinIn intergranularcontrast,to parMnmodeltitionforandcoefficient,predicting the parDtitioning of aliovaX lent=Xions. , decreases with decreas- by a melt, an aqueous fluid or a gas phase . 2 ¼ ¼ 2 GM=GB GM GB amorphousþ films with thicknesses of 0.5 to 10 nm observed in some The partition coefficients for isoCa¼valent elements are well increases with increasingTo test whetherconcentrationthis thermodynamicin the olivmodeltheinegrain-boundardescribesgrains,segr(2)egatiothe yn plane.Ni , whichThis doconcentrnot segregate,ation mayhave, ionicin fact,radiirep-(Mn þing0t:083empenmrature, and (3) D increases with increasing Ca polycr2ystalline oxides are not present at olivine grain boundaries¼inexplained by the strain energy, U, causedGM=GBby the misfit of trace to grain boundaries in mantle rocks, we fabricated aggregates of and Ni þ 0:069nm) close to those of the host cations. For these resent the real grain-btheseounspecimens,da¼ry coasncdemonstratedentration,byastheolatticebservfringeed fimageor usingconcentration in the matrix. All of these features are predicted by olivine, olivine anorthite (10 vol.%), and olivine diopside isovalent cations, we conclude that segregation is controlled by þ segregation ofþ Ca to withinhigh-rea singsolutionle atomicelectron planemicroscopyat grain(HREM)boundariesin Fig. 1b (ref. 11).equation (1). (10 vol.%). Powders of San Carlos olivine (Mg1.8Fe0.2SiO4) with differences in ionic radius. Similar partitioning behaviour takes or without synthetic anorthite or syntheticin Ca-dopeddiopside glassMgO(crystal-(ref. 15).Concentrations of Ca were extracted from measurements oStrainf energy minimization and space charge compensation are placecharacteristicbetweenX-racoexistingy counts mineralsabove backgrou(for ndexample,using theolivstandardine, pyroxenes lized at 1,373 K) were isostatically hot-pInressFiged .at21, ,4C7a3 Kmonolayand er coverage is plotted versus grain-matrix17,18 two important thermodynamic driving forces for segregation to andCliff–Lorimerplagioclase)14 approaand betweench. For thismineralsmethod, andsensitivmeltity factors. for 300 MPa for 4–7 h in a gas-medium apparatus. The aggregates Ca 16 Ca concentration (molartheAliovamajoroccupancylentelementselementsofwereoctahedralsuchdeterminedas K, usingCrM, Asites),thel andmatrixTiXcanGMas asegregate,standard,graintoboundariesgrain . A survey of all elements measured in the grain- were then annealed at 1,373, 1,473 or 1,523 K for ,250 h at a total pressure of one atmosphere with theforoxygenthefugacitsamplesy fixeddescribednear boundarieswith theabovcompositiontoe ccompensateombinedas determinedwspacithethecharge.by EPMA.resuAlltsComparablecommonlyfrom sensi-boundarsegregatesytoprofiles indicates an enrichment of Ca, K, Cr, Ti, Al and the Ni/NiO buffer. Finally, the samplessampleswere quenchedof twoin naturalwater. olivtivperidotites,ineity factorsgrain boundariesfor aCafine-grainedwere takenandasocthecasionallylherzolitemean of theexandcfactorseedsanthefor SiamountoandccasioofnCa,ally Fe, depletion of Mg and occasionally Si, and no Fe, effectively a linear interpolation of the factorsarticlfor these majores Cracks formed owing to thermal shock, especiallyolivine alongphensomeocrofyst-thebearas showning bainsFigaltic. 1a.rock,Becauseas Siwandell Feas conca synentrationsthetic ardete constantectableinenrichment or depletion of Mn and Ni at the grain grain boundaries; however, many boundaries remained closed. The elements versus atomic number. The accuracy of this interpolation, olivine diopside atheggwrheprigcofile,hatyei1e.ldAlIendmustdkividsubstituteua1l.17dâta0.for0p8 oifMgonr tasatll rmtheepearesgrainusremnetnboundaries.ts, bwoaus ndarTheies1,11 (T.H., unpublished work). These observations 1–20 mm grains are polygonal in shape. A very smallþamount of melt CaSi ¼ G(,1rvoal.%)iformedn bin osomeusamples.ndtheByausingrcoivdiffererageesentfrommineralasasolubilitsingdeemedrele sygrainadequateoef Arlboundarivngivenolivotheiineyrstatisticalsigrainsn theolimitationsissamples.fbelow oftheThethedetectionmeasure-suggestlimitthatof differences in ionic radius and valence state from that assemblages and annealing temperatures, we obtained a series of EPMA,ment ofindicatingthe low CathatX-rayit parintensities.titionsTmoreo comparestrontheglydegreeto grainof bound- observed variation in Ca coverage among grain boundaries in a 3of host cations (that is, Mg and Si) control the enrichment of aggregates with different concentrations of Ca in the olivine grains ariessegrthanegatiodoesn amCa,ongevspenecithougmens, hththee toionictal excradiusess CaofinAltegrþat(0.054ed nm) 2 given sample can be explained by the variation of grain-boundary elements at grain boundaries. Divalent cations such as Ca þ, which as determined by electron probe microanalysis (EPMA). articles is throucloserghthanthe wthatidthofof Cathe toprofilethe valuewas determinedof eitherandthe cradiionvertedof hostto cations incompatible elemenetffesctive imnonolayterhs atethe boundary. The procedure for this Grain-boundary compositional profilesstrwerucetacqureuiredforbydSTEM/iffereorntthemitheoretisorientcalatiostrain-freens betweradiusen neidiscussedghbourinlaterg . areLikewenrichedise the at grain boundaries, have much larger ionic radii GEDXrai(scanningn bountransmissiondaries electronas resmicroscervoopy/energirs ofy dispersive calculationFigure 1 Chemistryis welland structuredescribed3 of olivine-olivinein ref. 1.grainBecauseboundariestheinprofilesa sample ofindicate grains and1,11 grain-boundarionicyradorientation.ius of Cr þHo(0wever.062 nm), the isplotsapprodemon-ximately equa(0.1l00tonmthe) than those of the host isovalent cations X-ray) spectrum profiling, as detailed elsewhere . The incident thatolivine Caanorthitesubstitutesannealedforat 1,473MgK.ata, X-raythe boundarintensity profiley, thefrom STEM/EDXgrain-boundary 2 2 2 incompatible elements in the þ Eelecatronrptrobhe com’sprisedm,1.a5 nAnofstratet20l0-ekeVthatelect(1)rons thein anconcthecanalysisooretiventrationerageforcalelementsofstrain-freeCa inisoftheexpressedCavicinityradiusatof theasthegrainmolarforboundaries.grainM2occupancysitesboundariesThe profilein olivofis theoctahedraline,sum ofyet(MgsegregatM þ ion0.072 nm and Fe þ 0.078 nm). In contrast, Mn þ and 2 profiles measured from five grain boundaries. The negativeCa compositional1 values of trace Figure 2¼Calcium concentrations in olivine grain boundaries¼ versus concentration in 2 Einacirdtehnt’spmrobae notfle,1.4 nm full-wincreaidth asest hwalfith-maincreasingximum to(M1oclioncentrationvandineM2)grariegularn boinulatticenthedarsites,iolives wineXaGBs :grains,dThisetecmeasurted(2). eTtheprohesveidesoNibanseþr,vawhichtions do not segregate, have ionic radii (Mn þ 0:083nm 1 2 1 elements with low concentrations in the olivine grains are indicative of the statistical olivine grain matrices. Open symbols, olivine only; grey symbols, olivine anorthite; ¼ (FWHM).1 For each2 boundar1 y, multiple profiles were acquired in upper bound for the grain-boundary concentration, because it 2 Takehiko Hiraga *, IanHiragaM. Anderson*&, DavidIanL.M.KohlstedtAnderson & David L. Kohlstedt indicate that aliovalency is also a strong driving factor for partition- þ scatter in the spectral-fitting procedure. b, HREM image and diffraction pattern of the andblackNisymbols,þ olivine0:069diopside.nm)Theoreticalclosecurvesto thoseare shown foroffourthetemperaturhostescations. For these 1Department of Geology and Geophysics, University of Minnesota, Minneapolis, Minnesota 55455, USA ¼ þ 2 NATURE | VOL 427 | 19 FEBRUARY 2004 | www.nature.com/nature 699 1Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA © 2004 Nature Puingblolivineishtoinggrain-boundargrainGroup boundaries.y area. However, the combination ofisovalentstrainfrom equationsandcations,(1) and (2). we conclude that segregation is controlled by * PDepartmentresent address: Graduate School ofofEngineering,GeologTohoku Uyniversiandty, SendaiGeophy980-8579, Japan sics, University of Minnesota, Minneapolis, Minnesota 55455, USA 2 Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennesseeva700len37831,ce fUSAactors complicates the development of a quantitative NATURE | VOL 427 | 19 FEBRUARY 2004 | www.nature.com/nature ...... © 2004 Naturedifferenc Publishing Gesroupin ionic radius. Similar partitioning behaviour takes The* Prconceesentntrationsaddressand: GlocationsraduateofSelemchoolentsofthatEngineering,strongly partitionTohokuinto theUfluidniversiphatsey,inSendairocks provide980-8579,essentiJapanal constraints on model for predicting the partitioning of aliovalent ions. geochemical and geodynamical processes in Earth’s interior. A fundamental question remains, however, as to where these place between coexisting minerals (for example, olivine, pyroxenes incompatible elements reside before formation of the fluid phase. Here we show that partitioning of calcium between the grain The partition coefficients for isovalent elements are well 17,18 interiors and grain boundaries of olivine in natural and synthetic olivine-rich aggregates follows a thermodynamic model for and plagioclase) and between minerals and melt . equil...... ibrium grain-boundary segregation. The...... model predicts that grain boundaries can be the primary storage sites for elements explained...... by the strain energy, U, caused by the misfit of trace with large ionic radius—that is, incompatible elements in the Earth’s mantle. This observation provides a mechanism for the selective extraction of these elements and gives a framework for interpreting geochemical signatures in mantle rocks. Aliovalent elements such as K, Cr, Al and Ti can segregate to grain

SegregationThe conceof incompatiblentrationselements resultsandin a markedlocationsdifference succofessionelemand visuallyentsinspectedthatto ensurestrongthe absencely ofpartbeam ition into the fluid phase in rocks provide essential constraintsboundarieson to compensate space charge. Al commonly segregates to between the composition of the interiors and the boundaries of damage. These profiles were then summed to obtain composite ggeocherains within tmicalhe rocks thatandcomprisegeodynamthe Earth’s mantle1.icaAt lprofilesprocessewith a total acquisitions intimeEarof 2.5th’ss per pixel.interior.Characteristic A fundamental question remains, however, as to where these chemical equilibrium, the molar concentrations of solutes in X-ray intensities were calculated by fitting reference spectra for olivine grain boundaries and occasionally exceeds the amount of Ca, grainincompatibmatrices, XGM, andlethe grainelementsboundaries, XGBreside, can be relatedbefofamiliesreof elementalformationlines to theseofcompositethespectra.fluid phase. Here we show that partitioning of calcium between the grain through the grain-boundary segregation energy, g: A grain-boundary profile characteristic of the olivine as shown in Fig. 1a. Because Si and Fe concentrations are constant in interiors and grain boundaries of olivine in natural and syntheticþ olivine-rich aggregates follows a thermodynamic model for XGB XGM g anorthite sample is shown in Fig. 1a. All profiles exhibit segregation exp 1 of Ca and concomitant depletion of Mg at the olivine grain XGB0 2 XGB ¼ 1 2 XGM RT ð Þ equilibrium grain-bounda ry segregatboundaries.ion.In contrast,Thethemodelprominent Alpredictssegregation shownthatin grain boundaries can be the primary storage sites for elementsthe profile, Al must substitute for Mg at the grain boundaries. The where XGB0 is the saturation level of XGB, R is the gas constant, and 2 Fig. 1a was observed only in the olivine anorthite specimen. Twitis thehtemperaturelarge. Thisionicthermodynamicradiusmodel—hasthatnot beenis,Similarincompatenrichments atibleolivine graelemin boundariesentsþ have beeninreportedthe Earth’s mantle. This observation provides a mechanism forsolubilitthe y of Al in olivine grains is below the detection limit of applied to grain boundaries in Earth materials; grain boundaries previously1,7,11–13. The width of the peaks (,5 nm FWHM) is havselee simplyctivebeen treaextrted as possibleactionsinks forofimpuritiesthesein rocks.elemcharacteristicents ofandelectron-givesbeam spreadinga framewin the specimen;orksmallerfor interpreting geochemical signatures in mantle rocks. Many studies that have inferred the presence of trace elements at widths were observed for data acquired from thinner regions of the grain boundaries in rocks attributed their observations to secondary EPMA, indicating that it partitions more strongly to grain bound- specimen. Also, the change in composition at the grain boundary effects such as contamination in intergranular regions introduced 3 does not indicate the presence of a second phase. Thin intergranular by a melt, an aqueous fluid or a gas phase3–10. amorphous films with thicknesses of 0.5 to 10 nm observed in some aries than does Ca, even though the ionic radius of Al þ (0.054 nm) To test whether this thermodynamic model describes segregation polycrystalline oxides are not present at olivine grain boundaries in toSegregationgrain boundaries in mantleof incompatiblerocks, we fabricated aggregaelementstes of results in a marked difference succession and visually inspected to ensure the absence of beam these specimens, as demonstrated by the lattice fringe image using olivine, olivine anorthite (10 vol.%), and olivine diopside is closer than that of Ca to the value of either the radii of host cations þ þ high-resolution electron microscopy (HREM) in Fig. 1b (ref. 11). (10betweenvol.%). Powderstheof SancCarlosompositionolivine (Mg1.8Fe0.2ofSiO4)thewith interiors and the boundaries of damage. These profiles were then summed to obtain composite or without synthetic anorthite or synthetic diopside glass (crystal- Concentrations of Ca were extracted from measureme1 nts of lgizerdaait n1,3s73 Kw) witerhe isnostatitchallyehort-oprecsskedsat t1,h47a3 Kt ancdomcharacteristicpriseX-rathy ceountsEabovartehbackgrou’s mndausingntlthee standard. At profiles with a total acquisition time of 2.5 s per pixel. Characteristiorcthe theoretical strain-free radius discussed later. Likewise the 14 300 MPa for 4–7 h in a gas-medium apparatus. The aggregates Cliff–Lorimer approach. For this method, sensitivity factors for 3 werechethenmannealedical ateq1,373,uil1,473ibroriu1,523m,K forth,e250mh atoalather majorconelementscenwertreadeterminedtions usingofthesmatrixoluastaestandard,s in X-ray intensities were calculated by fitting reference spectraiofornic radius of Cr þ (0.062 nm) is approximately equal to the total pressure of one atmosphere with the oxygen fugacity fixed near with the composition as determined by EPMA. Comparable sensi- thegrainNi/NiOmatricbuffer. Finallyes,, theXsamplesGM,werande quenchedthein grainwater. tivboundaries,ity factors for Ca were takenXasGBthe,meancanof thebefactorsrelafortedSi and families of elemental lines to these composite spectra. Cracks formed owing to thermal shock, especially along some of the Fe, effectively a linear interpolation of the factors for these major theoretical strain-free radius for M2 sites in olivine, yet segregation grainthroughboundaries; howevtheergrain-boundar, many boundaries remained closed.y segregationThe elements versusenergatomic numbery, g. The: accuracy of this interpolation, A grain-boundary profile characteristic of the olivine which yielded k 1.17 ^ 0.08 for all measurements, was 1 1–20 mm grains are polygonal in shape. A very small amount of melt CaSi ¼ þ (,1 vol.%) formed in some samples. By using different mineral deemed adequate given the statistical limitations of the measure- anorthite sample is shown in Fig. 1a. All profiles exhibit segregationto olivine grain boundaries was detected . These observations assemblages and annealing temperatures,XweGBobtained a series of XmentGMof the low Ca X-raygintensities. To compare the degree of aggregates with different concentrations of Ca in the olivine grains segregation expamong specimens, the total excess Ca integra1ted of Ca and concomitant depletion of Mg at the olivine grindicain ate that aliovalency is also a strong driving factor for partition- as determined by electron probe microaXGB0nalysis2(EPMA).XGB ¼ 1 2throuXgGMh the width of theRTprofile was determined and convertedð toÞ Grain-boundary compositional profiles were acquired by STEM/ effective monolayers at the boundary. The procedure for this boundaries. In contrast, the prominent Al segregation shown in EDX (scanning transmission electron microscopy/energy dispersive calculation is well described inref. 1. Because the profiles indicate ing to grain boundaries. However, the combination of strain and X-ray)wherespectrumXGB0profiling,isasthedetailedsaturationelsewhere1,11. The incidentlevel ofthat XCa GBsubstitutes, R isfor theMg at thegasboundarconstant,y, the grain-boundarandy electron probe comprised ,1.5 nA of 2002-keV electrons in an coverage of Ca is expressed as molar occupancy of octahedral M Fig. 1a was observed only in the olivine anorthite specimen. Ca þ valence factors complicates the development of a quantitative iTncidisent theprobe temperatureof ,1.4 nm full-width .atThishalf-maxthermodynamicimum (M1 and M2) regular latticemodelsites, XGB:hasThis measurnote probeenvides an (FWHM). For each boundary, multiple profiles were acquired in upper bound for the grain-boundary concentration, because it Similar enrichments at olivine grain boundaries have been reported applied to grain boundaries in Earth materials; grain boundaries 1,7,11–13 NATURE | VOL 427 | 19 FEBRUARY 2004 | www.nature.com/nature 699 © 2004 Nature Publishing Group previously . The width of the peaks (,5 nm FWHMmodel) is for predicting the partitioning of aliovalent ions. Figure 1 Chemistryhavande simplystructurebeenof treaolivine-olivineted as possiblegrain boundariessinks for impuritiesin a sampleinof rocks. characteristic of electron-beam spreading in the specimen; smaller Many studies that have inferred the presence of trace elements at The partition coefficients for isovalent elements are well olivine anorthite annealed at 1,473 K. a, X-ray intensity profile from STEM/EDX widths were observed for data acquired from thinner regions of the þ grain boundaries in rocks attributed their observations to secondary explained by the strain energy, U, caused by the misfit of trace analysis for elements in the vicinity of the grain boundaries. The profile is the sum of specimen. Also, the change in composition at the grain boundary effects such as contamination in intergranular regions introduced does not indicate the presence of a second phase. Thin intergranular profiles measuredbyfroma melt,five grainan aqueousboundaries.fluidTheornegativea gas phasecomposition3–10. al values of trace Figure 2 Calcium concentrations in olivine grain boundaries versus concentration in amorphous films with thicknesses of 0.5 to 10 nm observed in some elements with low concentratTo test whetherions in thethisolivinethermodynamicgrains are indicativemodel describesof the statisticalsegregation olivine grain matrices. Open symbols, olivine only; grey symbols, olivine anorthite; polycrystalline oxides are not present at olivine grain boundariesþin scatter in the spectral-fittingto grain boundariesprocedure. inb, HREMmantleimagerocks,andwediffractionfabricatedpatternaggregaof thetes of black symbols, olivine diopside. Theoretical curves are shown for four temperatures these specimens, asþdemonstrated by the lattice fringe image using olivine grain-boundarolivyinarea.e, olivine anorthite (10 vol.%), and olivine diopside from equations (1) and (2). (10 vol.%). Powdersþ of San Carlos olivine (Mg Fe þSiO ) with high-resolution electron microscopy (HREM) in Fig. 1b (ref. 11). 1.8 0.2 4 Concentrations of Ca were extracted from measurements of 700 or without synthetic anorthite or synthetic diopside glass (crystal- NATURE | VOL 427 | 19 FEBRUARY 2004 | www.nature.com/nature © 2004 Nature Publcharacteristicishing GroupX-ray counts above background using the standard lized at 1,373 K) were isostatically hot-pressed at 1,473 K and 14 300 MPa for 4–7 h in a gas-medium apparatus. The aggregates Cliff–Lorimer approach. For this method, sensitivity factors for

were then annealed at 1,373, 1,473 or 1,523 K for ,250 h at a the major elements were determined using the matrix as a standard, total pressure of one atmosphere with the oxygen fugacity fixed near with the composition as determined by EPMA. Comparable sensi- the Ni/NiO buffer. Finally, the samples were quenched in water. tivity factors for Ca were taken as the mean of the factors for Si and Cracks formed owing to thermal shock, especially along some of the Fe, effectively a linear interpolation of the factors for these major grain boundaries; however, many boundaries remained closed. The elements versus atomic number. The accuracy of this interpolation, which yielded k 1.17 ^ 0.08 for all measurements, was 1–20 mm grains are polygonal in shape. A very small amount of melt CaSi ¼ (,1 vol.%) formed in some samples. By using different mineral deemed adequate given the statistical limitations of the measure- assemblages and annealing temperatures, we obtained a series of ment of the low Ca X-ray intensities. To compare the degree of aggregates with different concentrations of Ca in the olivine grains segregation among specimens, the total excess Ca integrated as determined by electron probe microanalysis (EPMA). through the width of the profile was determined and converted to Grain-boundary compositional profiles were acquired by STEM/ effective monolayers at the boundary. The procedure for this EDX (scanning transmission electron microscopy/energy dispersive calculation is well described in ref. 1. Because the profiles indicate X-ray) spectrum profiling, as detailed elsewhere1,11. The incident that Ca substitutes for Mg at the boundary, the grain-boundary electron probe comprised ,1.5 nA of 200-keV electrons in an coverage of Ca is expressed as molar occupancy of octahedral M Ca incident probe of ,1.4 nm full-width at half-maximum (M1 and M2) regular lattice sites, XGB: This measure provides an (FWHM). For each boundary, multipleFigureprofiles1 wereChemistryacquiredandinstructureupperofbounolivine-olivined for thegraingrainboundaries-boundary inconcentratia sampleonof, because it

NATURE | VOL 427 | 19 FEBRUARY 2004 | www.nature.com/naolivineture anorthite annealed at 1,473 K. a, X-ray intensity profile from STEM/EDX 699 þ © 2004 Nature Publishing Group analysis for elements in the vicinity of the grain boundaries. The profile is the sum of profiles measured from five grain boundaries. The negative compositional values of trace Figure 2 Calcium concentrations in olivine grain boundaries versus concentration in elements with low concentrations in the olivine grains are indicative of the statistical olivine grain matrices. Open symbols, olivine only; grey symbols, olivine anorthite; þ scatter in the spectral-fitting procedure. b, HREM image and diffraction pattern of the black symbols, olivine diopside. Theoretical curves are shown for four temperatures þ olivine grain-boundary area. from equations (1) and (2).

deformation regimes are first introduced. Then, the 1983a, 1983b), and Dgb is the diffusion coefficient for influence on rheological behavior of two important the slowest species diffusing along the boundaries. fluids, water and melt, are discussed. For grain boundaries

ÁEgb PÁVgb D Do exp – þ 2.14.3.1 Constitutive Equations gb ¼ gb RT Constitutive Equations, Rheological Behavior, and Viscosity of Rocks 399 ÁHgb 2.14.3.1.1 Diffusion creep Do exp – 36 ¼ gb RT ½ Diffusion creep can be divided into two regimes, one deformation regimes are first introduced. Then, the 1983a, 1983b), and D isinthwhe icdiffusioh diffusniocoefficin throuentghforthe interiors of grains con- gb where Do is a material-dependent parameter and influence on rheological behavior of two important the slowest species diffusingtrols thealongrate othef deboundarieformations.(Nabarro, 1948; Herring, gb Á , Á , and Á are the activation energy, fluids, water and melt, are discussed. For grain boundaries 1950) and the other in which diffusion along grain Egb Vgb Hgb boundaries governs the rate of deformation (Coble, activation volume, and activation enthalpy for grain- boundary diffusion, respectively. Again, the strain rate 1963Á).EgbThePreÁlaVgbtive importance of these two mechan- D Do exp – þ 2.14.3.1 Constitutive Equations gb ¼ gb isms depeRTnds primarily on grain size and temperature. increases linearly with increasing differential stress and increases exponentially with inverse temperature Constitutive Equations,ÁHgbRheological Behavior, and Viscosity of Rocks 399 2.14.3.1.1 Diffusion creep Do exp – 36 ¼ gb 2.14RT.3.1.1.(i) Grain m½atr ix diffusion The rate while decreasing exponentially with increasing pres- Diffusion creep can be divided into two regimes, one sure. However, in contrast to the Nabarro–Herring in which diffusion through the interiors of grains con- of deformation in the Nabarro–Herring creep regime deformation regimes are first introduo ced. Then, the 1983a, 1983b), and Dgb is the diffusion coefficientcasfore, strain rate for Coble creep varies inversely as trols the rate of deformation (Nabarro, 1948; Herring, where Dgb is a materiainl-dewhichpenddiffusionent paramethrougter ahndgrain interiors limits the influence on rheological behavior of two important the slowest species diffusing along the boundariethe cs.ube, rather than the square, of the grain size. 1950) and the other in which diffusion along grain ÁEgb, ÁVgb, and ÁHgbrateareofthcreepe activisatdesioncribedenergy,by the relation (Nabarro, fluids, water and melt, are discussed. For grain boundaries Comparison of eqn [33] with eqn [35] reveals the boundaries governs the rate of deformation (Coble, activation volume, and 1948activat; Herring,ion entha1950lpy fo) r grain- boundary diffusion, respectively. Again, the strain rate following points: (1) Both creep mechanisms give rise 1963). The relative importance of these two mechan- ÁE PÁV 1 'oVm Dgm– gb þ gb to Newtonian viscous behavior ("_ ' ) with viscos- isms depends primarily on grain size and temperature. increases linearly with increasing differen"_NHtiaDl gbstreNHssDgbexp 33 _ 2.14.3.1 Constitutive Equations ¼ ¼ RT d 2 RT ½ Constitutiveand increaEquations,ses exponentiaRheologicallly with inversBehavior,e temperatandure Viscosity of Rocks 399 ity, X '/"_, independent of stress. (2) As long as the o ÁHgb 2.14.3.1.1.(i) Grain matrix di2ff.u1s4i.o3n.1.1 Diffusiowhn cilreedeepcreasing exponewherentiallywiNHth isincareageometricalsing presD- expterm– and Dgm is the grain36 size is small enough that diffusion creep dom- The rate ¼ gb RT ½ of deformation in the Nabarro–HerrinDiffusgiocreepn creeregimep can be disuvire.deHod inwetovetwor, inregcoimntesdiffusionras, otneto thecoeNafficientbarro–Hforertheringslowest speciesdiffusing inates over dislocation creep, Nabarro–Herring deformation regimes are first introducase, straced.in raThen,te fortheCoble1983acreep, 1983bvaries),inandversDelgby isasthe diffusion coefficient for 2 in which diffusion through grainininteriwhicorsh dilimitsffusiothen through the interiors of grainthroughs con- the grain omatrix (gm), that is, the grain creep ("_ _ 1/d ) is more important than Coble influence on rheological behavior of two important the sloweswhert especiesD isdiffusinga materiaalongl-depethendenboundariet parames.ter and 3 rate of creep is described by thetrorelslationthe ra(Nteabarro,of deformattheioncub(Nae,bararrtho,er19tha48n; tHeheinterior.rrsqingua,re, Theof theimpograinrtantgbsize.points to note are that strain creep ("_ _ 1/d ) at larger grain sizes. (3) Nabarro– fluids, water and melt, are discussed. For graÁinEbou, ndariesÁV , and ÁH are the activation energy, 1948; Herring, 1950) 1950) and the other in whComich parisondiffusionofaleqnong[33]rategrawithinis lineareqnly[35]gbproportionalrevealsgb theto thegb differential stress, Herring creep dominates at higher temperatures boundaries governs thefollowinrate ofgdepointsform:a(1)tionBoth(Coinverselycreepble, mechaproportioactivnismsationnalgivevotolurisemethe, squareand actiofvattheiongrainenthasize,lpy for gandrain-Coble creep at lower temperatures because 1 'Vm Dgm to Newtonian viscous behavior ("_boun' )dawithry dioviscos-ffusion, reÁsEpegb ctiPveÁlyV.gbAgain, the strain rate "_NH NH 1963). The relativ33e importance of these two meandchan-exponentiall_ Dgb y dependeD exp –nt on þtemperature and ÁHgb < ÁHgm. ¼ 2.14.3.1RT d 2 Constitutiv½ e Equations increases¼ ligbnearly with inRTcreasing differential stress isms depends primarily onity,graiXn'si/"ze_, indepenand temdentperpressureatofure.stresthrougs. (2) Ash thelongdiffusioas the n coefficient since The analyses of Nabarro (1948) and Herring and increases exponeÁntHially with inverse temperature where NH is a geometrical2.14.term3.1.1andDDifgmfusisiothen creegrainp size is small enough that diffusion creepo dom- gb Dgbexp – 36 (1950) and of Coble (1963) strictly apply to deforma- diffusion coefficient for the slowes2.t1spe4.3cies.1.1diffusi.(i) Gngrain minatesatrix overdiffusdislocation ion creep, whNabarro–Hile de¼o creaerringsingÁexEgmpoRTnePntÁiaVllgmy with increa½sing pres- Diffusion creep can be divided into two regimes, oneThe rate Dgm D exp – þ tion of a single spherical grain. Subsequent analyses _ 2 sure.¼ Hogmwever, in contRTrast to the Nabarro–Herring through the grain matrixin (gm),whichofthadideffftusormationis,ionthethrougraininghthetheNabaricreepnterro–Herriniors("o_f gr1/ading s)creepconis -morregimee important than Coble pointed out the necessity of grain-boundary sliding 3 o ÁH interior. The important points to note are that strain creep ("_ 1/d ) at larger whgraerinesizes.cDasgbe,is(3)straoaNabarro–inmaraterteiafol-rdegmCopeblndeencrteepapravameriesterinavendrsely as trols theinrawhichte of dediffusionformationthroug(Nabahrrgraino, 19_ 4interi8; Heorsrringlimits, the Dgmexp – 34 in diffusion creep of polycrystalline materials Herring creep dominatesÁat higher, Áthe¼cub,temperaae,ndrathÁtureser thaRTarnethethesquaactrive,atofiotnheeng½raeringy,size. rate is linearly proportional195to0)theanrateddiffethoferentialcreepotherstresisindeswhs, cribedich diffbyusiothen alreolationng gra(inNabarro,Egb Vgb Hgb (Liftshitz, 1963; Raj and Ashby, 1971). The fact o Comparison of eqn [33] with eqn [35] reveals the inversely proportional to thebousquarendari1948esofgo; theHerring,vergrainns thesize,1950rate) ofanddefoCoblermationcree(Cop blate, lowerwhereactivtempDatioeraturnisvoalumaterialesmebecaus, and-depeacetivndentation enparameterthalpy forandgrain- gmfollowing points: (1) Both creep mechanisms givethatrisegrains tend to be equiaxed in samples deformed and exponentially depende196nt3).onThtempere relativaturee impoandrtanceÁofHtgbhedeformation mechanism is referred to in which the creep rate is limited by diffusiono ÁHgm 3 Dgmexp –NH 14 and C 14%. creepS34ince g(in"r_ain-1/diffusioboun)datarnylargeacreepno dr graofinÁsizes.Hpolycrystgb (3) Nabarro–alline materials interior.2.14The.3.1.impo1 Drtantif¼fuspointsion ctorenoteeRT¼p are that strain¼ 1985Constitutive½ ; Carter_ anddSass, 1981D Equations,exp; Ricoult– and Kohlstedt,Rheologicalsimply36 as diffusionBehavior,creep. and Viscosity of Rocks 401 through grain boundaries (Coble, 1963): grain-matrix diffusion are independent/p¼aralgblel RT ½ rateDiisfflinearusionlycrpreeoportionalp can be ditovidethed diffeintorentialtwo regstresimess,, oneHerring(Liftshicreeptz,dominat1963; esRajatandhigherAshby, tempera1971tures). The fact where Do is a materialpr-depeocessendents, eqn [parameter33] and [35and] can be combined to give inversely'VminDwhgb icproportioh diffgmusionaln ttohroutheghsquarethe intofertheiorsgrainof gracombined:size,ins con-and Coblethat gracreeinsptendat lowerto be equiaxedtemperaturin essamplebecauss defoe rmed "_C C 3 35 o ¼ andRT exd ÁponentiallEgm, ÁVygmdepende, an½ d ÁntHgmonaretemperthe activatioature andn energy,ÁH wh (2/3)Tm, where Tm is the melting ð Þ temperature) deformation of crystalline materials is Nabarro (1967) formulated a model for steady- the observed one-to-one correlation between the state dislocation creep based solely on dislocation activation energy for creep and the activation energy climb. Bardeen–Herring sources (Bardeen and for self-diffusion of the slowest ionic species. This Herring, 1952) generate dislocations that form a net- correlation has been observed for a large number work and continuously climb. Dislocation of metallic and ceramic materials (e.g., Dorn, multiplication occurs by operation of dislocation 1956; Sherby and Burke, 1967; Mukerjee et al., 1969; sources thus increasing their density, while climb of Takeuchi and Argon, 1976; Evans and Knowles, dislocations of opposite sign toward one another 1978) as well as for olivine (Dohmen et al., 2002; results in annihilation thus decreasing their density. Chakraborty and Costa, 2004; Kohlstedt, 2006). A balance between multiplication and annihilation Two climb-based models of dislocation creep results in steady-state creep described by the flow merit particular attention because of their pioneering law (Nabarro, 1967; Nix et al., 1971) contributions in this area and because they incorporate most of the elements found in subsequent GV 3 D 1 "_ 2 m 47 models. Weertman (1955, 1957a) developed a flow ¼ RT G b2 ln 4G= ½ law for steady-state dislocation creep in which dis- ð Þ location glide produces the strain but dislocation Although the steady-state strain rates obtained climb controls the stain rate. In this model, a disloca- from the models developed by Weertman and tion source generates a dislocation loop that expands Nabarro, eqns [47] and [46], respectively, differ in by gliding until it encounters an obstacle such as a magnitude, these flow laws share fundamental ele- dislocation loop that was generated on a parallel glide ments. Both yield a cubic dependence of strain rate plane. The two dislocations interact to form a on differential stress and a linear dependence of dipole that prevents further glide until the two dis- strain rate on diffusivity. That is, both result in locations comprising the dipole climb to annihilate power-law equations similar to that given in eqn one another. Once the dipole is annihilated, the dis- [43] with m 0. The exponential dependence of ¼ location sources produce new dislocation loops strain rate on temperature enters through D as does (i.e., dislocation multiplication) to continue the at least part of the dependence of strain rate on DIFFUSION-ACCOMMODATED FLOW AND SUPERPLASTICITY*

M. F. ASHBYt and R. A. VERRALLt

Polycrystalline matter c8n deform to large strains by grain-boundary sliding with diffusional eocommo- detion. A new mechanism for this sort of deformation is described end modelled. It differs fundamentally from Nabarm-Herring and Cable creep in 8 topological sense: grains switch their neighbors 8nd do not elongate significantly. A constitutive equation describing the mechanism is derived from the model. The strain-rate may be diffusion controlled, in which case the constitutive equation resembles the Naberro-Herring-Coble equation but predicts strain-rates which are roughly 8n order of magnitude faster. Or it may be controlled by 8n interfece reaction-roughly speaking, by the restricted ability of a boundary to act 8a a sink or source for point defects, or by its restricted ebility to slide. The flow behavior of superplastic alloys css be explained 8s the superposition of this mechanism and ordinary power-law creep (“dislocation creep”). The combined mechanisms appear to be cspable of explaining not only the observed relation between strein-rate and stress, but most of the microstructural and topological fe8tures of superplastic flow 8s w6ll.

SUPERPLASTICITE ET PLASTICITE AVEC ACCOMMODATION PAR DIFFUSION La mati& polycristalline peut p&enter des d6formations t&s important@ par glissement 8ux joints de grains avec 8ocommodetion par, diffusion. Un nouveau m6canisme pour cett.e sorte de dbfor- mation est dbcrit et un modAle est p&sent& 11 dil%re fondamentelement du fluage de Nabarro-Herring et Coble au se118t opologique: les grains bousculent leurs voisins, et ne subissent p&s d’allongement sign&&if. Une 6quetion decrivant le m6c8nisme est d&iv& du modble. La vitesse de dbformstion peut 6tre contr616e par la diffusion, auquel c&s 1’6quation ressemble B 1’Bquation de Nabarm-Herring-Coble mais pr6voit des vitesses de d6formetion plus &levbs d’environ un fectaur dix. Elle peut dtre aussi contr&l& par une r&&ion d’interfeca, approximativement, par l’aptitude limit6e d’un joint 8, agir comme un puits ou une source pour les d6feuts ponctuels, ou par son aptitude limit& B,g lisser. Le comportement plaatique des &ages superplastiques peut 6tre expliqub comme r6sultant de la superposition de ce m6cenisme et du fluage chbssique (“fluage de dislocations’ ). Les deux m6canismes combin6s semblent Btre cap8bles d’expliquer non suelement la relation observ6e entre 18 vitesse de d6formetion et 18 contrainte, mais aussi l8 plupart des caractbres topologiques et de microstructures de 18 superpl8sticit.6.

DIFFUSIONSAKKOMODIERTES FLIEDEN UND SUPERPLASTIZIT~T Man kann polykrist8llines Materid aufgrund von Korngrenzengleiten (Akkomodetion durch Diffusion) his zu groDen Abgleitungen verformen. Fiir diese Art der Verformung wird ein neuer Mechenismus besahrieben. Er unterscheidet sich in topologischem Sinn fundamental von Nabarm-Herring-und Coble-Krieohen: Kijrner &ndern ihre Nachbarn und erleiden keine wesentliche Verltigerung. Aus dem Modell wird eine Materialgleichung abgeleitet, die den Mechanismus beschreibt. Die Abgleitung kenn diffusions-kontrolliert sein; in diesem Fall gleicht die Materialgleichung der Naberro-Herring- Coble-Gleiahung, eagt aber Abgleitgeschwindigkeiten voraus, die etwa eine GrBDenordnung hiiher sind. Die Abgleitung kann such durch eine Grenzfl&chenreaktionk ontrolliert eein-grob gesprochen aufgrund der begrenzten Fi%higkeite iner Korngrenze aIs Punktdefektsenke oder-quelle zu wirken oder zu gleiten. Das l?lieBverhslten superpl86scher Legierungen kenn 8ls Superposition dieses Mechanismus mit ASHBY AND VERRALL: DIFFUSION gAeCwCiOihMnMlicOhDeAmT EKDri echenF LO(“WVe rseAtNzuDn gsSkUrPieEcRhePnL”A) STeIrCkIlT&Yt werden1. 51 Die kombinierten Mechanismen kijnnen nicht nur den beobachteten Zusammenhang zwischen Abgleitgeschwindigkeit und Spannung sondem such die Gefiige- und topologischen Eigensohaften des superplestiechen FlieBens e&l&en.

1. INTRODUCTION it usually suffer roughly the same strain as that This paper describes a deformation mechanism for imposed on the specimen as a whole (Fig. 1). All polycrystalline matter which has not previously been well-eccepted deformation mechanisms lead to this + STRAIN-0 55 -+ modelled in detail. It is based on grain-boundary quasi-uniform deformation: “quasi” because, on a s&&g with diffu-sional aomnmdation, and is impor- scale comparable with the spacing of slip lines, the tant when temperatures are high enough to permit deformation is not uniform, but on a scale comparable diffusion, end when strains are large (as in superplastic with the size of grains, it is. flow). The mechanism is described by a model, from This sort of deformation could be defined more which an approximate constitutive equation (a re- exactly as follows. Suppose the external shape (‘4 INITIAL STATE -(b) INTERWEDIATE STATE (0 FINAL STATE lation between strain-rate, stress and temperature) is change of the specimen is described by the deformation FIG. 4. The unit step of the defo~aderived.ti~ pro cessThe. A grocharacteristicsup of four grain s mofov es thefrom tmechanismhe initial, th rougmatrixh the i ntBij,er- defined by mediate state to the final state. The initial and final states of the polycrystal are thermodynamically identical, although it has suffered a true strresembleain so = 0 .6those5. In sobservedo doing, th eduring grains s usuperplasticffer accommo dflow.ation strains and translate past each other by sliding at their boundaries. Xi’ = BiiXi 2. QUASI-UNIFORM AND NON- UNIFORM FLOW where xi, a point on the specimen surface before array. Then the translations of the grains involve The fundamental topological featuredeformation, of this non- transforms to xj’ after deformation. When a polycrystalline specimen is deformed in also their rotation. The rotation is not linked in any uniform flow is that grains in the agThen,gregat eif cthehan gdeformatione is quasi-uniform, all points tension (to take a simple example), the grains within simple way to the translation, but is related to the their neighbors ; this permits a largwithine, per mthean ecrystalnt, suffer the same transformation, at DIFFUSION-ACCOMMODATED FLOW AND SUPERPLASTICITY* probability of one gram being surrounde dR ebcye ivoethd eJrus nofe 23,s h19a7p2e;- crheavnisgeed Nofo vtheem bsepr e1c3im, 1e9n7 2w. ithoutleast req uapproximately.iring any In particular, the coordinates of t Division of Engineering and Applied Physics, Harvard particular sizes and in particular angUunliavre rpsMiot.sy i,t FiCo. namsA.S bHriBdYgte l,a Mrgacent sdca haahRnu. gseeAt .ti sn. VsEhRaRpAeL Loft its grains. Thisthe fact centers-of-mass was first of the grains (crosses on Fig. l), Figure 5 shows a larger area of the deforming emul- recognized by Raohinger’s) in a paper which has been Polycrystalline matterAC TcA8n MdeformETAL tLoU lRarGgIeC sAtr,a insV bOy Lg.r ai2n1-,b oFuEnBdaRrUyA sRlidYi ng w19it7h3 d iff1u4s9io nal eocommo- 160 ACTA METALLURGICA, VOL. 2sion.1, 1 97S3w itching eventsdetion . liAk en ethew m eonecha nidisemal fiozre dth iisn s oFritg of. 4d eformation is described end modelled. It differs fundamentally 5 ooeur locally, producingfrom Na baa rlmo-cHael rrsitnrga ma nofd Caboutable c re0.55.ep in 8 topological sense: grains switch their neighbors 8nd do not elongate significantly. A constitutive equation describing the mechanism is derived from the model. and the shape of the grains themselves, change in a The surrounding Tshter usctrtuairne- rartel amxaeys btoe dtransmitiffusion c onthetro lled, in which case the constitutive equation resembles the way described by pij Fig. l(b)]. Flow by dislocation strain to the speciNmabener rsou-Hrfearcreinsg. -CAfteroble eaq ustraination butin thepredicts strain-rates which are roughly 8n order of magnitude faster. Or it may be controlled by 8n interfece reaction-roughly speaking, by the restricted ability glide or by ~slocation creep show these two charaeter- specimen of 0.55 ofa lal gborauinndsa rhy atvoe a cpta 8rat iac ispiantke dor sionu ronece f or point defects, or by its restricted ebility to slide. istics. So also do Nabarro-Herring and Coble creep switching event, onT hthee flow av ebreahgaev.i orAt of ssumpearllpelra stsitcr alilnosy,s css be explained 8s the superposition of this mechanism and ordinary power-law creep (“dislocation creep”). The combined mechanisms appear to be cspable of (although in the case of these two diffusional flow a correspondinglye xpslaminailnlegr n oftr aocntliyo nth e ofob segrrvaeidn sre lahtaiovne between strein-rate and stress, but most of the microstructural processes a grid scribed onto a single grain would show switched neighborasn.*d topoTWloOg icCLaAlS Sf e8PtLuATrEeS s of superplastic flow 8s w6ll. SEPARATED BY SPACER no distortion). The important points are: (a) flow One way of identifyiSnUgP EthisRPL sAoSrTtI CofIT fElo w EbTe haPvLiAoSr TisIC ITE AVEC ACCOMMODATION PAR DIFFUSION by any one of these mechanisms causes the grai?aa to to map the trajectorLieas mofa tthei& gpraoliync criesntatellrins ea sp tehuets ppe&oein- ter des d6formations t&s important@ par glissement 8ux joints de grains avec 8ocommodetion par, diffusion. Un nouveau m6canisme pour cett.e sorte de dbfor- e~~ga~e (strictly, to suffer the same shape transforma- OIREmenCTIO~ deOFf orFmLOWs . Fimgautrieo n 6(a)est d bschroitw ets usnc hmeomdAaltei ceastll py& stheent& 11 dil%re fondamentelement du fluage de Nabarro-Herring tion as the specimen itself) ; (b) grains z&c& are nearest FIG. 2. The omelI ovinem wehnicth ofth eg roaiiletn e- mcCeuonlbtselieor nsa uwd uasers1i ndeformed.18gt oqpuoalosgii-q uunei:f orlems gfrloawin.s bousculent leurs voisins, et ne subissent p&s d’allongement The channel has the shape ofs igan l&og&aifr.i thmUnice s6pqiureatli. on decrivant le m6c8nisme est d&iv& du modble. La vitesse de dbformstion peut neighbors remain nearest neighbors; (c) the number of The trajectories 6atrree cloonctarl6l1y6 ep paara llal edl iffeuvseiroynw, ahueqrue;e l ci&f s 1’6quation ressemble B 1’Bquation de Nabarm-Herring-Coble equation (1) is obeyedmais pwr6ivthoitn dthees v istpeessceims deen , dthis6form metuiosnt plus &levbs d’environ un fectaur dix. Elle peut dtre aussi grains across the cross-section of the specimen diamonds-which were not prceovnitoru&sll&y panre aurnees tr &n&eiiognh - d’interfeca, approximativement, par l’aptitude limit6e d’un joint 8, agir remains constant.* be so. By contrcaosmt,m ed uunr ipnugi tsn ooun -uunnei fsoorumrc e pourflow lethes d6 feuts ponctuels, ou par son aptitude limit& B,g lisser. bors come togtertahjecrt,o rfioersc inofg agpraairLntes twocwomh picoohtr hteeundergomresn tat p lra iagtthehiqtu e sdweist c&haignegs superplastiques peut 6tre expliqub comme r6sultant de la The mechanism described below is one of non- superposition de ce m6cenisme et du fluage chbssique (“fluage de dislocations’ ). Les deux m6canismes angles. This eventsneigh bareor- sl~otccahlliyn g~ eT~e~v~edn~t c~isT the(Fig . m4).ai n Figure 6(b) uniform flow. The grains do not suffer the same shape combin6s semblent Btre cap8bles d’expliquer non suelement la relation observ6e entre 18 vitesse de featureem ulsion:of theis f lanow . actual tracidn6gfo rofm etgiorani net t1r8a jceocnttorraiienst e, inm atheis a uosisli- l8 plupart des caractbres topologiques et de microstructures de 18 superpl8sticit.6. Ioiln flo w(ginrains)g tehmrouulsg i+ho n the model, chan nespanningl, theemu lsai onstrainsuffe reofd about 0.1. t t t a todetergtal strainentF ofiv e aboutswitc h10.ing Afterevents aaren i nrDiitnIiFagFle UdstrainS: IOwNSit AhofKinK OtheMO DrIiEnRgTs,E S FLIEDEN UND SUPERPLASTIZIT~T 0.5 during wghriacihn s shomavee gmovedrain-eMan loonng kaaotnirontnh p ooglyookncraciuls tr8rlpleiadnt,eh*ss M. ateOurrid aouiflg- rund von Korngrenzengleiten (Akkomodetion durch Diffusion) (grain boundaries) his zu groDen Abgleitungen verformen. Fiir diese Art der Verformung wird ein neuer Mechenismus the grains ceaesmedu lstoio ne lomodelngate ebisve esanah rtihwebooeu-ndg.i hmE rethe nusni otsnearamslc hpelone;eid et sicihn inthe to pologischem Sinn fundamental von Nabarm-Herring-und as a whole continueddeformation to de foofrm a.C otSbhltre~e-Kcetr-~die~iomlhyee nns: iosKnpiaejrla nkeirsn tgr&u,n cdteurrne ,i hrseu bN-a chbarn und erleiden keine wesentliche Verltigerung. Aus dem Modell wird eine Materialgleichung abgeleitet, die den Mechanismus beschreibt. Die Abgleitung the %ow reacshuerdf acae sgteraidnys scantakte ena,n p ipndei affwru hsiatiocnh s the-kuonn tlsirmuolrlifitaeercdte seorin ; suinr fadciees em Fall gleicht die Materialgleichung der Naberro-Herring- grams can disappCeoabrl ei-Gntleoi athehun gi, netaegrti oarb.e r ButAbgl enoitg esnewchw indigkeiten voraus, die etwa eine GrBDenordnung hiiher sind. deformation could occur with noDie furtherAbgleitu nggr akiann ne lsounchg ad-u rch eine Grenzfl&chenreaktionk ontrolliert eein-grob gesprochen aufgrund tion. The samphpelen ostrainmena wareas ithendernvo bl veegdnre-ttnihrzetel yn Fdiessential%uhei gtoke ithete i ns etre Kpo rinng rethenze aIs Punktdefektsenke oder-quelle zu wirken oder zu gleiten. flow is still that shoDwans iln?l iFeBigv.e r4h. slten superpl86scher Legierungen kenn 8ls Superposition dieses Mechanismus mit relative translations of grains gpeawsitih eachnliche mother Kri ecinhe nsu c(h“V ersetzungskriechen”) erkl&t werden. Die kombinierten Mechanismen a way that the number of graiknisjn naelnon nichtg one n udimensionr den beob achteten ZusammeFnIhQa. n5g. Pzwhoistcohgernap Ahbs g(laei)t-(gee)s cformhwin dai gskeeqiut eunnade iSnp oarndneurn ofg * In pm&ice, a sstoranidne mof sourcdhe rd i0e.5 G eisf iipgreo-b uanbdly tonpeoeldoegdis ctoh en Eiginecnrseoahsainftge snt rdaeisn .s uwpeirgphleesrt-ideicshheinng F lieBeevnesn ets& hl&aevne. occur- of the specimcehanng e inctrheea ssetrdu catunrde fromthat itas leoqnugil ibar iduimre scttaitoen i n the absence red in the areas printed with high contrast. The surround- at right angleofs dflowecre taos eidts. steady state while deforming. This point is ings relax, transmitting the loaally generated strain to discussed further in Section 4(c). the specimen surfaces. An idealization of this is shown in 1F. iIgN. T4.RO DDUuCTriInOgN a it usually suffer roughly the same strain as that 4 I I neighbor-switching eventThis, like paperthat sdescribeshown here, a deformationadjacent mechanism for imposed on the specimen as a whole (Fig. 1). All (0) (bl grains follow perpenpolycrystallinedicular traje ctmatterories. whichIn r ehasalit ynot previously been well-eccepted deformation mechanisms lead to this Fro. 1. Quasi-uniform flow. A grain suffers roughly modelled in detail. It is based on grain-boundary grains are not all the same size and even in two quasi-uniform deformation: “quasi” because, on a the same shape change as the specimen as a whole and s&&g with diffu-sional aomnmdation, and is impor- does not switoh its neighbors. Dislocation creep, Nabarro- dimensions do not usually form a perfectly hexagonal scale comparable with the spacing of slip lines, the Herring and Coble creep all have these characteristics. tant when temperatures are high enough to permit deformation is not uniform, but on a scale comparable diffusion, end when strains are large (as in superplastic with the size of grains, it is. change as the specimen, and grains switch their flow). The mechanism is described by a model, from This sort of deformation could be defined more neighbors. It is most easily described and studied by which an approximate constitutive equation (a re- exactly as follows. Suppose the external shape using an oil-emulsion analog, made as follows. If lation between strain-rate, stress and temperature) is change of the specimen is described by the deformation mineral oil is shaken with a dyed detergent (15% derived. The characteristics of the mechanism matrix Bij, defined by resemble those observed during superplastic flow. ammonium lauryl sulphate) and the emulsion is Xi’ = BiiXi centrifuged, a structure consisting of “grains” of oil 2. QUASI-UNIFORM AND NON- where xi, a point on the specimen surface before separated by very thin “gram boundaries” of deter- UNIFORM FLOW deformation, transforms to xj’ after deformation. gent solution results, Its deformation can be studied When a polycrystalline specimen is deformed in Then, if the deformation is quasi-uniform, all points tension (to take a simple example), the grains within by causing it to %ow through a channel between two within the crystal suffer the same transformation, at glass plates (Fig. 2). The arrangement simulates ! Received June 23, 1972; revised November 13, 1972. least approximately. In particular, the coordinates of the topology (though not the mechanics) of flow Fro. 3. A sequence of pt hDoitvoigsrioanp hofs ofE nggrinaeinesr ining tahned oAilp-p lied Physics, Harvard emulsion showing thUen ivdeerfsoitrym, aCtaiomnb rimdegceh, Mancitssmaa. huTsehtets . the centers-of-mass of the grains (crosses on Fig. l), in polycrystalline or polyphase materials made up sample had suffered a total strain of order 8 before these ACTA METALLURGICA, VOL. 21, FEBRUARY 1973 149 of incompressible grains which can slide at their photographs were taken; the grains show their steady- state strain of 0.3. B5e tween states (a) and (b) the boundaries. sample has suffered a further small increment of strain. Figure 3 shows what happens when the emul- The grams translate past each other: the two marked by diamonds have become nearest neighbors, separating sion deforms. Pairs of grains-those marked by the two marked by dots. The double print (c) gives some idea of the local strain.

! Since completing this study, a paper by Cannonu’ has appeared which emphasizes this, and certain of the other, * The steady-state grain elongation depended on grain geometric properties described in this section. size and was between 0 and 0.5. ASHBY AND VERRALL: DIFFUSION ACCOMMODATED FLOW AND SUPERPLASTICITY 151

160 ACTA METALLURGICA, VOL. 21, 1973 and the shape of the grains themselves, change in a way described by pij Fig. l(b)]. Flow by dislocation glide or by ~slocation creep show these two charaeter- istics.pre viousSo also dmodelso Nabarro-H earerring and Coble creep + STRAIN-0 55 -+ unifor(although imn the ﬂo cawse of these two diffusional flow processes a grid scribed onto a single grain would show TWO CLASS PLATES SEPARATED BY SPACER no distortion). The important points are: (a) flow thisby any isone of these mechanisms causes the grai?aa to e~~ga~e (strictly, to suffer the same shape transforma- OIRECTIO~ OF FLOW non-unifortion as the specimenm its eﬂolf) ; w(b) grains z&c& are nearest FIG. 2. The oelI in which the oil emulsion was deformed. (‘4 TINhITIeAL chaSTnATnE el has the shape of a log-a(br) ithINmTERiWcE DsIApTiE ral. STATE (0 FINAL STATE neighbors remain nearest neighbors; (c) the number of grains across the cross-section of theFIG. sp 4ec. imThene unit step of the defo~ati~ process. A group of four grains moves from the initial, through the inter- diamonds-which wereAS HnotB Tp revioAuNsDly nVeaErResRt AnLeiLgh: - DIFFUSION ACCOMMODATED FLOW AND SUPERPLAASTICITT 153 remains constant.* mediate state to the final state. The initial and final states of the polycrystal are thermodynamically identical, although it hasb soursff ecomered a torgueet hsterra, info sroc i=n g0 .a6p5.a rItn twoso d ooitnhge,r sth eat g rraiginhst suffer accommodation strains and translate The mechanism described below is one of non- angles. This neigphabsto re-as~chtc hoitnhge r bye vselnidt inisg athet th emira bino undaries. uniform flow. The grains do not suffer the same shape and of magnitude Api. Then, since the current I feature of the flow. array. Then the translationsIn flowing tofhr outhegh thegra cihnasn ninelv, otlhveee mulsioThensuf fefundamentalred topological feature of thisor inon-gina tes at a source and flows into a sink, the power t t t a total strain of about 10. After initial strain of also their rotation. The rotation is not linked in anany uniform flow is that grains in the aggregate dcihsasnigpea ted in driving the boundary as a source or sink 0.5 during which some grain-elongation occurred,* simple way to the translation, but is related to the their neighbors ; this permits a large, permanent, the grains ceased to elongate even though the sample is 21Apui. probability of one gram being surrounded by others of shape-change of the specimen without requiring any as a whole continued to dGReAfINo rm1. St~ct~~ly speaking, 3. Grain boundary sliding. As well as the normal particular sizes and in particular angular positions. large change in shape of its grains. This fact was first the %ow reached a steady s/ tate, in which unlimited displacements which diffusion produces, shear dis- Figure 5 shows a ldeformationarger area ofcould the occurdefor mwitnhg noe mfurtherul- rgercaoing neilzoendg a-b y Raohinger’s) in a paper which has been sion. Switching eventstion .l ikThee the sa moneple idstrainealiz ewda si nthen Fig .e n4t irely due to the placements occur in the boundary plane. It is these ooeur locally, producingrelat ivae ltranslationsocal stram ofof g raboutains p a0.55.st each other in such which permit the grams to translate past each other. The surrounding star uwcatuy rthate r ethelax enumbers to transmitof grains athelon g one dimension I Their magnitude can be seen by examining the motion strain to the specimeofn thesu rsfapecceism. enAfter incr eas estraind and ithatn the a long a direction EiDUNDARl at right angles decreased. of grains relative to a fixed origin-say the center of specimen of 0.55 all grains have participated in one K OlffUSlVE FLUX An idealization of this is shown in Fig. 4. During a 4 I I switching event, on the average. At smaller strains, mass of grain 1, as shown in Fig. 8. Work is done neighbor-switching event, like that shown here, adjacent (0) (bl a correspondingly gsrmaianlsle rf ollforwac tipoenrp enofd icuglraari nstr ahjeactvoer ies. In reality against the boundary viscosity when sliding occurs, Fro. 1. Quasi-uniform flow. A grasinw istucfhfeerds rnoeuigghhlyb ors.* FIG. 7. The accommodation strains required when grains the same shape change as the specimen as a whole and grains are not all the same size and even in two leading to a second, independent, irreversible process. does not switoh its neighbors. DislocationOne cree pw, Naayb aofrro -i dentifdyiminegn sithisons dso rnott ofm uofslvuoeawl l yfromb eformha v itaoh rpe e isrifn ecitliya lh teoxa gtohnea li ntermediate states. These Herring and Coble creep all have these characteristics. may be obtained by bulk diffusion and by boundary diffu- If the local shear stress which appears in the boundar! to map the trajectories of the grain centers as thespeoi- sion. Not,e that the flow, particularly in grain 2, is local, men deforms. Figure 6(a) shows schematically the plane is 7, and a total area A of boundary slides with a change as the specimen, and grains switch their confined to the surface regions of the grain. movement of grain-centers during quasi-uniform flow. relative velocity ti, this process dissipates power equal neighbors. It is most easily described and studied by allowing the external stress to do work. The work The trajectories are locally parallel everywhere; if to Simple geometry shows that the total sliding using an oil-emulsion analog, made as follows. If done per second on the group is rAti. mineral oil is shaken with a dyeedq udetergentation (1) (15%is obeyed within the specimen, this must displacement at each of the four sliding boundaries of ammonium lauryl sulphate) andbe the so . emBuyls iocno nistr ast, during non-uniform flow the Fig. 4(a), during a unit step, is u = 0.46d, and that the centrifuged, a structure consistingt rofaj e“cgtroariiness” of ogilr ains which undergowhere the u issw ithetchi ngt ensile stress, g the tensile strain-rate separated by very thin “gram boueventsndarie s”are of lodceatelrly- ~eT~~~d~c~T (Fig. 4). Figure 6(b) net sliding area A is ,3d 2. These large sliding displace- and V the volume of the group. gent solution results, Its deformationis an can ac betu aslt utdriaedci ng of grain trajectories in the oil- ments are an essential part of the model; they are by causing it to %ow through a channel between two emulsion model, spanning a strainThis of waboutork drives0.1. four irreversible processes. They glass plates (Fig. 2). The arrangement simulates larger than those which accompany Nabarro-Herring Five switching events are ringedare: w :i thin the rings, the topology (though not the mechanics) of flow Fro. 3. A sequence of photographs of grains in the oil- or Coble creep. In spite of this, the power dissipated grains have moved onem ulosriotnh osghoonwailn g1. pt ahetThe hsd.e fo rOurdmiaftifo unos iilm-v eec hapnirsomc. esTsh e by which the grains tem- in polycrystalline or polyphase materials made up sample had suffered a total strain of order 8 before these in driving them is surprisingly small-indeed, it is of incompressible grains which ecanmu lslidesion atmodel the ir is a phtowtog-draipmhes nwspeiroenr taken;aalr ilone;y the cghrianai nnsthe gseho w tshheiar pstea,d y- suffering accommodation deformation of a thrsetaet-ed ismtreanins ioofn a0l. 3. stBreutwcteuenr e,s tasteusb -( a) and (b) the almost always negligible. This point is discussed boundaries. sample has sufsfetrread ian fsu.r theTher sma llv ionlcuremmeent ofof s trmatterain. which must be moved Figure 3 shows what happenss uwhenrface thegra inesm ucanl- apTpheea rg raatms thetran slsauter fpaacset eorac h soutrhfear:c et he two marked by diamonds have become nearest neighbors, separating further in the Appendix. sion deforms. Pairs of grains-thosegrams markedcan di sabpyp ear tihnet ot wothe marked incante rbioy r be.dots. Butobtained T hnoe d onewub leb pyr instu (pc)e griivmes posing, as shown in Fig. 7, some idea of the local strain. 4. Fluctuations of boundary area. The final pro- phenomena are involved-the essentialthe cen tseterps iofn mthea ss of a grain in two states and exam- ! Since completing this study, a paper by Cannonu’ has cess is less-obviously an irreversible one. The grub appeared which emphasizes this, and fcleorwta inis ofst itlhl ethat othe rs, hown* Tinh e Fsitge.a d4y. -sitnatien gg rathein e lolnogcaatiol n rdeeapednjdueds tomn egnratisn which must be made in geometric properties described in this section. size and was between 0 and 0.5. boundary area increases as the cluster of four grains its shape. (StraightforwardFIQ. 5. P hogtoegormaphest r(ya) -(es)h formows athat sequ entheae i n order of * In pm&ice, a strain of order 0.5 is probably needed to increasing strain. weigher-dishing events have movesoccur- from the initial to the intermediate state, change the structure from its equilibriauvme srtatge ein vthoel uabmseen ceof matterred in th e maroeavse pdri/ngtread iwnit h hpieghr cuonntriats ts. tTehpe surround- of flow to its steady state while deforming. This point is ings relax, transmitting the loaally generated strsaitno rtoin g free energy in the system. As the cluster discussed further in Section 4(c). is 0.075d3, and that the weightedt-hme espaenc imdenis stuarnfacces . over moves from the intermediate to the final position the which this transport occurs is 0.3d.) This diffusive boundary area decreases again, releasing the energy. process, which may occur either by bulk or by grain- There is no way in which this energy can be made boundary transport, inserts or removes material into the boundary separating two grains: that is, it produces normal displacements there. It requires a [email protected] current, I, to flow from one set of sites on the boundaries of the grains, to another set, as shown in Fig. 7. The quantity I is the total number of atoms or vacancies flowing/second, from sources to sinks within the group of four grains. If the (average) chemical potential difference between a sink and source is Ap.

then the power dissipated by this process is simply GRAIN I GRAIN 3 IA/L \ / \ / 2. The interface reaction. Grain or phase boundaries may be imperfect sinks or sources for point defects. When this is so, an interface barrier exists when a vacancy is removed from, or added to, the

boundary. In principle, the magnitude of this chemi- FIG. 8. The relative shear translations of grain boundary cal potential barrier* may differ at the source and the sliding. These transformations are a fundamental feature of the model. Because of them, the total diffusional sink, but we shall assume it to be the same at both, transport/unit strain is lower than in classical treat- ments of Nabarro-Herring creep and the diffusional flow * Further discussion of the problem is given by Ashby.‘6.r’ becomes merely an accommodating mechanism. MEI AND KOHLSTEDT: INFLUENCE OF WATER ON DIFFUSION CREEP 21,461

Table 1. WaterFugacity Exponent, s in Equation(11), Describingthe Dependence of theConcentrations of IntrinsicPoint Defects in Olivineon WaterFugacity for VariousCharge Neutrality Conditions Charge-Neutrality Diffusion Species Conditions // [VMe] [Me['] IVy'] [Oi//] rv////1'--si' [Sii""] [ e/] [h'] [(OH)•]: 2 [VMc]// 1/3 - 1/3 - 1/3 1/3 2/3 -2/3 1/6 - 1/6 21,460 MEI [(OH)•]AND= [e/]KOHJ_,STEDT: 1/2 -1/2IN•UENCE -1/2 1/2 OF1 WATER-1 1/4ON DIFFUSION-1/4 CREEP [(OH)•]= [H Me]/ 0 0 0 0 0 0 0 0 [Fe•=]= [HM=] ' - 1/2 1/2 1/2 - 1/2 - 1 1 - 1/4 1/4 [Fe•] =3 [H///• si• - 1/4 1/4 1/4 - 1/4 - 1/2 1/2 - 1/8 1/8 indicatesthat the temperature[Fe•] =2[(2H)s//i]fluctuates by -2/3 no 2/3more 2/3 than -2/3 -4/3To verify4/3 -1/3 that the1/3 water fugacitywas properlybuffered, it [Fe•c]= [(3H)/si] -3/2 3/2 3/2 -3/2 -3 3 -3/4 3/4 rpl///1 _+2K in the21,458 hot zoneMEI AND KOHLS•DT:and variesINFLUENCE byOF WATER <2 ON DIFFUSIONK along 0CREEP the 0 length0 of 0 was0 important 0 0 to determine0 if the samplewas water saturated. [(OH)•]- 2[(2H)s//,] -1/3 1/3 1/3 -1/3 -2/3 2/3 -1/6 1/6 the sample.dependence of Deformation viscosityon water fugacity or waterexperiments concen- were conductedat To test this point, oriented single crystals of olivine trationmust be quantified.Second, the mechanism[(OH)•]:by which [(3H)'s•] -1 1 1 -1 -2 2 -1/2 1/2 water weakens olivine must be identified. Both are treated as .•.::i.:..::.?•a,,,..ß ...:: temperaturesprimarygoals between of this studyso that laboratory 1473results andcan be 1573 K, confiningpressures ~3x3xl mm were embeddedin three of the polycrystalline extrapolatedto mantle conditions. To attain thesegoals, a ' .,...:::•'fi'::::'"".... ' .?!( ...... sampleassembly was designedto maintainwater-saturated ß. '5? in the range 100 to 450 MPa,diffusion creep differentialis the'.:•:"... dominantstresses mechanism .. of ofdeformation 20 to squares300 fit to thedata yields n = 1.1_+ 0.1, in goodagreement conditionsduring a deformationexperiment. Water fugacity :•:"::'.i*::i''::' ...... ;::.4:'!'i:.':-. '"'. .... samplesto act as indicatorsfor water content,since OH when differential stress • 100;? :.: :.:':3': MPa.ß': ;' :.::.:'2. was systematicallyvaried by changingthe confiningpres- with the valueobtained for samplesdeformed under hydrous MPa, andsure axial in orderto quantifystrainthe relationshiprates between Tofrom evaluatecreep rate 7.0x10the effect of'? pressureto 1.2x10on creep rate,'4 sq. three conditions. solubilityin olivine crystalsas a functionof water fugacity and water fugacity. These data impose constraintson the 3.1.2. Grain size dependence.Since the rate of diffusion point defect mechanismby which waterexperiments weakensolivine. were conductedJOURNALunder anhydrous OF GEOPHYSICALconditions RESEARCH,in VOL. 105, NO. B9, PAGES 21,457-21,469, SEPTEMBER 10, 2000 In FigureThe present 1paper ofreports Meithe influence andtheof waterdiffusion Kohlstedton diffusioncreep regime. [this Onesample issue], was deformed raw at data two creep aredepends hasstrongly previously on the grainsize of been the sample, quantifiedthe [e.g., Kohlstedtet al., 1996]. creepof olivine. Resultson the effectof confining water on dislocationpressures, 5. 100Experimental and 300 MPa, while two samples effect dataof grainsize (andon creeprate was empiricalmeasured. The average ﬂow laws) creep are reported in the companionpaper by Mei and plottedin Kohlstedt the [this issue].form of wereload deformed versusat a single time conEming andpressure, displacement300 MPa. grainsizes of After a sample abefore deformationand after deformationexperiment were the watercontents in both the Becauseresults from a singlesample provide optimal resolu- determinedfrom opticalmicrographs using the line intercept versustime 2. Experimental for one Details experiment. tion, creepdata Influencefor The the sample load,of deformed water whichat on both plastic was 100 and moni- deformation method with asingle correctionof olivinefactor crystal of aggregates 1.5 [Gij'kins,and 1970]. the The polycrystalline aggregate were deter- 2.1. Specimen Preparation 300 MPa are shownin Figure6. On thebasis of theseresults, grainS. Mesize i of & a D sample. Kohlsteat a particulardt time during an experiment tored by an internal loadthe activation cell,volume 1. was Diffusion is 15 held _+5x10 creep '6 m3/mol.constant regime A linear least in each minedby FTIR spectroscopy. 2.1.1. Starting material. To minimize the presenceof was estimatedfrom the initial and final grain sizes of the spinelinclusions in our samples,grains of SanCarlos olivine, JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105,specimen NO. B9,using PAGES the 21,457-21,469, graingrowth SEPTEMBERlaw [e.g., 10,Karato 2000 et al., deformation(Mg0.91Fe0.09)2SiO4,step ~3 to 6by mm in maximummeansdimension were of S. Meia servo-controlledmotor. 1986; Hirth and Kohlstedt, 1995] hand-pickedbefore grinding to a f'me powder in a fluid energymill. This materialwas stokes settled in distilledwater Department of ChemicalEngineering and Materials i Science, University of Minnesota-TwinCities Differentialto obtainstresses a powder with particlewere sizes rangingdetermined from10-4 2 to _ i byi i dividingi i i i i I load i i by the 2.5. dtData2 - di2 - kAnalysist , (3) Minneapolis 21,462 MEI AND KOHI, STEDT: INFLUENCE OF WATER ON DIFFUSION CREEP 18 ktm. The particle size distributionis shownin Figure 1.- T = 1523 K instantaneousThe powder areawas then stored ofin thea vacuum deformed drying oven at sample. Reportedvalues 425 K. - d- 15 •m -_ InfluenceD. L.of Kohlstedt water on plasticdeformation of olivine aggregates I I ' - * ' * I ' *''l 3.2. Water Content of Deformed Samples 2.1.2. Sample fabrication. Mixtures of olivine plus 5%- all20• 1 - Strain rate versus stress results were calculatedP = 450 MPa from _ the for differentialenstatitepowder were stress cold pressed intoare nickel 1. cansaccurate Diffusion26 mm inDepartment to creep of _+2 Geology regimeMPa. and Geophysics, CreepUniversity of data Minnesota-Twin Cities length,11.6 mm in outerdiameter, and 10 mmin innerdiame- - T = 1523 K FTIR spectraobtained from olivine singlecrystals embed- Minneapolis T = 1523 K ter. The enstatitelimits graingrowth during hot pressingand recordedd= 15gm load and displacement versus time data. Differentialded in polycrystallineaggregates deformed under hydrous were obtaineddeformation,and by it buffers varying the silica activitythe of the olivine. stress__ in a stepwisefashion at 10-4 :- o= 26 MPa S. Mei10 -5 -- The cold pressingwas performedlayer by layer in ~2 mm 10-5 conditionsat P = 100, 300, and 450 MPa are presentedin Abstract. The influenceof water on diffusioncreep of olivine- aggregatesall20• 0 was investigated constanttemperatureincrements at a pressureof ~200and MPa so pressure; Departmentthat the initialof Chemicalthe Engineering sampleand Materials was Science, unloadedUniversity of Minnesota-Twin stress- Cities was correctedfor the strengthof theall20 Fe• 1 jacket, _' NiFigure 9a. The spectrumfrom a relatively dry, as-received porosity was homogeneouslydistributed throughoutthe .

by performinghigh-temperature creep experiments under both hydrous and anhydrous - undeformedsingle crystal is includedfor comparison.The sample The skinof eachlayer was carefullyremovedMinneapolis with a n=l.1 - betweeneach step. For most samples,conditions.Deformation steady experiments state deforma- were conductedon fine-grainedsleeve,- samples and using talca gas- sleeve as well as for the changein cross-hydroxyl concentrations of the four crystalsare around110, -

D. L. Kohlstedtmedium apparatus at confiningpressures of 100 to 450 MPa andtemperatures between 260, 380, and<10 H/106Si,respectively. On thebasis of tion was achievedby --0.5% strain1473for and1573 diffusionK. Water wascreep suppliedbyand the dehydration 1% of sectional talc,n=l.1 which occursarea near of the sampleduring deformation. To makepublished results for water solubilityin olivine as a function Departmentof Geologyand Geophysics, ß ß W University P=100MPa_of Minnesota-Twin Cities - ßo0 10-5 0.15 10-6 1075K. Water fugacitiesof-85 to 520 MPa wereobtained by varyingthe confining - of waterfugacity (confining pressure at fixed oxygenfugaci- strain for dislocationcreep. Minneapolis To preserveO [] the 'h' P=300MPa- deformation- thesecorrections, the flow laws for_ Fe and Ni were taken- from pressureunder water-saturated conditions. - Under both hydrous and anhydrous conditions ty) [Bai and Kohlstedt, 1992c], our sampleswere fully . q- x P=450MPa- _ saturated with water. deformationwas dominated by grainboundary - diffusion. 10-6At Frost a waterfugacity andof Ashby-300ß MPa,P=300MPa [1982], - while the strengthof talc was inducedmicrostructure, deformed Abstract. Thesamples influenceI i of . waterIwere , ,•l on diffusion cooled•creep , of under olivine aggregates was investigated Infraredspectra from four deformedolivine aggregates are 0.10 samplescrept -5 timesfaster than those deformed under anhydrous conditions at similar[] P=lOOMPa differential stress at a rateby of--2 performing101 differentialK/s high-temperature tostresses 900and creepK; temperatures. 10 experiments 2this decreaseWithin under theboth range hydrous of waterdeterminedand anhydrousfugacity investigated, in theour laboratory[Mei, 1999]. The sleevespresented in Figure 9b on an absorptioncoefficient versus conditions.strainDeformation rate is proportionalexperiments to were water conductedI•gacity toon the fine-grained 0.7 to 1.0 power,samples assuming using a gas-values for the wavenumberplot. One sample was deformedunder anhy- I I I I lllll , 10-6 , , drous conditions and three under hydrous conditions. mediumapparatus activationat volumeconfining of pressures 0 to 20xl of0'6 100m3/mol, to 450respectively. MPa andtemperatures We proposebetweenthe following point in temperatureresulted in a decreasein pressureof--50 MPa supported~3 to 20% of the largestand smallestapplied Hydroxyl concentrationsin the samples from hydrous 0.05 1473 and1573defectK. model Waterto owas explain (MPa) suppliedthis by water-weakening the dehydrationeffect:of talc, In which going101 occursfrom annear anhydrous to a 101 3x101 as the gas confining medium1075K. Water cooled. fugacities of-85 The to 520load MPa werewasobtained thenby varying loads,the confiningrespectively. experimentsare approximately 2100, 4300, and 5800 H/106 Si, Figure 5. A hydrouslog-log plotenvironment of strainthe ratecharge versus neutralitystress for condition changes from[ Fete ] --2 [ VM//e ] to for confiningpressures of 100, 300, and 450 MPa, respec- pressureunder water-saturated conditions. Under both hydrous and anhydrous conditions d (gm) samplesdeformed [Fete] at - three[H•e ].conEming Asa consequence, pressures forunder olivine aggregates theconcentration o of(MPa)silicon tively. Under "dry" conditions,the hydroxyl concentration removed,and 0.00 the samplesdeformationwere was broughtdominated by to grain ambientboundary diffusion. condi-At a water fugacity Theof -300 deformation MPa, results Figurewere 7. A log-loganalyzed plot of strainin rate terms versusgrain of size a for flow hydrousconditions. interstitials, Data theare ratefrom of samples silicondiffusion, PI-107 (openand therefore the rate of diffusioncreep increase is<100 H/106 Si. Thesevalues are 15-20 times higher than the 0 5 10 circles),15 PI-184 20 (solidFigurecircles), 2. TransmittedPI-295 plane (pluses), light photomicrographsPI-308 (solidof hot- Figure 6. A log-log plot of strain rate versus stressfor PI-507. samplescrept systematicallypressed-5 timessamples: faster (a)with drythan increasingsamplethose and (b)deformed water wet sample.fugacity under Scale (i.e.,anhydrous OH concentration).conditions law ofat similarthe form concentrationsin the singlecrystals, indicating that a large tions at a coolingrate ofstars), ~ 1PI-333 K/s (open barand represents squares), a20/•m. depressurizationPI-351 These micrographs (open stars),are from PI-507 polished ratesample PI-220 deformedunder anhydrous conditions at two Particle Size (gm)differential stresses and temperatures. Within therange of waterfugacity investigated, the portion of the water residesin grain boundariesand fluid (crosses),strainPI-568rate is proportional and(solid etched squares) sectionsto andwatercut normal PI-569I•gacity to (asterisks).the long to the axes 0.7 Theof tothe 1.0confining power,pressures,assuming values100 (openfor thesquares) and 300 MPa (solid of ~ 1 MPa/s.Figure 1. Particlesize distribution of thestarting powder. 1. samples.Introduction circles).Linear study,least-squares water refers fits toto the any data water-relatedyield n = 1.1species for detected as inclusions. straightactivationlines are volume linearofleast 0 tosquares 20xl 0fits'6 m3/mol, to the datarespectively. from PI- We proposethe following point whered i and dt are grain size at the start (initial) of and at 107, PI-184, and PI-569. the stressexponent. O-H stretching bands with an infraredspectrometer. sometime Al- t during the experiment,respectively. The con- After removalfrom thedefect apparatus,model toThe explain plasticthe this deformation sample water-weakening behaviorjackets effect: of olivine, Inwere going thefrom most an anhydrousthoughit hasto abeen established that evena traceamount of 4. Discussion hydrousenvironmentabundant component thecharge of neutrality the Earth'scondition upper mantle,changes plays from[ an Fete ] --2 [ VM//e ] to stantk hasthe form k: k•exp(-Eo/RT), whereEg is the water weakens•(o,d,T,P, olivine [Carterfi_t2o) and Avd- Lallemant, Aactivation 1970;fI•2Oenergy expfor grai• growth Q*+ and PV k ø is a material [Fete]- [H•eimportant]. Asrole a consequence, in determiningforthe olivine dynamics aggregates of thethe Earth'sconcentration Chopraofand silicon Paterson, 1984; Mackwellet al., 1985;d Karatoe RT ' (2)4.1. Dependence of Creep Rate on Water Fugacity dissolvedusing aqua regia (50% HC1 + 50% HNO3). The parameter.A valueof Eg= 160kJ/mol was used based on interstitials,interior.the rate To of silicondevelopdiffusion, an understandingand therefore of the thecreep ratebehavior of diffusion et al.,creep 1986],increase the relationship between creep rate and water publishedresults for grain growth in olivine aggregates As illustrated in Figure 5, the creep rate of samples systematicallyof olivinewith increasing at conditionswater appropriate fugacity (i.e.,for the OH Earth's concentration). mantle, specimens were cut into sections foi analysis of the fugacityor waterconcentration has never been quantified [Karato,1989a],for and k ø: 3.8x10-9m2/s was obtained from our deformed under hydrous conditionsdepends strongly on numerousstudies have been conductedsince the early 1970s olivine. In addition, the mechanismresponsible for water data. confining pressure. At the same differential stressand microstructure and determination in orderof tothe quantify water the high-temperature concentration.and high-pressure where• is axialstrain rate, o is differentialstress, d is grain 1. Introduction study,water weakeningrefers to inany olivine water-relatedhas not yet species been identified.detected as rheologicalproperties of this major rock-forming mineral The dependenceof strainrate on grain size is shownin O-H stretchingRecentbands analyses with anof infrared the waterspectrometer. content in Al-mantle-derived [Kohlstedt and Goetze, 1974; Durham and Goetze, 1977; Figure7. A least squaresfit of the data yieldsa grain size The plastic deformation behavior of olivine, the most thoughit hassize,rocksbeen demonstrateA established is athat thatmaterials water even contenta tracecanamount parameter, varyof significantly O is activationenergy, P is Chopra and Paterson, 1981; Kohlstedtand Hornack, 1981; exponentof ~3, indicatingthat deformationinvolves grain abundantcomponent of the Earth's upper mantle,plays an water weakensfrom olivine one environment [Carter andto theAvd next. Lallemant, For example, 1970; it hasbeen T (øC) Karato and Ogawa, 1982; Poumellec and Jaoul, 1984; boundarydiffusion, often referredto as Coble [1963] creep. 2.4. Water Concentrationimportant Analysesrole in determining the dynamicsof the Earth's Chopraand confiningestimatedPaterson, that1984; olivine Mackwell pressure,grains etin al., rocks 1985;at V a Karato depthis of activation ~ 120km volume, T is absolute Ricoult and Kohlstedt, 1985; Borch and Green, 1989; Bai et 3.1.3. Activation energy for creep. Two experimentswere 1350 1300 1250 1200 interior. To developan understandingof the creepbehavior et al., 1986],beneath the relationshipa mid-oceanbetween ridge creep are ~20%rate and saturated water with OH al., 1991; Bai and Kohlstedt, 1992a,1992b]. Throughthese temperature,and R is theconducted gasto constrain constant.the activation The energy foractivation diffusion 0-4 of olivine at conditionsappropriate for the Earth's mantle, fugacityor water[Hirthconcentration and Kohlstedt,has 1996].never been Inquantified contrast,forSobolev and I ,,,,i,,,,I,,,,_ I , , , , I , , , •.- studies, the dependenceof the creep rate of olivine on creepunder hydrous conditions. In Figure8 theresults from The concentrationof water-relatednumerousstudies have speciesbeen conductedinsince olivinethe early 1970sprior olivine. In Chaussidonaddition, the[1996] mechanismargued thatresponsible nominally for anhydrous water minerals - d= 15•m - temperature,confining pressure, flow stress,partial pressure energy and activation volumean experiment termsthat was conducted includeat a confining contributionspressure of - P = 100 MPa in order to quantify the high-temperatureand high-pressure weakeninginsuch olivineas olivinehas not are yet fullybeen saturatedidentified. with waterin themantle of oxygen,and activity of a constituentoxide phase has been 100 MPa and temperaturesfrom 1493 to 1573 K (PI-568) all20• 1 - to and after deformationrheological wasproperties determinedof this major rock-forming using mineralFourier Recent analyses wedgeabove of thea subductingwatercontent slab. in mantle-derivedInfrared(IR) spectroscopy ' O=60 MPa - determinedexperimentally. from the effect of temperatureillustrate theand effect pressure, of temperaturerespectively,on strain rate. Data on [Kohlstedt and Goetze, 1974; Durham and Goetze, 1977; rocks demonstrateanalysisthatof major waterminerals content can in mantle vary significantlyxenoliths obtainedreveals thatat differential stresses near 60 MPa were normalized - Q = 300 kJ/mol ' By comparison,current understanding of the influenceof waterconcentration in olivineranges from about10 to 2000 transform infrared (FTIR)Chopra spectroscopy. and Paterson, 1981; Kohlstedt Spectraand Hornack, in 1981; thefrom one environmentto the next. For example,it hasbeen to a single differential stress of 60 MPa using a stress wateron the creepbehavior of olivineis very limited. In this hydroxylH/10s Si [Bell and solubility Rossman1992]. asSince wellwater has as a pro- on point defect (e.g., silicon Karato and Ogawa, 1982; Poumellec and Jaoul, 1984; estimatedthat olivine grains in rocksat a depthof ~ 120km exponent of n = 1. A linearleast squares fit to the datayields wavenumberrange 2000 to Ricoult 4000 and Kohlstedt,cm 4 1985; were Borch collectedand Green, 1989; Baiwith et a nouncedeffect on creeprate, the mechanicalbehavior of beneatha vacancy mid-oceanridge or are ~20%interstitial) saturatedwith OH concentrationQ = 300 kJ/mol. Data from anand experiment mobility.that was conducted For 10-5 _ _ al., 1991; BaiCopyright and Kohlstedt,2000 by the 1992a, American1992b]. Geophysical ThroughUnion. these olivinein theupper mantle may vary significantlyfrom one $•/ _ _ [Hirth and Kohlstedt,1996]. In contrast,Sobolev and at a confiningpressure of 400 MPa at temperaturesfrom 1473 Nicolet-100 high-resolutionmicro-FTlR spectrometer.Room location to the next. studies, the dependenceof the creep rate of olivine on Chaussidondiffusion[1996] argued that creep nominally theanhydrous stressminerals to exponent 1533 K (PI-544) aren also =plotted 1. in IfFigure grain8. Creep matrixdata temperature,Paperconfining number pressure, 2000JB900179. flow stress, partial pressure To understandthe influenceof water on the creepbehav- suchas olivineare fully saturatedwith waterin themantle were normalizedto a single differentialstress of 27 MPa temperatureabsorption spectra of oxygen,and0148-0227/2000JB were activity of measured a constituent900179 $09.00oxide phase on has doublybeen ior of olivine,two topicshave to be investigated.First, the - % - wedgeabove diffusiona subductingslab. dominates, Infrared(IR) spectroscopythe usinggrain n = 1. sizeThesedata exponentyield an activationp energy = of 2, O =and 290 if determinedexperimentally. polishedsections (150 pm thick, 3 mm diameter). On theanalysis 21,457of major minerals in mantlexenoliths reveals that kJ/mol. No correction was made to the strain rate for the By comparison,current understanding of the influenceof waterconcentration grainin boundary olivineranges from difthsion about10 to 2000change dominates, in water fugacityp with= change3. Forin temperature; dislocationthis - P = 400 MPa wateron the creepbehavior of olivineis very limited. In this H/10s Si [Bell and Rossman1992]. Sincewater has a pro- basisof previousstudies [Beran and Puthis, 1983,' Miller et correctionwould increasethe calculatedactivation energy by Q = 290 kJ/mol nouncedcreep,effect on creepn =rate, 3-4 the mechanical for olivine.behavior of < 10 kJ/mol.In our Likewise, experimentsno correctionwas madethe for transitionthe change 10-6 ,,,,,, ,,, al., 1987; Bell and Rossman,Copyright 20001992; by the AmericanLibowitzkyGeophysical Union. and Beran,olivine in theupper mantle may vary significantlyfrom onein strain rate due to the increase in oxygen fugacity with 0.60 0.65 0.70 location tofrom the next. diffusion creep (n increasing= 1) temperature; to dislocationin this case, the dependencecreep of (n strain • 3) 1995],infrared absorption bandsPaper number between2000JB900179. 3300 and 3700 cm 4 To understandthe influenceof water on the creepbehav- rate on oxygenfugacity is not known for hydrousconditions. 1000/T (K) 0148-0227/2000JB 900179 $09.00 ior of olivine,occurstwo topicsat have ato differential be investigated.First, stressthe of ~100 MPa. In this paper, If the creep rate increasesas oxygen fugacity to the l/6th Figure 8. Arrhenius plot of strain rate versus inverse result from the stretchingvibrations of O-H bonds. The21,457 power as it does under anhydrousconditions, then quoted temperaturefor samplePI-568 (solidcircles) and PI-544 (open only the difthsion creepvalues datafor activation wereenergy selectedoverestimate the truefrom values by each circles) deformed under hydrous conditions at P = 100 and hydroxylconcentration was calculatedfrom infraredspectra, experiment. The dislocation~25%. creep results are discussedby400 MPa, respectively.. after making a backgroundcorrection using a third-order Mei and Kohlstedt [this issue]. polynomialfit to the backgroundabsorbency by integrating across the hydroxyl bands. Hydroxyl concentrationwas determined from the relation [Paterson, 1982] 3. Experimental Results 3.1. Mechanical Data

3.1.1. Creep results. Diffusion creep results (for Cøn_ 1507 f 3780K(v) -vdv ' (1) differentialstresses z 100 MPa) from morethan 60 creeptests conductedon 14 samplesare summarizedin Table 1 of Mei whereCon is molarhydroxyl concentration, K (v) is the and Kohlstedt [this issue]. Strain rate versus stressresults absorptioncoefficient (in cmq) at wavenumber v (in cm'•), from deformation experimentsconducted under hydrous and y is an orientationfactor accountingfor the distribution conditionsat three differentconfining pressures are plotted of the OH groups. For the isotropicbroad absorption bands in Figure5 onlog-log axes. Data werecorrected to a common foundin polycrystallinesamples, a valueof != 1/3 wasused; grain size of 15 •m usinga grain size exponentof 3. A least for the anisotropicsharp bands from singlecrystals, a value squareslinear regressionanalysis of the data at eachconfin- of != 1/2 wastaken for theinfrared beam parallel to the [010] ing pressureyields a stressexponent of n = 1.1 _+0.2 at all direction. three pressures (water fugacities), demonstrating that 6. Effects of water

7. Effects of melt

B06205 TAKEI AND HOLTZMAN: VISCOUS CONTIGUITY MODEL, 1, GENERAL B06205

Figure 5. Schematic illustration showing the significant difference in the diffusion paths between (a) CK model and (b) contiguity model, under shear deformation shown by 3-D arrows. Thick arrows show the quasi-2-D approximation of the 3-D diffusion paths.

Figure 6. One quarter of a circular grain domain with four pores in the directions of (a) q = p/4, 3p/4, f f 5p/4, and 7p/4, and (b) q = 0, p/2, p, 3p/2. Tractions under a given shear strain rate (e_ xx = e_yy = e_(>0)) calculated from the contiguity model (thick line), from the CK model (thin line), and fromÀCoble creep (dashed line) are shown as functions of position q (0 q p/2) on the grain surface. The traction and shear viscosity calculated from the contiguity model aresignificantly different between Figures 6a and 6b.

10 of 19 B06205 TAKEI AND HOLTZMAN: VISCOUS CONTIGUITY MODEL, 1, GENERAL B06205

f and A are on the spherical grain surface. P is mapped to P0 where z direction is taken to be normal to the circular on the circular plane, where P0 is in the same plane as plane. Then, fz0(P0) satisfying equations (26) and (27) is f f triangle OA P and the length of line segment A P0 is taken solved analytically on this circular plane and is again to be equal to the length of segment AfP along the great mapped to the contact patch on the spherical surface by circle (Figure 7b). Radial velocity at the spherical surface, f 0r(P) = f 0z(P0). In this manner, at procedure B in Figure 1, R C R R u_ r(P), is mapped to the circular plane by u_ z(P0) = u_ r(P), fr0(r ) at X (r ) = 1 is obtained analytically from u_ r(r ) at XC(rR) = 1. [37] Unit vectors for the spherical coordinate system f f f f (q, f, r) at A are written as eq, ef and er, detailed descriptions for which are presented in Appendix B. Position P0 on the circular plane is represented by polar coordinate (0 r 1, 0 F 2p); the length of line f segment A P0 is represented by afr, where af represents the radius of the circular plane, and the angle between vectors f f A P0 and eq is represented by F. The af and wf are related by 1 af = R cosÀ (1 2wf). For P0 represented by (r, F), outward unit normal toÀthe spherical grain at the corresponding position P is written as

a r a r nf r; F cos f e f sin f cos F e f sin F e f : 38 ð Þ ¼ R r þ R q þ f ð Þ For areal segment rdrdF on the circular plane, the corresponding area on the spherical grain is b(r)rdrdF, where b(r) is given by

R a r b r sin f : 39 ð Þ ¼ af r R ð Þ

[38] Cijkl is obtained by following procedures A–C in Figure 1 in clockwise order. At procedure C, the integral in equation (29) over the spherical grain is converted to the integral over the circular plane by using equations (38) and (39). Then, the solution for Cijkl can be written as

3 nf 1 2p 1 2p 3NAkTR af 4 4 C À ijkl 32DWd R p2 ¼ f 1 r 0 F 0 r0 0 F0 0 X¼ Z ¼ Z ¼ Z ¼ Z ¼ f f f f n r; F n r; F n r0; F0 n r0; F0 G r; F; r0; F0 i ð Þ j ð Þ k ð Þ l ð Þ ð Þ b r r r0 dr dF dr0 dF0; 40 Á ð Þ ð Þ

Figure 4. Solutions for the 2-D model. (a) Radial velocity at the contact face (XC(q) = 1) under a given shear strain rate f f (e_ xx = e_ yy = e_(>0)) versus position q along the 2-D grain surfaceÀfor 1/4 of a circle (0 q p/2). (b) Differential traction on the grain surface calculated from the CK model at various values of 82D and from the Coble creep model at 82D = 1. Arrows show the direction of matter flux. Top gray and white bars show the regions acting as matter source and sink, respectively. Positions qm (m = 1–4) are shown for the case of 82D = 0.92. (c) Differential traction on the grain surface calculated from the contiguity model at various values of 82D (bold solid lines) and from the Coble creep model at 82D = 1 (dotted line). Top gray and white bars show the regions acting as matter source and sink, respectively, for the case of 82D = 0.92. (d) Flux of component A at the contact face (XC(q) = 1) calculated from the contiguity model (bold lines) and from the CK model (thin lines).

9 of 19 Viscosity as a function of contiguity and melt fraction at textural equilibrium (a) 1 semi-empirical theoretical 0.8 J

, 0.6 y t i u g i t n

1 o 0.4 c Qd20, 14 A=2

d 30, 12 0.3 A=2.3 Q 0.2

Qd30, 14

0 compressional 0 0.05 0.1 0.15 0.2 melt fraction, F 0

0 (b) 10 R=5Mm

0.6 c c H " H

-1 tensional , magnitude of normal component 10 y t i s

o Qd0, 14 c s -1 i v

a L=18 e h s -2 d L=23 A=2 e 10 d 30, 12

z Q i l a A=2.3 m r o n Qd30, 14 0.9 10-3 0 0.05 0.1 0.15 0.2 melt fraction, F

Fig.9 Summary

1. Diffusion is a random walk process of vacancies, coupled to atoms. To change the shape of a grain, the diffusion is driven by stress gradients.

2. The rate (kinetics) of shape change depend on the diffusion path and its diffusivity that contribute to the ﬂux of atoms. The diffusivity depends strongly on T (the thermally activated process is the jump frequency).

3. Diffusion Creep, from models and experimental result has a non-linear dependence on grain size (d^(-2-3)) and a linear dependence on stress.

4. High strains are achieved by grain switching, requiring grain boundary sliding and less strain per grain (can be referred to as “superplasticity”).

5. Water aids diffusion by weakening bonds and/or by increasing the number of vacancies.

6. Melt aids diffusion by reducing diffusion path lengths if the porosity is connected. The clockwise path: strain to stress (Elastic case)

3.2 Stress calculA newation model for grain-boundary diffusion creep with melt Takei and Holtzman, 2009a., (based on Cooper-Kohlstedt model) The constitutive relation is S L vi σij + p δij = [Cijkl]ε˙kl (19) Sign convention is tension = positive, compression = negative for stress and strain rate, and the opposite for the fluid pressure, pL. Strain rates: 3 Calculations: 3-D contiguity and diffusion creep model Pure Shear, Plane Strain: Overview of the calculation: ε˙ 0 0 now in 1. Calculate Cijkl from geometry of faces. 11 steady state, 2. Strain rate applied to Cijkl to get stress. 0 and0the lo0cal displacement vector at each point on grain-grain contact surfac(20)e: 3. Calculate tractions on each face of grain. small strain 0 0 ε˙33 u (rR) = εf rR at XC (rR) = 1 YT98 eqn(11) (2)   i ij j limit   where ε˙11 = 1 and ε˙33 = 1. − 3.1 The calculation of Cijkl Pure Shear, 3-D: YT’s notes explain the derivation of the viscosity tensor. The big C tensor: Path B. Grain-ijklscale governing equations ε˙11 0 0 FiguThereequ1:3 ilinfbrium cond4 ition: (1 φ)3RT (d/2) af f f f f vi ∂τij Cijkl = − 0 ε˙ 0 ni nj n n (14) (21)  32DδΩ 22 d/2 k l = 0 at r < R i.e. τij,j = 0) (3) f=1 ∂x ! " # j 0 0 ε˙33 nf is the number of the face. We are using a 14-faceThemconodel.stitutivd is graierelationn size, (aelastic):is the area of the contact patch,   f 1 which will1 be calculated with some function of the contiguity tensor (ϕij) and position, D is diffusivity, etc. where ε˙11 = 2 ε˙22 = 2 and ε˙33 = 1. ∂up ∂ui ∂uj − − τij = λ δij + µ + or τij = λup,pδij + µ(ui,j + uj,i) at r < R (4) ∂xp ∂xj ∂xi Compression axes rotatedIn my 45code,◦,thePlfilanee constraitaininn:g the angular locations of each face !is read in": angle!s 14.dat. The"face radii, 3λ+2µ af , are calculated with a subroutine called radiusandisoλ14.mis thfore Lamisotréopicconsandtantradi(?)usandanisµo 14.mthe shforearanmoisotropduluics. The bulk modulus k = 3 and Poisson’s ratio 3k 2µ model (variable called ALP). This model inε˙tro11ducesν =0a6kf−+2actorµ .0that is a ratio between large and small faces:

0 ε˙22 0 (22) Then calculate scalar contiguity forisotropic is Path C. Grain-scale stress to macroscopic framework(?) stress 0 0 ε˙ nf 33  1 a2  1  f=1 f  σS = τ (r)dV (5) ϕ = 2 ij (15) ij 4 4π(d/2) Vs r

f F u˙ r = (d/2)ε˙ijnjni (25) 4 The tangential component is not expressed so easily, because it has components itself, but

u˙ t = u˙ f u˙ f nf (26) i i − r i

7 and the local displacement vector at each point on grain-grain contact surface:

R f R C R ui(r ) = εijrj at X (r ) = 1 YT98 eqn(11) (2)

Path B. Grain-scale governing equations The equilibrium condition: ∂τij = 0 at r < R i.e. τij,j = 0) (3) ∂xj The constitutive relation (elastic):

∂u ∂u ∂u τ = λ p δ + µ i + j or τ = λu δ + µ(u + u ) at r < R (4) ij ∂x ij ∂x ∂x ij p,p ij i,j j,i ! p " ! j i " 3λ+2µ and λ is the Lam´e constant (?) and µ the shear modulus. The bulk modulus k = 3 and Poisson’s ratio 3k 2µ ν = 6k−+2µ .

Path C. Grain-scale stress to macroscopic framework(?) stress

1 σS = τ (r)dV (5) ij V ij s #r

1 σS = (f rR + f rR)dS (6) ij 2V i j i i s #r=R

To solve this path, boundary conditions are mixed! Where XC = 1, displacement is constrained; when XC = 0, traction (i.e. pressure in the fluid) is constrained. These mixed boundary conditions mean that this path must be solved numerically for the elastic case. The viscous case is different because of the different constraints due to GBS? Grain Boundary Diffusion creep (from Cooper-Kohlstedt / Coble creep model) 2.2 Diffusion creep model symbol 2.2 Diffus2.2ionDicrffeepusiomnocrdeleep model2.2 Diffusion creep model J matter flux (mol/m2/s) 1. No 2melt Jv vacancy flux1.(molNo/mmel/st1.) No melt 1. No melt This is basically the Coble Creep model for diffusion through grain boundaries being the rate-limiting step. Cv mole fractionThiofs visacbanasciciesThiallinys tgrainishebCobasbicounallle yCreedarythep(GCobmBo)led.elCreefor pdimffThiuosdisonelisfthrorbasdioughicffallusyigrontheainthrCobboughounledariCreegrainespbbmeingounodedarilthforeerate-limitinsdibffeingusionththregrate-limitinoughstep. graingbsountep.daries being the rate-limiting step. 3 Ω molar volumMasseMassof mconseattconseervMassorationrvvationacconseancy r(vmation/mol) Mass conservationu˙ r P normal traction on the surface, compression positive (Pa) J =u˙ r u˙ r u˙ (7) J = δΩ J = r(7) (7) ∆P variation from P ◦ −∇−∇· · δΩ−∇ · δΩ J = (7) 3 −∇ · δΩ Nv number of vacancieMasssbbalancey mole per unit volume of GB (mol/m ) 2.2 Diffusion creep model2.2 Diffusion creepMass mbalanceodelMass balance Mass balance Uv formation energy of a vacancy (J/mol) J = J (8) 3 v J = J (8) N number of matter (atoms) in moles per unit volume of GB (mol/mJ =) J−v v J = Jv (8) (8) − − The ratio− of tangential to normal displacement: 1. No melt 1. No melNv t = NCv, numbVeracancyof molesfluofx vacancies Vacancy fluxV2acancy flux Vacancy flux Dv diffusivity of vacancy (m /s) This is basically the Coble Creep model for diffusion through grain boundaries being the rate-limitinJv =g sDtep.v Nv (9) f f This is basically the Coble Creep model 2for diffusion throughJgr=ainDbounJvNdari= Desv beingNv the rate-limiting step. (9) (9) u˙ n D 2.2 = DiCvDffv,usdiffuiosivnitycrof meepatter (mmo/s)del v − −v · ∇· ∇v − · ∇ Jv = Dv Nv (9) f i i 2.2 Diffusion creep model − · ∇ T Nu = f (27) Mass conservation Mass conservation and the definition of N The ratio of tangential to normal displacement: u˙ Tablean2:dFtheor ddiffefiniusionantiond cthereofepdNefiniconstituttion ivofeNrelation and the definition of N i u˙ r u˙ r Uv + P Ω U + P Ω | | 1. No melt Nv = NCv = N exp Uv + P Ω v U(10)v + P Ω J = JN == NC =NNv =expNCv =(7)N exp (7)(10)f f (10) 1. No melt −∇ · δΩ −∇ · v δΩ v − − RTRT − RT Nv = NCv = N exp u˙ n (10) This is basically the Coble Creep model for diffusion through! gr! ain boun!" "daries b"eing the3rate-limitin.3T N f T=r−acti gRiiosTtep.ns on each face (27) This is basically the Coble Creep model for diffusion forthrpresoughsuregrdrivainenbdiounffusdariion. Comes bbeingining ththee firate-limitinrst three givesg step. u ! f " Mass balance Mass balance for pressure fordrivpresen disurffuesdiorivn.enComdiffbuininsion.g theComThefibforrstininpresthrratiog ethesure giefivofrstedsrivtthranenedigene ffgiuvsteiosialn. Comto normalbining the displacemenfirst three gives ut˙:i Mass conservation 4 | | Mass conservation u˙ r Traction is the vector describing how the stress acts on each face. With the assumption that grain boundaries J = Jv J = Jv u˙ r u˙ r(8) u˙ r (8) u˙ Nv = N = u˙ r(11) f f(11) − r − ∇ ·JN∇v== −δΩDv v Nv = (11) u˙ n(7) (11) J = 3.3 ∇T·r∇actio−ns∇δ·Ω∇Dovn e−ac(7)δΩhDvface are inviscidf and tihusi GBS relaxes all the tangential component, the only stress acting on a face is the normal −∇ · δΩ ∇ · ∇ T−NδΩuD=v (27) Vacancy flux Vacancy flux −∇ · δΩ u˙ f SubstituteSuinbstitufor Ntev infromforeNquatifromon XXequatiandonlinXXearandize tlinheeearxpizeontehentialexpteonrmen(btialy atemermth(boydaI’mmecomnotthodpfuonenI’mlly cnotleart (fuon)pllylusclearitheon)assumptions that go into the clockwise trajectory through the derivation, i.e. starting Mass balance Substitute in for Nv from equativ on XX and linSuearbstituize theteeinxpforoneNntialv fromtermeq(buatiy aonmeXXthoandd I’mlinnotearizefullytheclearexponon)ential| ter|m (by a method I’m not fully clear on) Mass balance ∆P Ω Traction isUvthe+P ◦Ωvector describing how the stress acts on each face. With the assumption that grain boundaries Jv = Dv Nv ∆P Ω ∆P Ω U +P ◦Ω U +(9)P ◦Ω such that Cv = Cv◦(1 RTJv) and= CDv◦ v= expNv( RT ), andv get to ∆theP Ωresult U +P ◦Ω (9) such that Cvsuc=hCthatv◦(1 C−v = C) v◦and(1 Cv◦R=T e)xpand(sucC−v◦h=thatexp),Cand( =geRCTt (1to),theandresge)utltandto theC res= euxplt ( withv v),elandocitiesget toanthed assresultming continuity only in the normal component, then the value of interest is the integral − · ∇ − RT − −are in· ∇visc− J i=RdTanJdv−tvhus GBSv◦ relaxesRT allv◦the tangenRtiT al component, t(8)he only stress acting on a face is the normal J = Jv 3.3 T−ractions−on(8)each face − − RT of the magnitudes of all the normal components across the face, which is a scalar value and the definition of N and the definition of N P = RT u˙ RT (12) componenPt =( plus thP 2e=asu˙ sumptir onsu˙ that go into the clRocTkwise(12) trajec(12)tory through the derivation, i.e. starting Vacancy flux Uv + P Ω ∇U· ∇v + PNΩCv2Ω δDvr 2 r P = u˙ r (12) Vacancy flux ∇Tr·ac∇ tionNi∇sCv·theΩ∇ δDvectorv NCvΩdesδDcribv ing how ∇the· ∇stresNs CactsΩ2δonD each face. With the assumption that grain boundaries Nv = NCv = N exp Nv = NCv = Nwithexp velocities and ass(10)uming continuity only in the nor(10)vmal cvomponent, then the value of interest is thπeRinTtegral − RT Jv −= DRvT Nv (9) f f 4 J = DThis iNs the constitutive relation, relatinareg a instraviscin riated an(i.e.d tvhelouscitGBSy(9)dividedrelaxesby someallletnhegthtangen(δ)) to atipalrescomsure.ponent, the only stress actin∆Fg on= a face is theu˙normal(af ) (28) v ! Thisv is thevconThis"stitutis tivhee crelationonstitut, ivrelatinofe relationthg!eamagnitstraThen, relatin−in rsolvateudesg· ea∇(i.e.thstra"isofvinwitheloalratelcitthesphy(i.e.divericalnidedormalveloharmoncitbyysomecdivompicidedls,enonengthby some(tsδ))acrosstolenagthpresth(surδ))eetof.acae,preswsurhiceh. is a scalar value r −???????· ∇ − 8DδΩ for pressure driven diffusion. Comforbininpresg thesurefirstdrivthrenedie ffgiuv???seios????n. Com???binin????g the first three givcomes anpdonenthe solut (tionplusis writtenthe assumptiin termsonsof conthtacatt patgo cihesnto. the clockwise trajectory through the derivat!ion, i.e." starting and the definition of N and the definition of N This is theUcvon+stitutP Ωive relation, relating a straPhinysicratealπ(i.e.lyR,Tvthieloscit∆y Fdividedis theby somediffelreenngthce (bδ))ettoweena presthesure.integral of the normal components and the melt pressure, Uv + P Ω with velocities and assuming continuity fonly in the normalf comp4 onent, then the value of interest is the integral u˙ rIn the 2-D case, this reduNcves=to.N. Cv =u˙Nr exp ∆F = u˙ r ((10)af ) (28) Nv = NNCv v== N expIn the 2-D caseIn th, the is2-Dreducaseces, thto.Nis. redu= ces to.???. ???? − (11)RT (10) 8(11)DδΩ − RT v of the magnit! udes of all"the normal comp∆onenF−=tsPacrossPL.thSeee facPe,agewh4/6ich ofis Ya Tscalar’s Notesvalu. e ∇ · ∇ −δΩDv! "∇ · ∇ −δΩDv ! − " for pressure drivenIndithffuescioasen.withCommelt,bininthegsetheequatifirstonsthrdo eneotgichavenge,s only the diffusion lengths do, asOrI u,ndinerthestandnorit matalized version, for pressure driven diffusion. Combining the first threeIngithveescase withIn themelt,casethwithese equatimelt,Phthonsysicesedoalequatinlyot,Inconsthihathnge,sedo2-D∆nonlyFotcaseischath,thenge,ethdiisffdonlyreduuisffionecrtheselengthneto.cdie.ffbussetdo,ionweenaslengthI uthends do,erinstanastegraldI itundatπeofRrstanTthed itnormalat components and the melt pressure, Substitute in for Nv from equationSuXXbstituandteliinnearizefor NvthferomexptheonenqeuatimomtionalentteXX(andrmandpbossy ilintheblyetharappeizebounrotximationhedareyxconponditie(bnonstialy adrivtemeingrmthoth(bde ydiffaumesionth). od I’m not fully cflear on) f 4 ∆F f = u˙ f (a )4 (29) the momentth(ande momposseniblyt (andthepbossouniblydar∆thyFcone=bounditiP daronsPydconriv. ingSeeditithonsPeagedidrivffuing4/6sionth).ofe diYffTus’ions Notes). . ∆F = u˙ r (af ) r f (28) ∆P Ω ∆P Ω Uv +P ◦Ω U +P ◦Ω L u˙ r − 8DδΩ − I’m not fully clear on) such that C = C◦(1 ) and C◦ = exp u˙ r( ), andv get to the res−ult sucv h thatv Cv =RTCv◦(1 RvT ) and Cv◦ =RTexp ( RT ), and NgevtIn=tothethecasereswithultmelt, these equations do not change, only! the di"(11)ffusion lengths do, as I understand it at − − Nv2.3= Geom− etry − Or∇, in· ∇the nor−maliδΩzDedv version,(11) Plot this value as the color value on each face, using the same scale as present... ∇ · ∇2.3 Geom−δΩ2D.3etryv Geometry Phthysice momallyen, tthi(ands ∆pFossiisblythethe dbouniffedarrenyccone bditietwonseendfrivtheing inthftegrale diffu4sionof).the normal components and the melt pressure, RT2.3.1 The relation between conRtiguitT y and melt fraction ∆F = u˙ r (af ) (29) SubstituP =te in for2.3.N1 fTheurom˙ r2.3.relate1quatiionTheonbrelatetXXwPeenandion= cblinonettewiguitareenizeyctonandheu˙teiguitxmpeltony(12)eandfractionntialmteeltrmfraction(by a method I’m not (12)−fully clear on) Substitute in for N from equation XX and linearize t2he vexponential term (by a metho∆d2FI’m= Pnotr fuPLlly. SeeclearPageon)4/6 of YT’s Notes. v ∇ · ∇ NCvΩ δDv ∇ · ∇ NCvΩ δDv − 3.4 What do we vary? When surface∆P Ωtension is satisfied and Uavs+tati2P.3◦cΩequGeomilibriumetrymelt configuration is reached, contiguity can be ∆P Ω such that UCvv+P=◦ΩCv◦(1 When)surandfaceCtev◦n=sionexpis satisfied( Plotandthis),a vsandaltatiuecgeeasqut theilitobritheumcolorresmeltvualultconfie ongurationeach isfacree,ached,usingconthtiguie samty cane scalebe as present... such that Cv = C◦(1 ) and C◦ = exp ( Whe),n surandfacegeRteTtntosiontheis satisfiedresult and a statiOrRTc, inequtheilibrinorum mmeltalizconfied vgurersion,ation is reached, contiguity can be v RT v RT calculated− in terms of melt fraction−(ε(φ)) and2.3.vice1 vTheersa. relation between contiguity and melt fraction Then solve this with spherical− harmonThisics, is the con−stitutivcalce urelationlated incalcterms, relatinulatedof melting atermsstrafractioofinnmeltr(ateε(φfr))(i.e.acandtionvvice(eloε(φvcit))erandsya.divviceidedversba.y some length (δ)) tocona ptigures∆Fsurityf e(isotrop=. u˙ f (aic))4 (29) RT r f and the solution is written in terms of contact patches. RT 3.4 What do we vn1ary? − ??????? P Wh= en nsur1 face tensionun˙φ1r is satisfiedn1 and a staticconequtiguilibriitumy (animeltsconfiotr(12)opic)guration is reached, contiguity can be P = u˙ n1 2 nφ1 (12) φ 2 r ∇c(1·c∇(1 ϕ)ϕ=)Plφ=Ncot(1φCtvϕhisϕΩ=)ϕ=δ=vDalφ vue asϕ =1the1 color v1alue on each face, using(13)(13)the sam(13)e scale as present... This is the constitutive relation, relating a stra∇in ·r∇ate (i.e.NvCelovΩcitδyDdivv ided by some length− −(δcalc)) toulateda−presin −termssur− ec.ofc−melt−− fracctio−n (ε(φ)) and vicegeomveetryrsa (Tofakei,strai. n contiguity (isotropic!) ! " !" " n1 ??????? In the 2-DThiscasei,sththeis creduwhereonstcitutesc isto.ivsome. relatione constan,t relatinthat depgenadsstraon thein rwateetting(i.e.angleveloθwcitandy divn1 ided0.5 fboryθsomew < 60l◦e.nHegthre,(cδ=))0to.54.aFφporressure. This is the constitutive relation, relating a strainwhererate c(i.e.is somwhereveloe constancitc isysomdivt thateidedconstandepbenytdssomethaton thedeplenenwgthedsttingon(δtheangle)) toweθttawingpandresanglensur1 eθ.w0.5andfornθ1w <060.5◦f.orHeθwre,

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