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)