Diffusion Creep Poirier, Chapter 2 and 7, 1985. Gordon, 1985. Monday, Nov. 3, 2003 12.524 Mechanical properties of rocks Fick’s First Law: Driving Force ∇c z Chemical Diffusion ∇T z Thermal diffusion JD=−i ∇µ ∇V z Electrical conduction σ z Diffusion creep 12.524 Mechanical properties of rocks Types of Diffusion Mechanism Path Process Isotope Lattice Interdiffusion Self-diffusion Pipe Creep Vacancy Grain Boundary Ambipolar Interstitial Surface Ring Pore fluid 12.524 Mechanical properties of rocks Diffusion mechanisms Vacancy Diffusion Interstitial Ring Diffusion Collinear Direct Exchange Non -collinear Cyclic Exchange A B 12.524 Mechanical properties of rocks Vacancy–Assisted Diffusion From Site by Glicksman and Lupulescu, RPI, 2003 The FCC lattice geometry requires W = ( 3−1)Da = 0.73Da Images removed due to copyright considerations. For more information, see http://www.rpi.edu/~glickm/diffusion/ 12.524 Mechanical properties of rocks plan view Kinetics Equation for Vacancy Diffusion z Coefficient of Diffusivity for Self-diffusion not the same as Coefficient for Vacancy Diffusion DNDsd= vi v migration =−Nvoexp (∆∆ G vf / kT )i D v o exp − ( G vm / kT ) =−+DGGkTsd oexp (∆∆ vf vm )/ 12.524 Mechanical properties of rocks Diffusion Creep in Monatomic Solid z Basic Ideas Vacancies Supersaturated z Supersaturation of vacancies owing to stress z Diffusion results z Work done on mat’l Vacancies UndersaturatedVacancies by tractions z Energy dissipated in heat, entropy, and surface area 12.524 Mechanical properties of rocks Diffusion Creep z z Nabarro-Herring Creep Monatomic Lattice z Quasi static z Coble Creep z Vacancy Grain Boundary z Increasing length; Poisoining 12.524 Mechanical properties of rocks Critical Idea: Tension makes vacancy formation easier. z Tension=supersaturation ∆=∆−ΩGGσσ0 ffvv( ) ( ) ∆G 0 fv () Atom CC=−iexp vo kT ∆G 0 −Ωσ l fv () d CCv ()σ =−exp kT 12.524 Mechanical properties of rocks Gradient in Composition z Path Length: z Boundary: 2xd/4 z Lattice: (π/2)x(d/4) z Concentration Difference ∆G −Ωσ C exp− (fv ) o kT ∆+ΩG σ −−C exp (fv ) o kT ∆G fv σσΩ −Ω Co exp−− ( ) exp( ) exp( ) kT kT kT ∆G σΩ ∆=CC2exp( − fv )i o kT kT z Quasi-static Approx. 12.524 Mechanical properties of rocks Fick’s 1st Law ∆C J path=−Di path .) ∆L path Total flux =ΣFlux on each Path: d Φ=⋅⋅+⋅⋅JlJlδ δ d/2 vac flux L2 B Φ 2δ J total ave..)flux==+vac flux J J ii totaldl/2⋅ L d B Plugging ∆C and ∆L into i.) and inserting fluxes in ii.): ∆G fv σΩ 2exp(Co − )i ∆G fv σδΩ 12 kT kT JDC=−2exp( − ) ⋅ +() − D total L okT kTπ d82 d b d 12.524 Mechanical properties of rocks Total Flux on Both Paths z 82∆GGvfσδΩ 2DCBvo∆ vf σΩ JDCTotal=−+ L vo exp exp − 2 π dktkTddktkT G 16 ∆ vf σπδΩ DB =−DCLvoexp 1 + π dktkTdD2 L 12.524 Mechanical properties of rocks Converting flux to strain b z Each vacancy that travels through b the channel adds layer of depth b 1 2 ∆lvacsb# b2J Ω ==⋅⋅⋅=2 Jb2 total lt⋅ s dtotal d d ε12 z 32 σπδΩ DB ε22 ε11 ε =+2 DL 1 π dkT2 dDV 12.524 Mechanical properties of rocks Summary: N-H and Coble Creep z Diffusion Creep Constitutive Law: 32 σπδΩ D B ε =+2 DL 1 π dkT2 dDV z DDCL = VM V z Strain rate linear in stress z ε ∝ 1 d 2,3 z Other geometries change initial constant 12.524 Mechanical properties of rocks.