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 D = 0.73D ( ) a a
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
∆=∆−ΩGG(σσ) (0) 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 σΩ 2exp(C − fv )i ∆G σδΩ 12 o JDC=−2exp( −fv ) ⋅ +() − D kT kT total L okT kTπ d82 d b d
12.524 Mechanical properties of rocks Total Flux on Both Paths
z
82∆GGvfσδΩ σ2DCBvo∆ vf Ω 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