Transport Properties Lars Stixrude University of Michigan 7/12/04
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Transport Properties Lars Stixrude University of Michigan 7/12/04 CIDER/ITP Short Course Rheology: Overview Strain Yielding Creep Slope= Elastic Apply stress>strength Ý Recoverable Strain Unrecoverable Strain Time Release stress Maxwell Relaxation Time M G Maxwell Relaxation Time Viscosity (T) 1.00E+20 Viscosity Shear modulus Age of the Universe creep regime 1 byr crust (500°K) Time (s) For =1021 Pa s G=100 GPa M=1010 s=300 years 1.00E+15 1 myr 1.00E+10 >> M 1000 years Viscosity a strong function of temperature mantle (1500°K) 1 year 1.00E+05 0 500 1000 1500 Temperature (°K) 2000 2500 3000 Strength Topography supported by rock strength Stresses in mantle must be sufficient to overcome strength Ý For viscosity of 10 21 Pa s, strain rate of 10-15 s-1 Stress=1 MPa, or about 10-5G Simple analysis of strength t G Strain 3 •Maximum stress that a crystal can support •Ideal crystal 3 •Shear strain applied at a 2 uniform rate •Stress will vary •Periodic function of time •Zero when crystal structure is ideal •Maximum (strength) when planes are displaced by half an interatomic spacing a G a /2 G t G 3a /2 3 Observed Strength Much less than predicted Why? Crystal not perfect! Defects! Cation vacancy negatively charged Point Defects: Schottky Anion vacancy positively charged Cation vacancy negatively charged Cation interstitial positively charged Point Defects: Frenkel Point defects: thermodynamics Change in Gibbs free energy with respect to perfect crystal G f H f TS f Small finite concentration of defects lowers Gibbs free energy Defect concentration should increase with increasing temperature Hf Energy Defects are •Energetically unfavorable •Entropically favorable Gf -TSf Defect Concentration Defect Motion Activated process Energy barrier Height = EA Jump frequency Energy E A exp kT Activation Energy EA Distance i.e. motion is exponentially faster at high T Diffusion and Viscosity G f E A D ~ exp kT Depth (km) d kT DV d = grain size V = volume/atom Log(viscosity) Pa s 2 i.e. viscosity decreases exponentially with temperature Infer grain size in lower mantle from geophysical determination of viscosity? Dislocations Dislocations: Structure Putnis (1993) Dislocations: Motion Dislocations: Motion Dislocations move more easily in some crystallographic directions than others These are the easy slip planes To minimize stress, easy slip planes tend to align with the plane of shear (plane of flow) Consider a deforming polycrystal Crystals tend to rotate such that their easy slip planes align with shear plane Lattice preferred orientation Preferred orientation Lattice Shape Quartzo-Feldspathic Gneiss, Malton Range, BC (C. A. Giovanella) Easy glide plane Plane of lowest critical resolved shear stress Relevant factors: Spacing normal to plane (bond strength) Spacing within plane (distance dislocation must move for lattice to return to registry burgers vector) Elastic constants b axis length is greatest: easy glide plane a axis length is shortest: easy glide direction a axes (fast) align with flow b axes (slow) normal to flow Olivine Polarization Anisotropy •VSV<VSH •Explain by olivine •If olivine b axis aligned horizontally •Horizontally polarized S-waves faster than vertically polarized Ekstrom and Dziewonski (1999) Nature Upper Mantle Azimuthal Anisotropy Fast Direction ~ Flow Direction Tanimoto and Anderson (1984) Shape preferred orientation Slow Fast Slow P-waves Fast
