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9. Microphysics of Mantle Rheology

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9. Microphysics of Mantle Rheology 9. Microphysics of Mantle Rheology Ge 163 5/7/14 Outline • Introduction • Vacancies and crystal imperfections • Diffusion Creep • Dislocation creep • Deformation maps • Viscosity in the Deep Mantle • Visco-elastic relaxation and thickness of the lithosphere • Role of Water in influencing creep strength • Appendix A • Details of the theory of Diffusion creep Why does elastic thickness (h, Te) increases with age ? Watts [2001] Diffusion Creep Diffusion creep is the migration or diffusion of point defects through the crystal lattice. An example of a point defect is a vacancy, where an atom is missing from the normal highly regular crystal lattice. Another is an impurity, where some species (of a minor concentration) exists or is located within the regular lattice. So the diffusion of a minor species becomes: b 2 1/2 φo ⎛ φo ⎞ −2φo /kT D = 2 ⎜ ⎟ e 3π kT ⎝ m ⎠ In general, the diffusion coefficient is a function of both temeprature and pressure and is expressed in the form: ⎛ E + pV ⎞ D = D exp − a a o ⎝⎜ RT ⎠⎟ where R is the gas constant, Ea is the activation energy, and Va is the activation volume. The pVa term takes into account of the effect of pressure in reducing the number of vacanies and increasing the potential energy barrier between lattice sites. Diffusion of atoms in a crystal in the presence of differential stress can result in creep. compression Poirier [1991] 12V D ε = a σ xx RTh2 This is "Diffusion Creep" (also known as Nabarro-Herring creep) which resuslts in a linear relation between strain-rate and stress. Recall that we have previously described a "constitutive relation" for a fluid as: σ = 2ηεxx ∴ the "viscosity" for a crystalline solid is: RTh2 η = 24VaD But substituting the relation that we have already derived for the D, the Diffusion 2 RTh ⎛ Ea + pVa ⎞ η = exp⎜ ⎟ 24VaDo ⎝ RT ⎠ In Nabarro-Herring creep, the atoms and vacanies diffuse through the interior of mineral grains. However, if the diffusion coefficient for grain boundary diffusion, Db, is sufficiently large, then 3 RTh ⎛ Ea + pVa ⎞ η = exp⎜ ⎟ 24Vaδ Dbo ⎝ RT ⎠ where δ is the width of the grain boundary. This is called "Coble creep" and the magnitude δ D of the ratio b determines where grain boundary hD diffusion cominates. Notice that the dependence between stress and strain rate is not linear. What is going on because this relation does not seem to be well predicted by: 12V D ε = a σ xx RTh2 Dislocations and Dislocation Creep Dislocations are imperfections in the crystalline lattice structure. The two basic types are edge and screw dislocations. An edge dislocation is essentially an extra plane of atoms in part of the crystal. The dislocation deforms the crystalline lattice and produces stresses. Burgers Vector b* Dislocation climb Dislocation glide Using the relation that we developed before from dislocation creep. If hd is the mean spacings between dislocations, then it can be shown that 12VaD εxx = 2 σ RThd Now, empirically, it is known that the number of imperfections in the lattice, or dislocations, increases with stress, σ . Under a wide raneg of conditions: b*G h = d σ where b* is the "Burgers vector" for the dislocations and G is the shear modulus. 12V D ∴ ε = a σ 3 xx RTb*2G2 This is a nonlinear, or a non-Newtonian relation between stress and strain. Nevertheless, an "effective viscosity" can be defined: σ RTb*2G2 1 ηeff = = 2 2εxx 24VaD σ Using the diffusion coefficient: *2 2 RTb G 1 ⎛ Ea + pVa ⎞ ηeff = 2 exp⎜ ⎟ 24VaDo σ ⎝ RT ⎠ This 3 −Ea / RT εxx = −εyy = C1σ e now seems to fit the experimental data on Olivine with 5 -3 -1 C1 = 4.2 × 10 MPa s E = 523 kJ mol-1 a Experiments and theory suggest a general form of the relationship between strain rate ε and deviatoric stress σ [Karato and Wu, 1993]: n m ⎛ σ ⎞ ⎛ b⎞ ⎛ E + pV ⎞ ε = −ε = A exp − a a xx yy ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ ⎝⎜ ⎠⎟ G h RT Dislocation creep or diffusion creep within the upper mantle? Use a 'deformation map' to delineate different mechanisms for deformation. A deformation map gives stress as a function of temperature for several strain rates. This map (using equation from Karato and Wu [1993]) uses a constraint grain size (h = 3 mm) at p = 0. A point relevant for the upper mantle is shown: ε ~ 5 cm yr-1 / 700 km = -15 -1 2 × 10 s and T = 1600 K. Predicted Viscosities Viscosity of the Deep Mantle Ranalli [1987] Ranalli [1987] Viscoelasticity A material that behaves elastically on short-time scales and viscously on long time-scales A Maxwell viscoelastic material is that the rate of strain, ε, is the superposition of a linear elastic strain rate, εe ,and a linear viscous strain rate, ε f ε = εe + ε f A Kelvin viscoelastic material is a superposition linear elastic and linear viscous stresses, σ + σ e f For a Maxwell body ε = εe + ε f For a maetrial subject to uni-axial stress (only one principal stress is non-zero) dε 1 1 dσ = σ + dt 2η E dt Suddenly, a strain εo is applied at t = 0 σ o = Eεo t = 0 dε / dt = 0 t > 0 (i.e. subsequently, no change in strain) dε 1 1 dσ = σ + = 0 dt 2η E dt dσ E = − dt σ =σ t = 0 σ 2η o ⎛ Et ⎞ σ = σ exp − o ⎝⎜ 2η⎠⎟ 2η τ M = ≡ Maxwell Time E 2η τ = M E η = 4 × 1019 Pa s E = 70 GPa τ M = 36 years Why does elastic thickness (h, Te) increases with age ? Because the stresses relax in the lower lithosphere. But the Thermal thickness increases. Watts [2001] The role of OH (water) in effecting mantle strength and viscosity From Greg Hirth Appendix A • Details of the theory of Diffusion creep Diffusion Creep Diffusion creep is the migration or diffusion of point defects through the crystal lattice. An example of a point defect is a vacancy, where an atom is missing from the normal highly regular crystal lattice. Another is an impurity, where some species (of a minor concentration) exists or is located within the regular lattice. Detailed analysis will give the number of atoms per unit volume, dnφ , with kinetic energies between φ and φ + dφ (This is the Maxwell-Boltzman distribution of energy). 2πnφ1/2 ⎛ −φ ⎞ dnφ = 3/2 exp⎜ ⎟ dφ k is Boltzmann's constant (πkT ) ⎝ kT ⎠ If the kinetic energy exceeds the potential energy barrier φo confinning it to its lattice position, it is able to escape the site and move elsewhere in the lattice. The number of atoms per unit volume in a crystal that have kinetic energies greater than , n : φo φo ∞ n dn φo = φ ∫φo after some work 1/2 ⎛ φ ⎞ /kT ⇒ n = 2n o e−φo φo ⎜ ⎟ ⎝ πkT ⎠ ∴ Fraction of high energy atoms is a very sensitive function of temperature. Since at any finite temperature some of the atoms have enough energy to escape the lattice, some lattice positions will be empty. These are called vacancies. A vacancy is a point defect. Assume number of vacancies per unit volume, nv , in a crystal lattice is equal to the number of atoms that have sufficient energy to overcome the potential barrier n n v = φo If an atom jumps into an adjacent vacancy site, then the vacancy has effectively jumped in the oppsite direction. Γv,+ x = frequency which a vacancy jumps into an adjacent lattice site in the + x direction is the probability that the atom has the right energy (n / n) and the frequency which the atom moves φo the vacancy (1 / 6τ ). n Γ = φo v,+ x 6τn The net rate of flow (flux) of vacancies in the x direction can be shown to be: dn J = −D v v,x v dx This is known as "Fick's first law of diffusion". b 2 1/2 φo ⎛ ⎞ −φo /kT Dv = ⎜ ⎟ e (m is the mass of the atom). 6π ⎝ πmkT ⎠ But this only describes the diffusion of vacancies, not how some minor species, A, moves through the lattice. This diffusion is much smaller and effectived the product of the diffusion of vacancies times the probability that a minor species will be adjacent to a vacancy. So the diffusion of a minor species becomes: b 2 1/2 φo ⎛ φo ⎞ −2φo /kT D = 2 ⎜ ⎟ e 3π kT ⎝ m ⎠ In general, the diffusion coefficient is a function of both temeprature and pressure and is expressed in the form: ⎛ E + pV ⎞ D = D exp − a a o ⎝⎜ RT ⎠⎟ where R is the gas constant, Ea is the activation energy, and Va is the activation volume. The pVa term takes into account of the effect of pressure in reducing the number of vacanies and increasing the potential energy barrier between lattice sites. Diffusion of atoms in a crystal in the presence of differential stress can result in creep. compression Poirier [1991] A cube of dimensions h is subjected to a compressional stress σ xx =σ in the x-direction and a tensional stress σ yy = −σ in the y − direction. There is no σ zz . Under this stress, atoms diffuse from faces A & C to faces B & D: When a layer of atoms has been removed from faces A & C and added to faces B & D, the strain in the x − direction & y − direction are, respectively: 2b 2b ε = ε = − xx h yy h Because of the application of a compressional stress on face A, the number density of vacancies is decreased from nv to nvA as they added to the face). Similarly, the application of of a tensional stress on face B increases the number density of vacancies from nv to nvB It takes more work to extract an atom from the lattice and bring it to a crystal face under compression, ∴ the number density of vacancies is higher there [less work for a face under tension → higher density] Consequently, there is a flux of vacancies (per unit area and time) from face B to face A, etc: Dv 2 Jv,AB = Jv,AD = Jv,CB = JCD = (nvB − nvA ) h h / 2 is the mean distance between adjacent faces of the cube.
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