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Mineralogy of Mantle and Core: Thermodynamics, Linear Elasticity and Equations of State Daniele Antonangeli ([email protected])

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Mineralogy of Mantle and Core: Thermodynamics, Linear Elasticity and Equations of State Daniele Antonangeli (Daniele.Antonangeli@Impmc.Upmc.Fr) Mineralogy of mantle and core: Thermodynamics, linear elasticity and equations of state Daniele Antonangeli ([email protected]) Ins$tut de Minéralogie, de Physique des Matériaux, et de Cosmochimie (IMPMC) UMR CNRS 7590 Sorbonne Universités – UPMC Univ. Paris 6 Muséum Naonal d’Histoire Naturelle 4 Place Jussieu 75252 Paris cedex 05 Today’s plan - Theore:cal consideraons on materials proper:es at HP-HT - Linear elas:city and equaons of state Today’s plan - Theore:cal consideraons on materials proper:es at HP-HT - Linear elas:city and equaons of state Pressure and interatomic potenal Atomic packing at HP HP à coordinaon number increases to maximize density at a given interatomic distance If atoms can be considered as spheres, ul:mate packing are “close packed” (fcc-hcp) At very high pressure electronic delocalizaon, atoms cannot be treated as sphere, ul:mate packing by most stable Coulomb packing à bcc P-induced phase transi4ons (T=0K) internal energy enthalpy T-induced phase transions (at constant P) Liquids have higher entropy than crystals SoLer crystals have higher entropy than harder crystals S2-S1 ~kBlog(ω1/ω2) P-T phase diagrams (single constuent) P-T phase diagrams: GiBBs phase rule ν = C + 2−ϕ degrees of freedom i.e. number of independent number of phases intensive thermodynamic variables number of components Consequences (for single component system, C=1): For a single phase we can independently vary P-T At phase boundary (2 phases) P and T are not independent any more Triple point (3 phase) fix in P-T, no degrees of freedom Four phases are not allowed, no such a thing as a quadruple point Phonons at high pressure Exceptions: Phonons get stiffer... Phonons can “soften” due to, e.g., anomalous interaction with the electrons (Kohn anomalies, Charge density waves, etc) or due to packing effects. ...as vibrations are probing Softening can affect a the repulsive wall of the small area of the phonon interatomic potential BZ, or an entire phonon branch (elastic instability) high P low P Phonons at high pressure: reconstrucve phase transions Unstable phonon P > Pc P = Pc Mechanical instability following P < Pc thermodynamic instability à Hysteresis 1 2 à Pressure-induced amorphizaon Electrons at high pressure kine:c term Coulomb term At low P (large l) the Coulomb term dominates At high P (small l) the quantum kinetic energy term dominates HP à electron delocalizaon magne:sm, electron localizaon, correlaon disappear at HP metallic state becomes increasingly favored at HP Electrons at high pressure: electron-driven transions Isolated atom Crystal field Pressure induced HS-LS transitions Concluding remarks Classical thermodynamics is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts Albert Einstein - Bear in mind thermodynamics - Have the sense of the physics of the system/ process you are looking at Today’s plan - Theore:cal consideraons on materials proper:es at HP-HT - Linear elas:city and equaons of state Compression curve and elascity P-V relaons à Bonding à How bonding change with pressure N.B. elas:c behavior ≠ plas:c Elasc behavior Hooke’s law: F= -kx more convenient in terms of strain ε=x/l0 à F= -kl0ε and strain σ=F/A à σ=(-kl0/A) ε = cε ßà ε= sσ sffness compliance (or moduli) Linear infinitesimal elascity à 3 dimensions σ11, σ22, σ33 are normal stresses sings are negave for compression σ12, σ13, σ21… are shear stresses N.B. if shear stresses, not hydrostac compression Elascity in 3 dimensions Generalized Hooke’s law σij = Cijkl εkl elas:c moduli 3 for cubic symmetry: C , C et C axial compression 11 12 44 σ11 à ε11 (along same axis) shear σ23 à ε23 (within the face) dilataon upon compression σ11 à ε22 (over perpendicular face) If all independent element of the elas4c tensor are known the complete elascity of the system can be determined Elasc moduli and sound velocies Christoffel equaons à Symmetry helps… Cubic symmetry à … Single crystal moduli and aggregate proper4es Proper:es of the aggregate from Cij aggregate ≠ mean value (texture, grain-grain interac:ons, …) Upper bound: Voigt average (uniform strain) Lower bound: Reuss average (uniform stress) Voigt-Reuss-Hill (average) For random orienta4on and cuBic symmetry Bulk modulus: k= 1/3*C11+2/3*C12 Shear modulus: G= 1/5*(C11)-1/5*C12+1/5*(2*C44+1/2(C11-C12)) 2 Compressional velocity: ρ VP = k+4/3G 2 Shear velocity: ρ VS = G Most general case rank 2 tensor rank 4 tensor rank 2 tensor transverse isotropic isotropic Tensors are invariant with respect to point group symmetry - no translaons involved ! - only rotaons (new) (old ) Tijk... = ∑ Ria RjbRkc...Tabc... a,b,c,... Linear infinitesimal elascity and hydrostac pressure σ1=σ2= σ3= - ΔP σ4= σ5=σ6=0 Volume varia4on, Bulk modulus and Cij For all symmetries but mono or triclinic: δV/V≈ ε1+ε2+ε3 -1 Bulk modulus k=-V δP/δV = -δP/(ε1+ε2+ε3 ) = (s11+s22+s33+s12+s13+s23) N.B. true only for hydrostac compression Reuss bound Volume under high pressure -1 k=-V δP/δV = -δP/(ε1+ε2+ε3 ) = (s11+s22+s33+s12+s13+s23) Linear elascity: δV/δP=-k/V Equaon of state formulaons EOS defines the elas:c relaon between volume and intensive thermodynamic variables V=f(P,T, B, χ…) Isothermal EOS à V=f(P) Some thermodynamic formulaons k=-V δP/δV BUT NO EXACT THERMODYNAMIC FORMULATIONS FOR THESE DERIVATIVES! k’= δK/δP à ASSUMPTIONS k”= δ2K/δP2 EOS: why not polynomial? EOS: from interatomic potenal? Interatomic poten:al à E=f(R) à E=f(V) P=-δE/δV But which interatomic poten:al? Not universal one EOS: Murnaghan Assump:on: k linear with P Advantages: can be inverted, integrated great for parametric fing Disadvantages: k”=0, does not fit data for V/V0 > 0.9 EOS: finite strain development Assump:on: strain energy Ψ is a polynomial in strain f Ψ = af2+bf3+cf4+… P=-δE/δV --> P=-δΨ/δV = -δΨ/δf δf/δV = -δf/δV (2af+3bf3+4f3+…) Infinitesimal linear strain ε=(l-l0)/l0 à εV = (V-V0)/V0 = V/V0 – 1 (Lagrangian) But empirically observed that Eulerian strains work the best… 2/3 fE= ½ [(V/V0) -1] EOS: 2nd order Birch-Murnaghan 2/3 finite strain EOS, uses Eulerian strains fE= ½ [(V/V0) -1] EOS: 2nd order Birch-Murnaghan EOS: 2nd order Birch-Murnaghan EOS: 4th order Birch-Murnaghan Advantages: fits data for V/V0 =0.8 provides correct values for k0 Disadvantages: cannot be inverted PδV must be numerical Problemac for thermodynamic database and parametric fing Birch-Murnaghan EOS, where to stop? F-f plot: normalized stress vs. finite strain BM EOS polinomial in fE F-f plot F-f plot Examples Other EOS Vinet Mie – Grueneisen High temperature For isotherms 298 K à V0, k0, K’, … T (K) à KT0, K’T0, kT0’, … Diamond thermal expansion Diamond thermal soLening P-V-T EOS Thermal expansion: simple approach constant at all P Failures of simple approach Fo92 olivine extrapolaon… P-V-T EOS: thermal pressure P-V-T EOS: thermal pressure S:ll need to assume how αkT changes along an isochor Quasi-harmonic approximaon: αkT = constant P-V-T EOS: thermal pressure Fo92 olivine Concluding remarks - Do things carefully - Be cri:cal towards your experiments/calculaons - Never forget the assump:ons/approximaons .
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