Mineralogy of Mantle and Core: Thermodynamics, Linear Elasticity and Equations of State Daniele Antonangeli (Daniele.Antonangeli@Impmc.Upmc.Fr)

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Mineralogy of mantle and core: Thermodynamics, linear elasticity and equations of state Daniele Antonangeli ([email protected])

Instut 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

- Theorecal consideraons on materials properes at HP-HT

- Linear elascity and equaons of state Today’s plan

- Theorecal consideraons on materials properes at HP-HT

- Linear elascity 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, ulmate packing are “close packed” (fcc-hcp)

At very high pressure electronic delocalizaon, atoms cannot be treated as sphere, ulmate packing by most stable Coulomb packing à bcc P-induced phase transions (T=0K) internal energy

enthalpy T-induced phase transions (at constant P)

Liquids have higher entropy than crystals

Soer 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

kinec 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 magnesm, 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

- Theorecal consideraons on materials properes at HP-HT

- Linear elascity and equaons of state Compression curve and elascity

P-V relaons à Bonding à How bonding change with pressure

N.B. elasc behavior ≠ plasc 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

elasc 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 elasc 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 properes

Properes of the aggregate from Cij aggregate ≠ mean value (texture, grain-grain interacons, …)

Upper bound: Voigt average (uniform strain)

Lower bound: Reuss average (uniform stress)

Voigt-Reuss-Hill (average) For random orientaon 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 variaon, 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 elasc 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 potenal à E=f(R) à E=f(V)

P=-δE/δV

But which interatomic potenal? Not universal one EOS: Murnaghan

Assumpon: 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

Assumpon: 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 soening 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

Sll 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 crical towards your experiments/calculaons

- Never forget the assumpons/approximaons