66
“Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

“Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Embed Size (px)

Citation preview

Page 1: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

“Phase diagrams are the beginning of wisdom…”-- William Hume-Rothery OBE

Page 2: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Geophysical inversion for mantle composition and temperature

“It is unworthy of great (wo)men to lose hours like slaves in the labor of calculation.”

-- Baron Gottfried Wilhelm von Leibniz

A fast, robust method for the calculation of chase equilibria (Perple_X)

Thoughts on the use and abuse of phase equilibria in geodynamic models

Page 3: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

The Non-Linear Phase Equilibrium Problem

The stable state of a system minimizes its Gibbs Energy (G)

Page 4: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

The Non-Linear Phase Equilibrium Problem

The stable state of a system minimizes its Gibbs Energy (G)

Brown & Skinner 1974, Saxena & Eriksson 1983, Wood & Holloway 1984, deCapitani & Brown 1987, Bina 1998, etc. etc.

Page 5: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

The Non-Linear Phase Equilibrium Problem

The stable state of a system minimizes its Gibbs Energy (G)

Brown & Skinner 1974, Saxena & Eriksson 1983, Wood & Holloway 1984, deCapitani & Brown 1987, Bina 1998, etc. etc.

Page 6: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

The Linear Phase Equilibrium Problem

A B

Page 7: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

The Linearized Phase Equilibrium Problem: “Pseudocompound” Approximation

White et al. 1958, Connolly & Kerrick 1987

Pseudocompounds

Page 8: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

A Problem with Pseudocompounds

The number of pseudocompounds for a solution in c components at cartesian spacing δis:

1 1

22 1

1δ 1 1δ 1 1 !1Π 1

2 1 ! ! 1 !

i ic c

i c i

cc

i i c i

Garnet – c = 4, δ = 1 mol % → Π = 2∙10 5 Melt – c = 8, δ = 1 mol % → Π = 2∙10 10

A Solution: Iterative Refinement

Page 9: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Conclusions for Part I

But a monkey could do that…

Page 10: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Equation of State, Stixrude & Bukowinski 1990

0 0 0, , , ,c th thA V T A A V T A V T A V T Gruneisen model for Helmoltz Energy:

Birch-Murgnahan “cold” part:

0 0 0

0

2 3

2 3

94

21

12

cA KV f K f

f V V

Debeye “thermal” part:

θ3 2

0

0 0 0

9 θ ln 1 d

θ θ exp 1

Tt

th

q

A nRT T e t t

V V q

Seven parameters (0 0 0 0 0 0, , , , ,θ ,A K K V q ) + 3 parameters for seismic velocities

Data from Stixrude & Lithgow-Bertelloni (2005) augmented by

•Post-perovskite from Oganov & Ono (2004), Ono & Oganov (2005)•Ca-perovskite from Akaogi et al. (2004), Karki & Crain (1998)•Wuestite, perovskite from Fabrichnaya (1999), Irifune (1994)

θ3 2

0

0 0 0

9 θ ln 1 d

θ θ exp 1

Tt

th

q

A nRT T e t t

V V q

Page 11: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

10

20

30

1100 1300 1500 1700 1900T( C) °

P(G

Pa

)

2900270

0

2500

2300

210

0 pvwuscpv

ocpxgt

wadgt

wuswadcpxgto

cpxgt

c2c

ocpxgt

opx

ocpxsp

opx

ocpxpl

opx

pvgt

wuscpv

pvgt

rngcpv

gtrngcpvwus

akigt

rngcpv

wadrnggt

rnggt

cpv

gtwad

290

0

270

0

2500

isentropes (J/K/kg)

2100 2300 2500 2700 2900T( C) °

120

130

140

P(G

Pa

)pv

wuscpv

ppvwuscpv

Computed Pyrolite (CaO-FeO-MgO-Al2O3-SiO2) Phase Relations

Page 12: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Pyrolite P-wave velocity

“Phase diagrams are the beginning of wisdom not the end of it.”-- William Hume-Rothery

Page 13: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Part II: the beginning of wisdom?

Putting phase equilibria (g) into geodynamics…

Page 14: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Putting g into geodynamics:What is the geodynamic equation of state (EoS)?

Phase equilibrium doesn't just provide parameters, but also an EoS that is essential to close conservation and continuity equations, e.g.,

σ f ρ,s T f ρ,s

Homogeneous systems => common EoS choices:

Gibbs, g(P,T) Helmholtz, a(v,T) enthalpy, h(P,s)

internal energy, u(v,s)

u(s,v) is the only EoS for a heterogeneous system

Page 15: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Phase equilibrium doesn't just provide parameters, but also an EoS that is essential to close conservation and continuity equations, e.g.,

σ f ρ,s T f ρ,s

Homogeneous systems => common EoS choices:

Gibbs, g(P,T) Helmholtz, a(v,T) enthalpy, h(P,s)

internal energy, u(v,s)

u(s,v) is the only EoS for a heterogeneous system

Putting g into geodynamics:What is the geodynamic equation of state (EoS)?

Page 16: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Putting g into geodynamics:What is the geodynamic equation of state (EoS)?

Phase equilibrium doesn't just provide parameters, but also an EoS that is essential to close conservation and continuity equations, e.g.,

σ f ρ,s T f ρ,s

Homogeneous systems => common EoS choices:

Gibbs, g(P,T) Helmholtz, a(v,T) enthalpy, h(P,s)

internal energy, u(v,s)

u(s,v) is the only EoS for a heterogeneous system

Page 17: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Legendre Transform?

h g Ts

gg T

Th(T,P)

Optimization of exotic free energy functions: example h(s,P)

We need h(s,P), we have g(T,P)…

Page 18: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Optimization of exotic free energy functions: example h(s,P)

We need h(s,P), we have g(T,P)…

Legendre Transform?

h g Ts

gg T

Th(T,P)

A hidden virtue of linearization:

Discretization of h(T,P) for individual phases yields h(s,P) for the system, likewise u(T,P)

yields u(s,v).

Page 19: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

How do we put u(s,v) into geodynamic models? The energy (aka temperature) equation as an example

The parabolic equation we know and love:

PdT

ρc k T αTσ L Qdt

with mechanical and EoS parameters 2

P 2

g gρc T

PT

and

2g gα

T P P

Page 20: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

How do we put u(s,v) into geodynamic models? The energy (aka temperature) equation as an example

The parabolic equation we know and love:

PdT

ρc k T αTσ L Qdt

with mechanical and EoS parameters 2

P 2

g gρc T

PT

and

2g gα

T P P

is derived f rom the elliptic equation

dsρT k T Q 0

dt

which is discretized in time as

n 1 nδts k T Q s

ρT

with EoS+mechanical update rules

n 1 n 1 n 1 n 1

2n 1 n 1 n n 1 n 1

s ,ρ s ,ρ

u uT , ρ ρ εδt σ ρ

s ρ

Page 21: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

A morsel of wisdom?

Don’t put g(P,T), or any f(P,T) equation of state, into geodynamics

Use u(s,v), it is no more difficult than g and eliminates 1st order phase transformations and thereby the Stefan problem

Page 22: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Enter Amir Khan: Geophysical Inversions for Planetary Composition, Temperature and Structure

Allows joint inversion of unrelated geophysical data

P-wave velocities of cheese are 1.2 (Muenster) - 2.1 (Swiss) km/s, velocities in the lunar regolith are 1.2-1.8 km/s

Ergo the moon is a mixture of Muenster and Swiss cheese

secondary parameters primary dataprimary parameters

Page 23: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Inversion Strategy

i) Guess a physical configuration (T, c, d, …)

ii) Construct a forward model of the observed data

iii) Test against observations

iv) Generate a new configuration, go to ii)

repeat 107 times

d=f(m) => m=g(d)?

Page 24: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Searching for the Answer

Page 25: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Bayesian Inversion:Prior Probability, Likelihood and Posterior Probability

Page 26: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Bayesian Inversion:Prior Probability, Likelihood and Posterior Probability

Page 27: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Bayesian Inversion:Prior Probability, Likelihood and Posterior Probability

Page 28: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

What’s good about an EM inversion?

Sensitivity of seismic (vp) vs EM () signals

P-T

mineral composition

mineralogy

Perovskite 1.18

2 1.01

11

Wuestite 1.26

6 1.02

10

1.36

400

Test of ability to predict phase relations without requiring accuracy in high order derivatives necessary to calculate elastic properties

EM inversions are in principle a vastly superior method of probing planetary composition

P-T: from 1880K - 23GPa to 2750K - 100GPa

Mineral composition: 10 mol % change in Fe- or Al-content

Page 29: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

The Observations

Periodic ionospheric and magnetospheric fields induce secondary magnetic fields

within the earth

Transfer function between external and induced fields is a function of earth’s

conductivity

Sub-European soundings (Olsen 1999) for periods of 3 h to 1 year (depths of

200-1500 km)

Earth’s mass and moment of inertia

Page 30: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

What’s bad about EM inversion? The forward model.

Dependent on a poorly known transport property rather than thermodynamic properties (i.e., more difficult to measure), more sensitive to contaminants

and possibly texture.

0 0exp ,m H Px T x a bxkT

Upper mantle: conductivities after Xu et al. 2000a,b (Cpx as a proxy for C2/c & akimotoite), no correction for mineral composition or oxygen fugacity (Mo-MoO2)

Lower mantle: Wuestite 0(xMg) after Dobson & Brodholt (2000a); Perovskite 0(xAl) after Xu & McCammon (2002, Goddat et al. 1999, Katsura et al. 1998); Al-

free perovskite as a proxy Ca-perovskite

Aggregate conductivity computed as the volumetrically weighted geometric mean (Duba & Shankland 1990):

, volume f ractionmineraliaggregate fi i

i

fi

Page 31: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Parameterization of the Physical Model

•Spherically symmetric 1-D model

•3 silicate layers (crust, upper mantle and lower mantle)

• Parameterized by a composition thermal gradient and thickness

•Core parameterized by density

Compositional bounds (wt %)

•CaO[1;8]

•FeO[5;20]

•MgO[30;55]

•Al2O3[1;8]

•SiO2[20;55]

Page 32: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

1 day 1 year 11 years1 hour

Data fit I: Predicted transfer function components

Phase difference between magnetic and electric field

Apparent resistivity

Page 33: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

1 day 1 year 11 years1 hour

Data fit II: Mass (M) and Moment of Inertia (I)

Phase difference between magnetic and electric field

~106 models

Page 34: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Thermal models

T-z coordinates of the 410 and 660 discontinuities anticipated from phase eq expts (Ito & Takahashi ’89)

T660~1500±250oCT~0.5±0.1oC/kmTCMB~2900±250oC

Page 35: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

mantle composition

Page 36: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Is there a 660 layer?

priorposterior

Page 37: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Mantle Mineralogy

Page 38: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Mantle Conductivity Profile

Olsen (’99) inversion (model 3)

Page 39: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Density and Seismic Velocities

PREM (Dziewonski & Anderson ’81) – solid white line

AK135 (Kennett et al. ’95) – dashed white line

Page 40: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Resolution and Stability

Page 41: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Is there any hope of (at least) an inversion consensus?

Cammarano et al. ‘05: mantle is superadiabatic (if it’s pyrolite)

Lyon Group: Mattern et al. ‘05 revisited by Matas et al. (pers. comm. ‘06)

Khan et al. 08: travel time inversion, super-adiabatic non-pyrolitic; upper/lower mantle Mg/Si=1.05-1.20

Page 42: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE
Page 43: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Mundane Conclusion

The EM inversion results suggest a relatively homogeneous, superadiabatic mantle of chondritic composition

More generally terrestrial inversions yield low Mg/Si (1.05-1.2) and low bulk CaO and Al2O3

Paper Subject Data Conclusion

Khan et al '06, J GR Planets

Moon seismic lunar basalts consistent with inversion comp

Khan et al '06, EM Earth em superadiabatic, chondritic

Khan et al '06, GJ I Moon em consistent with seismic inversion

Khan et al '07, GJ I Moon seismic composition, T, lunar core.

Khan & Connolly '08, J GR Planets

Mars Love #, Q

SNC composition, large core

Khan et al '08, J GR Earth seismic superadiabatic, geochemically consistent PUM=> Fe-rich, Si-poor LM

Page 44: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Mars

What is sort of known:Composition from a set of “Martian” meteorites (e.g., McSween ’94)

Core, but only a paleo-magnetic field (e.g., Weiss et al. ‘02)

Page 45: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

What is known well: 4 Scalars (Yoder et al. ‘03)Mean mass and moment of inertia (distribution of mass)

Second degree tidal Love number (squishyness ~ f(S, KS, ))Tidal dissipation (inelasticity ~ g(S, qlocal))

What is not known well at all:Thermal structure, core size and state from forward models that assume the

SNC mantle composition and sensitive to crustal thickness

Page 46: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Martian Mantle Composition

Page 47: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Martian Mantle Mineralogy

No significant perovskite transition

Page 48: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Core Radius and Density

Page 49: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Areotherms and the Frozen Core Dilemma

After Stewart & Schmidt ‘06

Page 50: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Martian Conclusions

The Martian mantle is Fe-rich relative to Earth, but significantly less so than inferred from the SNC meteorites

(Dreibus & Wanke ’85)

The hot areotherm and large core radius preclude a Mg-perovskite phase transition in the lower mantle (bad news

for super-plumes? Not really)

The martian core is far above its liquidus

Page 51: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

“This is not the end, this is not even the beginning of the end, perhaps it is the end of the beginning.” –- Winston

Churchill

Free energy minimization provides the basis for a general physical model that permits joint inversion of a priori

unrelated geophysical data sets (seismic, gravity, electromagnetic)

Page 52: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Martian Temperature Distributions

PriorPosterio

r

Page 53: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Seismic Velocities and Thermodynamic Consistency

122 2

22, , ?S S

G GG G GN K

P P T TP P

KS SSobolev & Babeyko ‘94 no no no

Connolly & Kerrick ‘02 yes yes no

Stixrude & Lithgow-Bertelloni ‘05

yes yes yes

Stixrude &Lithgow-Bertelloni '05a,b fi nite strainmodel EoS ,S f G T

Does it really matter? Probably not, phase relations are most sensitive to integration constants and low-order derivatives, seismic velocities are most

sensitive to high-order derivatives.

Non-thermodynamic issues: anelasticity, aggregate modulii

Page 54: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Core Radius and CMB Temperature

Page 55: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Martian Inversion Resolution (T at 1200 km)

Page 56: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Trade-offs

Page 57: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Mapping Strategy

Page 58: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Free Energy Minimization by Linear Programming and Applications to Geophysical Inversion for Composition and Temperature

Free Energy Minimization – a method for predicting the thermodynamic (elastic) properties of rocks as a function of environmental variables (typically pressure and

temperature)

A forward model for rock properties: Geodynamic and Inversion calculations.

A robust and efficient method.

Some thoughts about cultural differences and data

Two inversions for planetary composition and temperature

Page 59: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

“Phase diagrams are the beginning of wisdom not the end of it.”-- Sir William Hume-Rothery

Page 60: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Optimize what when? And an Unexpected Virtue of the Linearized Solution

G(P,T,n) – inviscid, known temperatureA(V,T,n) – known strain rate, temperature

H(P,S,n) – inviscid, known heat fluxU(S,V,n) – known strain rate and heat flux

Thermodynamics provides stability criterion (i.e., an extremal function) for any choice of variables among the conjugate pairs P-V, T-S, -n

G(P,T,n) –> A, H, U as a f(P,T,n)

Stefan Problem

Page 61: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Forward Geodynamic Modelling: Subduction Zone Decarbonation

Closed system models suggest carbonates in slab lithologies remain stable beyond sub-arc depths (Kerrick & Connolly, 1998, 2001a,b).

Would infiltration-driven decarbonation alter this conclusion?

Page 62: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE
Page 63: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE
Page 64: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Slab fluid composition and production

Page 65: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Slab Properties

Is infiltration decarbonation

important?No.

Page 66: “Phase diagrams are the beginning of wisdom…” -- William Hume-Rothery OBE

Some thoughts about cultural differences and data

Geophysics/Mineral physics• Limited data, lots of theory• Individual minerals and phase

transitions• “A good experiment ****s any

computation” D. Yuen, 2005Pro: Amenable to simple parameterization

for geodynamic modelsCon: Ignores strong autocorrelation of

thermodynamic parameters

Petrology• Lots of data, little theory• Global averagePro: Objectivity, single mega-parameterCon: Difficult to: assess uncertainty;

separate first and second order affects; modify without access to primary data

D” and the Fe-Mg-Al Post-perovskite transition