Dynamics and Composition
of the Mantle: From the
Atomic to the Global Scale
Lars Stixrude and Carolina Lithgow-Bertelloni
University College London
Earth Materials
Processes governed by behavior of Earth,
environmental, and planetary materials
Seek fundamental understanding of this behavior
Earth as an Experimental
Sample
—  Pressure and temperature self-generated
—  How does it respond to changes in:
—  Energy
—  Temperature (Heat capacity)
—  Density (Thermal expansivity)
—  Phase transformation (Free energy)
—  Stress
—  Density (Bulk modulus)
—  Temperature (Grüneisen parameter)
—  Phase transformation (Free energy
Probing Extreme Conditions
—  For most of present Earth,
seismology is the most powerful
observational probe
—  Mineralogical Earth model forms
the link to thermal and
compositional state
Earth Materials
—  Heterogeneous on many scales
—  Equilibrium thermodynamics of
multi-component systems
—  Differentiation
—  Affects physical properties
Geotherm
Geotherm
Elemental Abundances
Cosmic Abundances
Mantle Mineralogy
Mantle Mineralogy
—  Minerals at ~200 km
depth
—
—
—
—
Olivine
Orthopyroxene
Clinopyroxene
Garnet
—  How do these change
with increasing pressure?
—  Polymorphic phase
transformations
Xu et al. (2008) EPSL
Mantle Mineralogy
Thermodynamics
—  Scope is very broad and
includes stars as well as
planets
—  Unusual for a physical
theory in making no
quantitative predictions
—  Instead very powerful
limits and relationships
—  Use knowledge of
material properties to
guide
—  Functional form
—  Values of parameters
He
Concept of Equilibrium
—  Show that any system at
constant P,T,Ni tends
towards the state of lowest
Gibbs free energy
—  Equilibrium state
—  Accessed in the limit of
infinite time
—  Time invariant
—  Path independent
—  Most important for Earth
because
—  Time scale is long
—  Temperature is high
Fundamental Relation
—  Gibbs free energy
—  G = G(V,T,Ni) not
fundamental
Internal Energy, U
—  P,T,Ni are natural
variables of G
Internal Energy, U
—  Complete information of
all properties of all
equilibrium states
Gibbs Free Energy, G
—  G = G(P,T,Ni)
Pressure,
Entropy,PS
Volume,
V=(dG/dP)
Temperature,
T=(dU/dS)
T,Ni V,Ni
One Component Phase
Equilibria
—  Relate Variation of G to extensive
properties of phases V, S
dG = VdP − SdT
—  Relate contrast in extensive properties to
slope of phase boundary
€
—  Clapeyron slope can be negative or
positive
dT ΔV
dP
=
ΔS
One Component Phase
Equilibria
Thermodynamic Variables
! ∂G $
1
V =# & =
" ∂ P %T ,Ni ρ
! ∂G $
S = −# &
" ∂ T %P,Ni
! ∂G $
µi = #
&
" ∂ N i %P,T
! ∂S $
CP = T # &
" ∂ T %P,Ni
1 ! ∂ρ $
α =− # &
ρ " ∂ T %P,Ni
!∂ P $
KS = ρ # &
" ∂ρ %S,Ni
γ=
ρ ! ∂T $
# &
T " ∂ρ %S,Ni
Thermodynamic Functions
G(P,T) = U + PV − TS
V=dG/dP
€
F
F(V,T)=G(P,T)-PV
Fundamental Thermodynamic Relation
H. B. Callen, Thermodynamics, Wiley, 1965, 1985.
Two Component Phase
Equilibria
—  Phase: Homogeneous in chemical
composition and physical state
—  Component: Chemically
independent constituent
—  Example: (Mg,Fe)2SiO4
—  Phases: olivine, wadsleyite,
ringwoodite, …
—  Components: Mg2SiO4,Fe2SiO4
—  Solid Phase with more than one
component is a solid solution
Properties of Ideal Solution 1
—  N1 type 1 atoms, N2 type 2 atoms, N total
atoms
—  x1=N1/N, x2=N2/N
—  Volume, Internal energy: linear
—  V = x1V1 + x2V2
—  Entropy: non-linear
—  S = x1S1 + x2S2 -R(x1lnx1 + x2lnx2)
—  Sconf = RlnΩ
—  Ω is number of possible of arrangements.
Properties of Ideal Solution 2
—  Gibbs free energy
—  Re-arrange
—  G = x1(G1 + RTlnx1) +
x2(G2 + RTlnx2)
—  G = x1µ1 + x2µ2
—  µi = Gi + RTlnxi
—  Defines chemical
potential
Gibbs Free Energy
—  G = x1G1 + x2G2 +
RT(x1lnx1 + x2lnx2)
Mechanical
Mixture
Ideal
Solution
µ2
µ1
0
1
Composition, x2
Gibbs Free Energy, G
!
Gibbs Free Energy, G
!
"
0
"
1
Composition, xB
0
Composition, xB
!+"
!
0
1
Gibbs Free Energy, G
"
Pressure
1
"
!
Composition, xB
0
1
Composition, xB
Two Component Phase
Equilibria
(wa)
(ol)
Multi-Component Phase
Equilibria
HeFESTo
—  Self-consistent computation
of phase equilibria and
physical properties
—  Formulation suitable for
aplication to entire mantle
pressure-temperature
regime
—  Anisotropic generalization
of thermodynamics permits
computation of full elastic
constant tensor
Cold Part
140
Pressure (GPa)
120
F=af 2
MgSiO3
Perovskite
300 K
100
80
60
40
20
0
0.70 0.75 0.80 0.85 0.90 0.95 1.00
Volume, V/V0
Thermal Part
%4
(
θ
S ≈ 3R' − ln + …*
&3
)
T
€
Stixrude & Lithgow-Bertelloni (2005) JGR
Thermal Part
—  Origin of Low velocity zone
—  Older results predict higher shear
wave velocity
—  Melt required?
—  Experimental data were limited
to low temperature
—  Velocity depends non-linearly on
temperature
—  Required by third law
—  HeFESTo captures correct
behavior with Debye model
# ∂S & # ∂ V &
% ( = % ( = Vα = 0
$ ∂P 'T $ ∂T ' P
Equation of State of Mantle
Solids
—  Thermal pressure
—  Decreases on compression
—  Thermal effects “squeezed
out”
—  PTH~γ3NkBT/V
—  Grüneisen parameter γ
—  Decreases on compression
—  Controls magnitude of
thermal pressure
—  Controls adiabatic gradient
—  γ=(dlnT/dlnV)S
—  Liquids? Virtually unknown
at lower mantle conditions
Thermal
pressure
Experiment
Experiments
Theory
Density Functional Theory
Charge density in Mg2SiO4 wadsleyite
Oxygen
Silicon
Magnesium
Density functional theory
Density Functional Theory