Equations of State
Ross Angel
Dipartimento di Geoscienze, Universita di Padova
[email protected]
www.rossangel.net
[email protected]
www.indimedea.eu
 Part I:
 Part II:
 Part III:
Linear Elasticity of Minerals
Equations of State
Equations of State Practical
 Volume variation with pressure



P-V curves
Bulk modulus derivatives
Practical EoS
 Volume variation with pressure and temperature
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Linear elasticity, hydrostatic compression
 In terms of the general strain-stress relationship at any one
pressure
ε n = smnσ m
Hooke law: F = -kx
Linear EoS
 The stress tensor for small hydrostatic P increase:


σ4 = σ5 = σ6 = 0
σ1 = σ2 = σ3 = -ΔP
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 ε 1   s11
  
 ε 2   s21
ε   s
 3  =  31
 ε 4   s14
  
 ε 5   s15
ε   s
 6   16
s12
s13
s14
s15
s22
s23
s24
s25
s32
s33
s34
s35
s24
s34
s44
s45
s25
s35
s45
s55
s26
s36
s46
s56
s16  − ΔP 


s26  − ΔP 
s36  − ΔP 


s46  0 

s56  0 
s66  0 
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Bulk modulus and elastic tensor
 The volume strain is the sum of linear strains:
 ε 1   s11
  
 ε 2   s21
ε   s
 3  =  31
 ε 4   s14
  
 ε 5   s15
ε   s
 6   16
s12
s13
s14
s15
s22
s23
s24
s25
s32
s33
s34
s35
s24
s34
s44
s45
s25
s35
s45
s55
s26
s36
s46
s56
s16  − ΔP 


s26  − ΔP 
s36  − ΔP 


s46  0 

s56  0 
s66  0 
∂V
= ε1 + ε 2 + ε 3
V
∂P
∂P
−1
K = −V
=
= (s11 + s22 + s33 + 2 s12 + 2 s13 + 2 s23 )
∂V (ε 1 + ε 2 + ε 3 )
 The bulk modulus of a single crystal under hydrostatic
pressure is the inverse of the sum of the top-left values of
the compliance matrix
 Volume response to equal normal stresses

Reuss bound on the bulk modulus
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Volumes under high pressure
 Under linear elasticity

Elastic constants
1.00
∂P
−1
K = −V
= (s11 + s22 + s33 + 2 s12 + 2 s13 + 2 s23 )
∂V
∂V
K
=−
∂P
V
V/V0
0.95
0.90
0.85
0 1 2 3 4 5 6 7 8 9 10
Pressure: GPa
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The diamond-anvil cell
 Apply hydrostatic stress, measure strain
quartz
+/-0.01 GPa
ruby
sample
+/-0.05 GPa
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Bragg equation determines 2θ
 Braggs law:

d(hkl) = λ/2sin θhkl
 Measure 2θ on
diffractometer
 d(hkl) cell parameters
and volume of the crystal
dd
2θ
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Volumes under high pressure
 Under linear elasticity

Elastic constants
1.00
K = −V
∂P
−1
= (s11 + s22 + s33 + 2 s12 + 2 s13 + 2 s23 )
∂V
V/V0
0.95
0.90
Material gets stiffer
0.85
 Elastic properties change
with pressure!
0 1 2 3 4 5 6 7 8 9 10
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Equation of State
 Defines the elastic relationship
of volume to intensive
variables:
 V = f(T,P,H,X….)
 Normally V = f(P,T)
 Isothermal EoS: V=f(P)
 Can also be defined as ρ = f(P)
 Or as the change in elastic properties
with pressure
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Elasticity basics: volume/bulk
 Elasticity is a material property:
1.00
 relates applied stress and resulting strain
 compliances describe “softness”
 moduli describe “stiffness”
 Stress is dP
 Strain is dV/V
V/V0
 For volume change with pressure:
0.95
0.90
0.85
−1
βV = (∂V ∂P )
V
 Compressibility
 Volume compliance
0 1 2 3 4 5 6 7 8 9 10
Pressure: GPa
K = −V (∂P ∂V )
 Bulk modulus:
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EoS formulations

The bulk modulus and its derivatives are
thermodynamic variables of precise
definition:
K = −V (∂P ∂V )
(
K ′′ = (∂K ′ ∂P ) = ∂ 2 K ∂P
)

An EoS says how these change with P

But there is no absolute thermodynamic
basis for specifying a correct form for an
EoS

All EoS are based upon assumptions
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K ′ = (∂K ∂P )
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Why not use a polynomial?
1.00
0.95
V/V0
V
= 1 + aP + bP 2 .. ????
V0
0.90
0.85
0 1 2 3 4 5 6 7 8 9 10
Pressure: GPa
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EoS of quartz
K = −V ∂P ∂V
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Equations of state: Murnaghan
 Actually derived from concept of finite strain

But also can be derived by assuming K is linear with P
−V
Bulk Modulus
K = K 0 + K ′P
∂P
= K 0 + K ′P
∂V
P
V
∂P
− ∂V
0 K 0 + K ′P = Vo V
−1
 K ′P  K ′

V = V0 1 +
K 0 

K′

K  V 
P = 0  0  − 1
K ′  V 

K0
 Advantages
 Can be inverted, easily integrated
 …great for parametric fitting, thermo
Pressure
 Disadvantages
 K’ is constant, K’’ = 0
 Does not fit P-V data for V/V0 < 0.9
 Can be extended K’’ <> 0 (Tait)
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Finite strain EoS
 Assumes strain energy is a polynomial in strain f
Ψ = af 2 + bf 3 + cf 4 + .....
 Pressure is then:
P=−
∂E
∂V
− dΨ
dΨ df
=−
dV
df dV
df
P=−
2af + 3bf 2 + 4cf 3 + ....
dV
P=
[
]
 So we need a definition of strain f
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Infinitesimal and Finite strain
 Conventional elasticity theory works with infinitesimal
strains:
ε=
l − l0
l0
εV =
V − V0 V
= −1
V0
V0
 Under compression the volume
changes are not small……finite
strain
εV = −0.142
f L = −0.049
fL =
1  V 
 
2  V0 

2/3

− 1

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Finite strain definitions
 Infinitesimal strains defined with respect to the initial state
are called Lagrangian:
εV =
V − V0 V
= −1
V0
V0
 Lagrangian finite strains are also defined with respect to the
initial state:
fL =
1  V 
 
2  V0 

2/3

− 1

 Eulerian strains are defined with respect to the final state:
fE =
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1  V0 
 
2  V 
2/3

− 1

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Finite strain definitions
Eulerian strains are positive when volume decreases!!
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Birch-Murnaghan 2nd-order EoS
 Finite strain EoS


Uses the Eulerian finite strain f
Do 2nd-order as example…
 V  2 3 
 0  − 1
 V 

fE = 
2
Ψ = af 2 + ...
dΨ df
− dΨ
P=
=−
dV
df dV
df
P=−
[2af ]
dV
df
−1
(1 + 2 f ) = 1 (1 + 2 f )5 / 2
=
3V0
dV 3V
P=
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
…errr…?
1
(1 + 2 f )5 / 2 [2af ]
3V0
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Birch-Murnaghan 2nd-order EoS
 Next step

Use definition of bulk modulus
P=
1
(1 + 2 f )5 / 2 [2af ]
3V0
K = −V

∂P 2a
(1 + 2 f )5 / 2 [1 + 7 f ]
=
∂V 9V0
At P= 0….what are the values of V, f, K..?
K0 =
2a
9V0
K = K 0 (1 + 2 f ) (1 + 7 f )
5/ 2
P=
1
(1 + 2 f )5 / 2 [9V0 K 0 f ] = 3K 0 (1 + 2 f )5 / 2 f
3V0
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Birch-Murnaghan EoS – full expression to 4th
 Finite strain EoS

Assumes strain energy is a polynomial in f
Ψ = af 2 + bf 3 + cf 4
− dΨ
dΨ df
=−
P=
dV
df dV
df
P=−
2af + 3bf 2 + 4cf 3
dV
[

 V  2 3 
 0  − 1
 V 

fE = 
2
]
Do derivatives up to K’’, substitute back and….
5 
3
3
35  
P = 3K 0 f E (1 + 2 f E ) 2 1 + (K 0′ − 4) f E +  K 0 K 0′′ + (K 0′ − 4 )(K 0′ − 3) +  f E2 
2
9  
2


5 
9
35  
K = K 0 (1 + 2 f E ) 2 1 + (3K 0′ − 5) f E +  K 0 K 0′′ + K 0′ (K 0′ − 4) +  f E2 
2
9 

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Birch-Murnaghan EoS – full expression
5 
35  
3
3
P = 3K 0 f E (1 + 2 f E ) 2 1 + (K 0′ − 4) f E +  K 0 K 0′′ + (K 0′ − 4 )(K 0′ − 3) +  f E2 
9  
2
2


5 
35  
9
K = K 0 (1 + 2 f E ) 2 1 + (3K 0′ − 5) f E +  K 0 K 0′′ + K 0′ (K 0′ − 4) +  f E2 
9 
2

 V  2 3 
 0  − 1
 V 

fE = 
2
 Advantages
 Fits P-V data for V/V0 to 0.8
 Provides correct K0
 Disadvantages
 Cannot be inverted
 VdP integrals must be numerical
 Problem for thermo databases, and parametric fitting
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Birch-Murnaghan EoS – truncations
5 
35  
3
3
P = 3K 0 f E (1 + 2 f E ) 2 1 + (K 0′ − 4) f E +  K 0 K 0′′ + (K 0′ − 4 )(K 0′ − 3) +  f E2 
9  
2
 2
5 
35  
9
K = K 0 (1 + 2 f E ) 2 1 + (3K 0′ − 5) f E +  K 0 K 0′′ + K 0′ (K 0′ − 4) +  f E2 
9 
2

Truncation:
 2nd order (in energy)
 Coefficient of fE must be zero
 K0’ = 4
 V0 and K0 are material parameters
P = 3K 0 (1 + 2 f )
5/ 2
Ψ = af 2
f
K = K 0 (1 + 2 f ) (1 + 7 f )
(4 + 49 f 3) = (K 0′ + 49 f 3)
K′ =
1+ 7 f
1+ 7 f
5/ 2
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Birch-Murnaghan EoS – truncations
5 
35  
3
3
P = 3K 0 f E (1 + 2 f E ) 2 1 + (K 0′ − 4) f E +  K 0 K 0′′ + (K 0′ − 4 )(K 0′ − 3) +  f E2 
9  
2
 2
5 
35  
9
K = K 0 (1 + 2 f E ) 2 1 + (3K 0′ − 5) f E +  K 0 K 0′′ + K 0′ (K 0′ − 4) +  f E2 
9 
2


Truncation:
 3rd order (in energy)
Ψ = af 2 + bf 3
 Coefficient of f2E must be zero
 V0 K0 K0’ are material parameters
 4th order (in energy)
K ′′ =
−1 
(3 − K ′)(4 − K ′) + 35 
9
K 0 
Ψ = af 2 + bf 3 + cf 4
 Coefficient of
non-zero
 V0 K0 K0’ K0’’ are material parameters
f2E
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Finite-strain EoS of quartz
K = −V ∂P ∂V
3rd order BM
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Birch-Murnaghan EoS for quartz
K ′′ = ∂ 2 K ∂P2
K ′ = ∂K ∂P
 V  2 3

  0  − 1
  V 

fE =
2
5 
3
3
35  
P = 3K 0 f E (1 + 2 f E ) 2 1 + (K ′ − 4 ) f E +  K 0 K ′′ + (K ′ − 4 )(K ′ − 3) +  f E2 
2
9 
 2
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Summary: Bulk modulus and elastic tensor
 Linear elasticity  Volume strain
 ε 1   s11
  
 ε 2   s21
ε   s
 3  =  31
 ε 4   s14
  
 ε 5   s15
ε   s
 6   16
s12
s13
s14
s15
s22
s23
s24
s25
s32
s33
s34
s35
s24
s34
s44
s45
s25
s35
s45
s55
s26
s36
s46
s56
s16  − ΔP 


s26  − ΔP 
s36  − ΔP 


s46  0 

s56  0 
s66  0 
∂V
= ε1 + ε 2 + ε 3
V
∂P
∂P
−1
K = −V
=
= (s11 + s22 + s33 + 2 s12 + 2 s13 + 2 s23 )
∂V (ε 1 + ε 2 + ε 3 )
 The bulk modulus of a single crystal under hydrostatic
pressure is the inverse of the sum of the top-left values of
the compliance matrix
 Volume response to equal normal stresses

Reuss bound on the bulk modulus
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Summary: EoS formulations

The bulk modulus and its derivatives are
thermodynamic variables of precise
definition:
K = −V (∂P ∂V )
(
K ′′ = (∂K ′ ∂P ) = ∂ 2 K ∂P
2
)
K ′ = (∂K ∂P )

An EoS says how these change with P

But there is no absolute thermodynamic
basis for specifying a correct form for an
EoS
All EoS are based upon assumptions




An assumed inter-atomic potential
An assumed relationship between EoS
parameters and P
An assumed relationship between free energy
and strain (and a choice of strain definition)
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Temperature: thermal expansion
 Expansion
 Softening
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Thermal expansion simple approach
 Expansion: simple polynomial
 Bulk modulus:
∂K
= constant
∂T
 These simple approaches do not work!
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Thermal expansion simple approach
 Isotherms of Fo92 olivine
25C
725C
1725C
∂K
= -.028GPa.K −1
∂T
1725C
25C
 Fixed dK/dT leads to negative thermal expansion

Hellfrich G, Connolly JAD (2009) Physical contradictions and remedies using simple
polythermal equations of state. American Mineralogist 94:1616-1619.
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Thermal pressure
 Heat a mineral:
 but keep the V fixed:
P>0
P=0
 Thermal pressure!
Holland & Powell (2011)
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Thermal pressure EoS of olivine
 No non-physical behaviour:
25C
725C
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1725C
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Summary: Equations of State
 Normal elasticity theory is linear



Hookes law, strain proportional to stress
Elastic moduli & compliances are constant
Bulk modulus is defined by the moduli and
compliances
 At high pressures:



Material becomes stiffer, moduli increase
Re-define strain
Need complex Equations of State
 At high P and T

Treat the T effects as thermal pressure
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Equations of State – Further reading

Angel et al (2014) Equations of state.


Angel R.J., Gonzalez-Platas J. & Alvaro M., 2014. EosFit-7c and a Fortran module (library) for
equation of state calculations. Zeitschrift Fur Kristallographie 229, 405-419.
Angel et al (2009) Elasticity measurements on minerals: a review

European Journal of Mineralogy, 21:525:550

Anderson OL (1995) Equations of State of Solids for Geophysics
and Ceramic Science. Oxford University Press, Oxford

Technical papers:
 Murnaghan (1937) Am. J. Maths 59:235
 Birch (1947) Phys. Rev. 71:809
 Stacy (1981) Geophys. Surveys 4:189
 Jeanloz (1988) Phys Rev B38:805
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