Equations of stateR. WentzcovitchU. of MinnesotaVLab tutorial

-EoS relates P,V,T in materials

-EoS of minerals are necessary to build Earth models

-In this lecture: isothermal EoS only (Eos parameters are functions of T)

Poirier’s “Introduction to the Physics of the Earth Interior”, Cambridge Press

The definition of the bulk modulus offers an EoS

ln

dP dPK V

dV d V

0

00

V P

Vo

dV dP

V K

PV V exp

K

(with K=cte=K0)

-This is only a naive example of how to generate EoS.

-K is not cte. It varies with P, except for really ifinitesimal volume changes.

Murnaghan EoS• It can be similarly derive assuming

'0 0K K K P

'0 0

dPK K P V

dV

0

0

1'

0

00 0 0 0

'1

'

KV P

V

K PdV dP V

V K K P V K

0 '

0 0

0

1'

KK V

PK V

0 'K is cte

Strains• Eulerian strain (f>0 for compression)

• Lagrangian strain (ε<0 for compression)

OK for ε→0

• Hencky strain (logarithmic strain)

0

1

20

(1 )

(1 2 )

l l f

l l f

0

1

20

(1 )

(1 2 )

l l

l l

0 0

lnl

H H

l

dl ld d

l l

0

1ln3H

V

V

23

011

2

Vf

V

23

0

11

2

V

V

For hydrostatic compression

• One more relation to be used:

320

11 3

1 2

Vf

V f

03dV

Vdf

0 00

00

0

lim lim

1 1lim

3 3

V V V V

P

dP VPK V

dV dV

dFK

f V df

0 00

19 lim

P

dFK V

f df

For f → 0

Therefore

Bulk modulus

3dV

fV

dFP

dV

with

Now we will expand the free energy in term of (eulerian)strains and derive relationships P(V), K(V), K’(V)…

F=af2+bf3+cf+…

Birch Murnaghan EoS (2nd order)

• 2nd order expansion of the free energy F=af2

• Recall that

• Therefore

0 0 0

19 limP

T

FK V

f f

0 0

9

2a K V

T

F F fP

V f V

23

011

2

Vf

V

with

with 52

0

11 2

3

Ff

V V

7 53 3

502

00 0

33 1 2

2

KP K f f

52

0 (1 7 )(1 2 )K K f f

7 53 3

0

0 0 0

0

7 52

KdP dPK

dd

32

0

/ (1 2 )w f

0

12 49'

3 21

K K f fK

P f P f

Therefore for f→0 0 ' 4K

2

( ' 4)

ndBM

M

PR

P K

LM assemblage

Murnaghan EoS overestimate P for non-infinitesimal strain

Birch-Murnaghan 3rd order2 3F af bf

T

F F fP

V f V

52

0

11 2

3

ff

V V

Take into account:

with

Then one gets:

At P=0 (f=0), K=K0

K’=K0’

0 0 0

19 limP

T

FK V

f f

52

0

33 1 2 1

2

bfP K f f

a

→ 2 eq.s for 2 unknowns, a and b

7 5 23 3 3

0

0 0 0

31 1

2

KP

0

3' 4

4K

If K0’=4 we recover the 2nd order BM

with

• One needs measurements in a larger pressure range to fit a 3rd order EoS

• There are trade offs between Ko and Ko’

• If the pressure range is small Ko’ is usually constrained to 4.

Vinet EoS

• This EoS is based on a different expansion of F

l is a scaling length

• Defining and changing variables (r→V) in F

• Replace a, l, K0, and K0’ from relations above and get

0

0

(1 )exp( )

( )

F F a a

r ra

l

34

3V r

02

00 2

0

exp( )4

4

FFP a a

V lrF

Kl r

00

2' 1

3

rK

l

1 12/3 3 3

0 00 0 0

33 1 exp ' 1 1

2

V V VP K K

V V V

Logarithmic EoS

• Expand F in powers of Hencky strains εH like Birch-Murnaghan

• To 2nd order in εH one gets

• And to 3rd order one gets

00 0

lnP K

00

0 0 0

' 2ln 1 ln

2

KP K

Comparison between parameters offered by various EoS

Summary

• EoSs based on expansions of F in terms of strain (finite strain EoSs) give order dependent parameters (trade offs).

• High order EoS require data in larger pressure ranges.

• The Vinet EoS is good for any pressure range.