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lecture3.ppt - What is VLab

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					               Equations of state
                         R. Wentzcovitch
                         U. of Minnesota
                          VLab 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




                                        (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

                                        is cte
                         Strains
• Eulerian strain (f>0 for compression)




• Lagrangian strain (ε<0 for compression)
  OK for ε→0



• Hencky strain (logarithmic strain)



      For hydrostatic compression
• One more relation to be used:
                        For f → 0


  Therefore

  Bulk modulus


                        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
           with

• Recall that

• Therefore

                   with
Therefore for f→0



          LM assemblage



         Murnaghan EoS
         overestimate P for
         non-infinitesimal
         strain
            Birch-Murnaghan 3rd order

Take into account:                           with

             with


Then one gets:


At P=0 (f=0), K=K0
                       → 2 eq.s for 2 unknowns, a and b
              K’=K0’




If K0’=4 we recover the 2nd order BM
• 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
             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
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.

				
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