Equations of state EOS

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Equations of state (EOS)
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An EOS is a relation between P, V and T. The EOS
known to everybody is the ideal gas equation
PVnRT Important thermodynamic definitions Incomp
ressibility KS-V(?P/?V)S KT-V(?P/?V)T Ther
mal expansivity a1/V(?V/?T)P Grüneisen
parameter (?P/?T)VgCV/V An important relation
is KS/KT1agT
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How does the density vary with depth
(pressure)? dP/Kdr/r Upon integration with
constant K r/r0exp(P/K0) This is different
from observations!
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We see that the increase of density with depth
becomes more difficult with increasing
compression. ? K must increase with P
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Empirical EOS

• Murnaghan (1951) KK0K0P
• Most EOS are truncated polynomials fit to some
  observations KK0K0P1/2 K0P2
• Birch (1961) for crustal rocks Vpa(m)br, m is
  the mean atomic weight

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Finite strain theory
Eulerian strain e?u/?x-1/2(?u/?x)2 V/V0r0/r(1-
2e)-3/2(12f)-3/2 Helmholtz free energy
Faf2bf3cf4 P-(?F/?V)T ? K ? r 2nd (linear
elasticity), 3rd, 4th order Birch-Murnaghan EOS
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Some remarks The assumption is that the strain
is Eulerian. The same theory can be applied to
Lagrangian strain which leads to different EOS.
Observations show that Eulerian strain best
describes Earths lower mantle. The shear modulus
(G) is more difficult because it is not as easily
defined thermodynamically, but equations take the
same form as for K
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Atomic potential representation
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A crystal is a lattice of oscillators (atoms) F-
?E/?r k(r-a0) The total vibrational energy
gives T The normal modes give the elastic
constants E can be expressed as the sum of an
attractive and a repulsive potential (Born-Mie
potential) E-a/rmb/rn where ngtm because
repulsive part has a shorter range
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Because ngtm, we have a non-linear
oscillation. With increasing pressure, the
interatomic spacing decreases and the restoring
force increases more rapidly. Compression becomes
more difficult, i.e. the bulk modulus increases
with pressure. At T0, we are the bottom of E. At
low temperature, we are near the bottom, and the
vibrations are nearly harmonic. At high T, the
vibrations are asymmetric and on average r is
bigger than a0 ? the volume of the atom
increases. This is thermal expansion.
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For the anharmonic oscillator, E is the potential
energy between two atoms. At ra0, P0 and
?E/?r0 The density r at a given pressure
relative to r0 at P0 is r/r0(a0/r)3 Consider a
crystal with N atoms each having 6f neighbours
(cubic f1, closed packed f2), the internal
energy is given by U3NfE and the volume is given
by VNgr3 (cubic g1, closed packed g1/?2)
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P-dU/dV (dU/dr)(dr/dV) f/g(1/r2)E K-V(dP/dV
) KdK/dP The EOS is given by the choice of E.
For the Born-Mie potential with m2 and n4, we
get the same results as with 2nd order
Birch-Murnaghan EOS.
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Thermodynamic approach to finite strain
The Grüneisen parameter can be expressed
by g(1/2 dK/dP 5/6 2/9 P/K) / (1 4/3
P/K) Other expression exist (so this is also
empirical) K(P) is known from seismology !!! And
this determines g
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The knowledge of g(P) determines the EOS g(P) is
sensitive to E, very precise thus P(r) is
sensitive to E The common assumption is
that gg0(r/r0)q ? density For the lower mantle
q1 is a good approximation
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The shear modulus
The thermodynamics of the shear modulus is
difficult, but to a good approximation GaKbP
along an adiabat, a and b are constants.
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Bulk sound (linear for small compression)
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Bulk sound (exponential for high compression,
closed packed)
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K(T, P0) exponential Anderson-Grüneisen
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G(T, P0) linear
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r(T, P0) integrate a a -1/r (?r/?T)P
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A practical approach with BM3