Physics of compression of liquids Implication for the evolution of planets

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Physics of compression of liquidsImplication for
the evolution of planets

• Shun-ichiro Karato
• Yale University
• Department of Geology Geophysics
• New Haven, CT
• (in collaboration with Zhicheng Jing)

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Outline

• Geological motivation
• How does a molten layer in a terrestrial planet
  evolve?
• Physics of compression of melts
• (bulk modulus, Grüneisen parameter)
• How is a liquid compressed?
• Compression behavior of non-metallic liquids is
  totally different from that of solids.
  Bottinga-Weill model does not work for
  compression of silicate liquids.
• Compression behavior of metallic liquids is
  similar to that of solids.
• The Birchs law is totally violated for
  non-metallic liquids but is (approximately)
  satisfied for solids and metallic liquids.
• --gt A new model is developed for non-metallic
  liquids.

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Motivation-I

• Melts are more compressible than solids --gt
  density cross-over
• Why is a melt so compressible?
• Could a melt compressible even if its density
  approaches that of solid?

density
Stolper et al. (1981)
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Motivation - II
How does a molten layer in a planet evolve?
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• Grüneisen parameter ???controls
• dTad/dz and dTm/dz

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Liquid-solid comparison bulk modulus
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metallic liquids
Liquid-solid comparison Grüneisen parameter
solids
non-metallic liquids
Boehler and Kennedy (1977), Boehler (1983)
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melt (peridotite) solid (perovskite)
K30 GPa
K260 GPa

• Thermal expansion in a melt is large.
• Thermal expansion in a melt does not change with
  pressure (density) so much, although thermal
  expansion in solids decreases significantly with
  pressure.

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SiO2 Karki et al. (2007)

• Densification of a (silicate, oxide) liquid
  occurs mostly
• not by the change in cation-oxygen bond length
• partly by the change in oxygen-oxygen distance
• mostly by something else

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Liquids versus solids

• (non-metallic) liquids are more compressible
  than solids.
• the bulk moduli of non-metallic liquids do not
  vary so much among various melts (30 GPa).
• the thermal expansion of liquids is larger than
  solids and does not change with pressure
  (density) so much.
• the Grüneisen parameters of (non-metallic)
  liquids increase with pressure (density) while
  they decrease with compression in solids.
• the bulk moduli of glasses are similar to those
  of solids (at the glass transition), but much
  larger than those of liquids.
• bond-length in (silicate) liquids does not
  change much upon compression.
• --gt compression mechanisms of (non-metallic)
  liquids are completely different from those of
  solids.

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Explanation of ???? relationship
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Entropy elasticity
For an ordered solid, the first term dominates (
small contribution from the second part
(vibrational entropy)) -gt compression behavior
is controlled by inter-atomic bonds, i.e.,
control by the bond-length Birchs law. For a
gas, (a complex) liquid the second term
dominates. Entropy elasticity --gt the Birchs
law does not apply.
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a hard sphere model

• Each solid-like element does not change its
  volume hard sphere model
• These elements (molecules) move only in the space
  that is not occupied by other molecules
  excluded volume
• Compression is due to the change in molecular
  configuration, not much due to the change in the
  bond length

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Consequence of Sconfig model of EOS(scaled
particle theory excluded volume effect)
(f packing fraction)
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• small KT (10-30 GPa)
• small ?T (large intrinsic T-derivative)
• positive density dependence of the Grüneisen
  parameter

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An extension to a multi-component system (MgO,
CaO, SiO2, Al2O3, FeO Na2O, K2O)
Bottinga-Weill model A hard sphere
model (Stixrude et al., 2005)
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• The Bottinga-Weill model (solid mixture model)
• does not work---gt what should we do?
• a silicate melt oxygen sea cations
• (van der Waals model of a complex liquid
  Chandler (1983))
• assign a hard sphere diameter for each cation
• determine the hard sphere diameter for each
  cation from the experimental data on EOS of
  various melts
• predict EOS of any melts
• modifications 1. Coulombic interaction, 2.
  Volume dependence of the sphere for Si, 3.
  T-dependence of a sphere radius
• compositional effect is mainly through the mass
  (?m)

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Jing and Karato (2009)
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Jing and Karato (2010)
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Some exceptions

• Metals behave differently.
• Little difference between solids and liquids
• lt--cohesive energy of a metal is made of free
  electrons screened atomic potential
  (pseudo-potential)
• --gt influence of atomic disorder is small

Ziman (1961)
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For metals ??solid?????liquid --gtsolidification
from below For silicates ??solid??? ??liquid,
??liquid becomes large in the deep interior Tad
increases more rapidly with P than Tm. --gt
Solidification from shallow (or middle)
mantle.
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Conclusions

• Evolution of a molten layer in a planet is
  controlled largely by the behavior of the bulk
  modulus and the Grüneisen parameter.
• The bulk moduli of silicate liquids are lower
  than those fo solids ad assume a narrow range.
• The dependence of the Grüneisen parameter of
  liquids on density (pressure) is different from
  that of solids.
• In non-metallic liquids, the Grüneisen parameter
  increases with compression.
• In metallic liquids, the Grüneisen parameter
  decreases with compression.
• Changes in configuration (geometrical
  arrangement, configurational entropy) make an
  important contribution to the compression of a
  (complex) liquid such as a silicate melt.
• A new equation of state of silicate melts is
  developed based on the (modified) hard sphere
  model.
• In metallic liquids, the change in free energy
  upon compression is dominated by that of free
  electrons, and consequently, the behavior of
  metallic liquids is similar to that of metallic
  solids.

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Stixrude-Karki (2007)

• liquid mixture of solid-like components
• (Bottinga-Weill model)

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Problems with a conventional approach

• Bond-lengths in liquid do not change with
  compression as much as expected from the volume
  change
• Bulk moduli for individual oxide components in a
  liquid are very different from those of
  corresponding solids, and they take a narrow
  range of values
• Grüneisen parameters of most of liquids increase
  with compression whereas those for solids
  decrease with compression.
• --gt fundamental differences in compression
  mechanisms

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Liquids versus glasses

• Glasses and solids follow the Birchs law.
• Liquids do not follow the Birchs law.
• Small K for a liquid is NOT due to small density.

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How to formulate an equation of state for a
multi-component system?

• Bottinga-Weill model
  does not work---gt what should we do?
• majority of silicate melt (MgO, FeO, CaO, Al2O3,
  SiO2) hard sphere model works, compositional
  effect is mainly through (mass) ?m

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(a) (b)
solid (or Bottinga-Weill model)
(oxide) liquid

1. Compression of a mineral (solid) can be described
   by the superposition of compression of individual
   components (a polyhedra model).
2. Compression of a silicate melt is mostly
   attributed to the geometrical rearrangement using
   a free volume. Individual components do not
   change their volume much. -gt compression of a
   silicate melt cannot be described by the sum of
   compression of individual components.

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Assign the size of individual hard sphere
components MgO, SiO2, Al2O3 ---- Determine the
size based on the existing data Use these sizes
to calculate the density at higher P (T)
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