Physics of compression of liquids�Implication for the evolution of planets

Shun-ichiro Karato

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Yale University

Department of Geology & Geophysics

New Haven, CT

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(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 Birch’s law is totally violated for non-metallic liquids but is (approximately) satisfied for solids and metallic liquids.

--> A new model is developed for non-metallic liquids.

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Motivation-I

Melts are more compressible than solids --> density cross-over

Why is a melt so compressible?

Could a melt compressible even if its density approaches that of solid?

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density

Stolper et al. (1981)

Motivation - II

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How does a molten layer in a planet evolve?

Grüneisen parameter γ controls

dTad/dz and dTm/dz

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Liquid-solid comparison: bulk modulus

metallic liquids

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Boehler and Kennedy (1977), Boehler (1983)

solids

non-metallic liquids

Liquid-solid comparison: Grüneisen parameter

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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melt (peridotite) solid (perovskite)

K~30 GPa

K~260 GPa

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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SiO2: Karki et al. (2007)

• (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.

--> compression mechanisms of (non-metallic) liquids are completely different from those of solids.

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Liquids versus solids

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Explanation of γ −ρ relationship

Entropy elasticity

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For an ordered solid, the first term dominates (+ small

contribution from the second part (vibrational entropy))

-> compression behavior is controlled by inter-atomic bonds,

i.e., control by the bond-length: Birch’s law.

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For a gas, (a complex) liquid the second term dominates.

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Entropy elasticity --> the Birch’s 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)

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(f: packing fraction)

• small KT (10-30 GPa)
• small δT (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)

• The Bottinga-Weill model (solid mixture model)

does not work---> what should we do?

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• 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 (ρm)

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Jing and Karato (2009)

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Jing and Karato (2010)

• Metals behave differently.
• Little difference between solids and liquids

<--cohesive energy of a metal is made of free electrons + “screened atomic potential (pseudo-potential)”

--> influence of atomic disorder is small

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Some exceptions

Ziman (1961)

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For metals γ solid~γ liquid -->solidification from below

For silicates γ solid γ liquid, γ liquid becomes large in the deep interior

Tad increases more rapidly with P than Tm. --> 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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liquid = mixture of solid-like components

(Bottinga-Weill model)

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Stixrude-Karki (2007)

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.

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--> fundamental differences in compression mechanisms

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Glasses and solids follow the Birch’s law.

Liquids do not follow the Birch’s law.

Small K for a liquid is NOT due to small density.

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Liquids versus glasses

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

• Bottinga-Weill model does not work---> what should we do?

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• majority of silicate melt (MgO, FeO, CaO, Al2O3, SiO2): hard sphere model works, compositional effect is mainly through (mass) ρm

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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. -> compression of a silicate melt cannot be described by the sum of compression of individual components.

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(a) (b)

solid (or Bottinga-Weill model)

(oxide) liquid

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Assign the size of individual hard sphere components:

MgO, SiO2, Al2O3 ----

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Determine the size based on the existing data

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Use these sizes to calculate the density at higher P (T)

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