March 2, 2010Global Network Symposium 1 Physics of compression of liquids Implication for the evolution of planets Shun-ichiro Karato Yale University Department

March 2, 2010 Global Network Symposium 1

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)


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.


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?

de

nsi

ty

Stolper et al. (1981)


Motivation - II

How does a molten layer in a planet evolve?


Grüneisen parameter controls dTad/dz and dTm/dz


Liquid-solid comparison: bulk modulus


Boehler and Kennedy (1977), Boehler (1983)

solids

non-metallic liquids 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.

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”

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.

Liquids versus solids


Explanation of relationship


Entropy elasticity

For an ordered solid, the first term dominates (+ smallcontribution from the second part (vibrational entropy)) -> compression behavior is controlled by inter-atomic bonds,i.e., control by the bond-length: Birch’s law.

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

Entropy elasticity --> the Birch’s law does not apply.


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


Consequence of Sconfig model of EOS(scaled particle theory: excluded volume effect)

(f: packing fraction)


• small KT (10-30 GPa)• small T (large intrinsic T-derivative)• positive density dependence of the Grüneisen parameter


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?

• 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)


Jing and Karato (2009)


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

Some exceptions

Ziman (1961)


For metals solid~liquid -->solidification from belowFor silicates solid liquid, liquid becomes large in the deep interior

Tad increases more rapidly with P than Tm. --> Solidification from shallow (or middle) mantle.


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.


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

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.

--> fundamental differences in compression mechanisms


Liquids versus glasses

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.



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

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

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


(a) Compression of a mineral (solid) can be described by the superposition of compression of individual components (a polyhedra model).

(b) 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.

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


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)


Documents

March 2, 2010Global Network Symposium 1 Physics of compression of liquids Implication for the evolution of planets Shun-ichiro Karato Yale University Department