Physics of compression of liquids
Implication for the evolution of planets
Shun-ichiro Karato
Yale University
Department of Geology & Geophysics
New Haven, CT
(in collaboration with Zhicheng Jing)
March 2, 2010
Global Network Symposium
1
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.
March 2, 2010
Global Network Symposium
2
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?
Stolper et al. (1981)
March 2, 2010
Global Network Symposium
3
Motivation - II
How does a molten layer in a planet evolve?
March 2, 2010
Global Network Symposium
4
Grüneisen parameter controls
dTad/dz and dTm/dz
=
rg
g
K liquid
=
rg
K
d logTad
dz
d logTm
dz
March 2, 2010
(
(ß d e f inition of Grüneisen parameter)
2g solid -
2
3
)
( Lindemann model)
Global Network Symposium
5
Liquid-solid comparison: bulk modulus
March 2, 2010
Global Network Symposium
6
Liquid-solid comparison: Grüneisen parameter
solids
non-metallic liquids
metallic liquids
Boehler and Kennedy (1977), Boehler (1983)
March 2, 2010
Global Network Symposium
7
melt (peridotite)
solid (perovskite)
K~30 GPa
K~260 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.
March 2, 2010
Global Network Symposium
8
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”
March 2, 2010
Global Network Symposium
9
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.
--> compression mechanisms of (non-metallic) liquids are
completely different from those of solids.
March 2, 2010
Global Network Symposium
10
Explanation of  relationship
g =
a KT V
CV
.
à
¶ log g
¶ log r
=
¶ log a
¶ log r
+
¶ log K T
¶ log r
-1-
¶ log CV
¶ log r
»
If the Birch’s law is satisfied, then
and
à
( ) =( ) ( )
¶KT
¶V
¶ log g
¶ log r
P
¶KT
¶P
T
¶P
¶V T
=-
( )
¶KT
¶P
¶ log a
¶ log r
+
¶ log KT
¶ log r
-1 .
( ) =( ) ( )
KT
T V
¶KT
¶T
P
¶KT
¶V
P
¶V
¶T P
.
= -1 for a material that follows the Birch’s law
For (non-metallic) liquids,
¶ log g
¶ log r
> 0 à the Birch’s law is violated.
à compression of non-metallic liquids occurs not much
by bond-length change (not by the change in the internal energy)
à importance of configurational entropy
March 2, 2010
Global Network Symposium
11
Entropy elasticity
P=-
( )
¶F
¶V T
=-
( )
¶U
¶V T
+T
( )
¶S
¶V T
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.
For a gas, (a complex) liquid the second term dominates.
Entropy elasticity --> the Birch’s law does not apply.
March 2, 2010
Global Network Symposium
12
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
u ex =
March 2, 2010
4 p ( 2 R)
3
3
Global Network Symposium
13
Consequence of Sconfig model of EOS
(scaled particle theory: excluded volume effect)
P
config
( r,T ) = r RT × F ( f ) = rm RT ×
KTconfig
æ ¶Pconfig ö
= rç
= rm RT
è ¶r ÷ø T
æ ¶KTconfig ö
çè ¶T ÷ø = rm R
V
(
config
¶KT
¶P
a
config
dTconfig
March 2, 2010
)
T
=
==
(
f 1+ 4 f + 4 f 2
(1- f )4
(
(
f 1+ f + f 2
(
1- f
)
3
f 1+ 4 f + 4 f 2
(1- f )4
) = Pconfig
)
( f ,T )
(f: packing fraction)
)
1+11 f +20 f 2 + 4 f 3
(1- f )(1+ 4 f + 4 f 2 )
( )=
¶r
r ¶T
1
(
(
2
1 (1- f ) 1+ f + f
T
1+ 4 f + 4 f 2
f 4+8 f +3 f 2 +8 f 3 + 4 f 4
)
)
(1- f )(1+ f + f 2 )(1+ 4 f + 4 f 2 )
Global Network Symposium
14
( )
• small KT (10-30 GPa)
¶K
dT º - a K1 ¶TT
T
• small T (large intrinsic T-derivative)
• positive density dependence of the Grüneisen parameter
March 2, 2010
Global Network Symposium
P
15
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)
March 2, 2010
Global Network Symposium
16
• The Bottinga-Weill model (solid mixture model) V ( P,T ) = å xiVi ( P,T )
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)
March 2, 2010
Global Network Symposium
17
Jing and Karato (2009)
March 2, 2010
Global Network Symposium
18
Jing and Karato (2010)
March 2, 2010
Global Network Symposium
19
Some exceptions
• Metals behave differently.
– Little difference between
solids and liquids
<--cohesive energy of a
metal is made of free
electrons + “screened
atomic potential (pseudopotential)”
--> influence of atomic
disorder is small
Ziman (1961)
March 2, 2010
Global Network Symposium
20
For metals solid ~ liquid -->solidification from below
For silicates solid ¹ liquid, liquid becomes large in the
deep interior
Tad increases more rapidly with P than Tm. -->
Solidification from shallow (or middle) mantle.
March 2, 2010
Global Network Symposium
21
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.
March 2, 2010
Global Network Symposium
22
Stixrude-Karki (2007)
liquid = mixture of solid-like components
(Bottinga-Weill model)
V ( P,T ) = å xiVi ( P,T )
March 2, 2010
Global Network Symposium
23
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
March 2, 2010
Global Network Symposium
24
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.
March 2, 2010
Global Network Symposium
25
March 2, 2010
Global Network Symposium
26
How to formulate an equation of state
for a multi-component system?
• Bottinga-Weill model V ( P,T ) = å xiVi ( P,T ) 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
March 2, 2010
Global Network Symposium
27
(a) solid (or Bottinga-Weill model)
(a)
(b)
(b)
(oxide) liquid
Compression of a mineral (solid) can be described by the superposition
of compression of individual components (a polyhedra model).
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.
March 2, 2010
Global Network Symposium
28
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)
March 2, 2010
Global Network Symposium
29
March 2, 2010
Global Network Symposium
30