Theory of the Earth
Don L. Anderson
Chapter 7. Nonelastic and Transport Properties
Boston: Blackwell Scientific Publications, c1989
Copyright transferred to the author September 2, 1998.
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Recommended citation:
Anderson, Don L. Theory of the Earth.
Boston: Blackwell Scientific Publications,
1989.
http://resolver.calt
ech.edu/CaltechBOOK:1989.001
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Abstract:
Most of the Earth is solid, and much of it
is at temperatures and pressures that
are difficult to achieve in the laboratory. The Earth deforms anelastically at small
stresses and, over geological time, this results in large deformations. Most
laboratory measurements are made at high stresses, high strain rates and low
total strain. Laboratory data must therefore be extrapolated in order to be
compared with geophysical data, and this requires an understanding of solid-
state physics. In this chapter I discuss transport and activated processes in solids,
processes that are related to rates or time. Some of these are more dependent on
temperature than those treated in previous chapters. These properties give to
geology the "arrow of time" an
d an irreversible nature.
Nonelastic
and
Transport
Properties
Shall
not
every rock be
removed
out of
his
place?
M
ost
of
the Earth is
solid,
and much
of
it is at temper
-
atures
and
pressures that
are
difficult
to achieve
in
the laboratory. The
Earth
deforms
anelastically
at
small
stresses and, over geological time,
this results
in
large de
-
formations.
Most
laboratory
measurements are made
at
high
stresses,
high
strain rates
and
low
total
strain.
Labo
-
ratory data
must
therefore
be
extrapolated
in
order
to
be
compared
with
geophysical
data,
and
this
requires
an un
-
derstanding
of
solid
-
state physics. In
this chapter
I
discuss
transport
and
activated processes in
solids,
processes that
are
related
to
rates or
time.
Some
of
these are more depen
-
dent on temperature
than
those
treated
in previous
chapters.
These
properties
give
to
geology the
"
arrow
of
time
"
and
an
irreversible nature.
THERMAL
CONDUCTIVITY
There
are
three
mechanisms
contributing to thermal
con
-
ductivity
X
in
the
crust
and
mantle. The lattice part,
X,,
is
produced
by
diffusion
of
thermal vibrations
in
a crystalline
lattice
and is
also
called the phonon
contribution.
3CR
is the
radiative part, due to
the
transfer
of
heat
by
infrared
elec
-
tromagnetic
waves.
If the mantle is sufficiently
transparent,
Y',
is significant.
3C,
is the exciton
part, due
to the
transport
of
energy
by
quasiparticles composed
of
electrons
and
posi
-
tive holes. This
becomes dominant
in
intrinsic semiconduc
-
tors
as
the
temperature
is raised. Thus,
thermal conduction
in
solids arises
partly from
electronic
and
partly from
atomic motion
and, at
high
temperature, from radiation
passing through
the solid.
Introduction
Debye
regarded
a solid
as
a system
of
coupled
oscillators
transmitting
thermoelastic
waves. For
an
ideal
lattice
with
simple harmonic motion
of the atoms,
the conductivity
would
be
infinite.
In
a real
lattice
anharmonic motion couples the
vibrations,
reducing the
mean
free
path and
the
lattice
con
-
ductivity. Thermal conductivity is related
to
higher order
terms in the potential
and
should
be
correlated
with
thermal
expansion.
Lattice conductivity can
be
viewed
as
the
ex
-
change
of
energy between high
-
frequency
lattice vibrations,
or elastic
waves.
A
crude
theory
for
the lattice conductivity,
consistent
with
the Griineisen
approximation,
gives
where
a
is the
lattice parameter,
y
is the Griineisen parame
-
ter,
T
is
temperature,
KT
is the isothermal bulk modulus,
and
p
is density. This is
valid
for large
T18.
This relation
predicts
that the
thermal
conductivity decreases
by
about
20
percent
as
one
traverses the mantle.
The
contribution
of
"
free carriers
"
such
as
electrons,
holes
and
electron
-
hole
pairs
to thermal
conduction can
be
estimated from
the Wiedemann
-
Franz
law
relating electri
-
cal,
X,,
and
thermal (lattice)
conductivity.
The
mantle is
a
good
electrical insulator,
and the
associated
thermal
con
-
ductivity
is almost
certainly negligible. The
Wiedermann
-
Franz ratio
is
where
k
is
Boltzmann's
constant
and
e
is the electronic
charge.
Using
estimates
of
electrical
conductivity,
X,,
of
the
mantle,
we
obtain
estimates
of
thermal
conductivity
due
to
this mechanism that
are
some six
orders
of
magnitude
lower than
observed lattice conductivities.
Excitons, or
bound
electron
-
hole pairs,
may
contribute
to thermal conductivity
at high
temperature
if the
excitation
energy
is of
the
order
of
1
eV.
Evidence
to date
suggests
a
much
higher energy
in silicates
and
oxides, so exitonic
ther
-
mal
transport
appears
to
be
negligible
in
the
mantle.
13
NONELASTIC
AND
TRANSPORT
PROPERTIES
The thermal conductivities
of
various rock
-
forming
minerals are given
in
'Table
7
-
1.
Note
that the crust
-
forming
minerals
have about
one
-
half
to
one
-
third
of
the conductiv
-
ity
of
mantle
minerals. This
plus the
cracks present
at
low
crustal pressures means that a
much
higher thermal gradient
is maintained
in
the crust, relative to the mantle, to sustain
the same conducted heat
flux.
The gradient
can be
higher
still
in sediments that
have
conductivities
of
the
order
of
0.4
to
2
x
callcm
s
°
C.
The thermal
gradlient
decreases
with
depth
in
the Earth
because
of
the increasing conductivity
and the
decreasing
amount
of
radioactivity
-
generated heat.
If the crustal radio
-
activity
and mantle
heat
flow
are constant
and
the
effects
of
temperature
are
ignored, regions
of
thick crust should
have
relatively
high
upper
-
mantle temperatures.
Horai
and
Simmons (1970) found a good correlation
between
seismic velocities
and
lattice conductivity,
as
ex
-
pected
from
lattice dynamics. They
found
with
X,
in
mcallcm
s
°
C and
V,,,
in
kmls.
They
also found
a linear relationship
between
the
Debye
temperature,
8
(in
kelvins),
and
XL:
Thermal conductivity
is strongly anisotropic,
varying
by
about
a factor
of
2 in
olivine
and
orthopyroxene
as
a
*
function
of
direction. The
highly
conducting axes are
[I001
for olivine
and
[OOl]
for orthopyroxene. The
most
conduc
-
tive axis for olivine is also the direction
of
maximum
P
-
velocity and
one
of
the
faster S
-
wave directions,
whereas
the
most
conductive axis for orthopyroxene is
an
intermediate
axis for
P-velocity
and
a fast axis for S
-
waves.
In
mantle
rocks the fast
P
-
axis
of
olivine tends
to
line
up with
the
TABLE
7
-
1
Thermal Conductivity
of
Minerals
Thermal Conductivity
Mineral
10-3callcm
s°C
Albite
4.71
Anorthite
3.67
Microcline
5.90
Serpentine
7.05
Diopside
11.79
Forsterite
13.32
Bronzite
9:
99
Jadeite
15.92
Grossularite
13.49
Olivine
6.7
-
13.6
Orthopyroxene
8.16
-
15.3
Horai
(1971),
Kobayzshigy
(1974).
intermediate
P
-
axis
of
orthopyroxene. These axes,
in
turn,
tend to
line
up
in
the
flow
direction,
which
is in the horizon
-
tal plane
in
ophiolite sections. The vertical conductivity
in
such
situations
is much
less
than
the
average
conductivity
computed for
mineral
aggregates,
which
is
about
7
x
lW3
callcm
s
°
C
at normal conditions. Conductivity de
-
creases
with
temperature
and
may
be
only
half
this value
at
the
base
of
the lithosphere. The implications
of
this
aniso-
trlopy
in thermal conductivity
and
the
lower
than
average
vertical conductivity
have not been
investigated.
Two ob
-
vious implications are that the lithosphere can support a
higher thermal gradient
than
generally supposed,
giving
higher upper
-
mantle temperatures,
and
that the thermal
lithosphere
grows
less rapidly
than previously
calculated.
For
example, the thermal lithosphere
at
80
Ma
can be
100 km thick for
YC
=
0.01
callcm
s deg
and only
30
km
thick
if the appropriate
'X
is
3
X
callcm
s
deg. The
low lattice
conductivity of
the oceanic
crust
is
usually
also
ignored in these calculations, but
this
may be
counterbal
-
anced
by
water circulation in
the
crust.
The lattice (phonon) contribution
to
the thermal con
-
ductivity decreases
with
temperature, but at
high
tempera
-
ture
radiative transfer
of
heat
may
become significant
de
-
pending on the opacities
of
the minerals.
If the
opacity,
E,
is independent
of
wavelength and
temperature,
then
'XR
in
-
creases strongly
with
temperature:
where
n
is the refractive
index and
u
is the
Stefan-
Boltzmann constant.
If
E
were
constant,
XR
would
increase
very rapidly
with
temperature
and would be
the dominant
heat conduction mechanism
in
the mantle. The parameter
E-I
decreases from
about
0.6
to
0.1
cm
in
the temperature
range
of
500
to 2000
K
for olivine single crystals
and
approaches
0.02cm
for enstatite (Schatz
and
Simmons,
1972). The
net
result is that
3'CR
is about
equal to
YC,
at high
temperatures
and
lower
-
mantle conditions.
Convection
is
the
dominant
mode
of
heat transport
in
the
Earth's
deep interior, but conduction is
not
irrelevant to
the thermal state
and
history
of
the
mantle as heat
must
be
transported across thermal boundary
layers
by
conduction.
Thermal boundary
layers
exist at the surface
of
the Earth,
at the core
-
mantle boundary and,
possibly,
at chemical
intterfaces
internal to the mantle. Conduction is also the
mechanism
by
which
subducting slabs cool the mantle. The
thicknesses
and
thermal time constants
of
boundary
layers
are controlled
by
the thermal conductivity,
and
these regu
-
late the rate at
which
the
mantle
cools
and
the rate at
which
the thermal lithosphere grows.
The
thermal
diffusivity,
K
,
is defined
and
the characteristic thermal time constant
of
a body
of
dimension
1
is
TABLE
7
-
2
Estimates
of
Lattice Thermal Diffusivitv
in the Mantle
Depth
(km)
K
(cm
Vs)
Horai
and
Simmons
(
1970).
The thermal diffusivities
of
mantle minerals are about
0.006
to
0.010
cm2/s
(see
Table
7
-
2).
In order
to
match
the
observed
elevation
and
geoid changes across oceanic
fracture zones, Crough (1979) derived a
diffusivity
of
0.0033 cm
2
/s
for the upper mantle. This
low value
may
be
related to the anisotropy discussed above or high
temperatures.
Lattice
Conductivity
Both
thermal conductivity
and
thermal expansion depend
on the anharmonicity
of
the interatomic potential
and
there
-
fore on dimensionless
measures
of
anharmonicity such
as
y
or
ayT.
The
lattice
or
phonon
conductivity,
9CL,
is
where
7
is the
mean
sound
velocity,
I
is the
mean
free path,
and
a
is the interatomic distance,
and
KT
is the isothermal
bulk
modulus. Therefore,
This gives
where
we
have used
the approximation
For
lower
-
mantle properties this expression
is
domi
-
nated
by
the
d
In
KTld
In
p
term,
and
the variation
of
X,
is
similar
to
the variation
of
KT.
The lattice conductivity decreases approximately lin
-
early
with
temperature, a
well
-
known
result, but increases
rapidly
with density.
The temperature effect dominates
in
the shallow mantle, giving
a
decreasing
X,,
but
pressure
causes
YCL
to
be
high
in
the
lower
mantle. This
has
impor
-
tant implications
regarding
the properties
of
thermal
boundary layers,
the
ability
of
the
lower mantle to
conduct
heat from the core
and
into
the
upper mantle,
and
the con
-
vective
mode
of
the
lower mantle.
The general correlation of
YC,
with
the elastic
wave
velocities suggests that pressure
-
induced
phase
changes
in
the transition
region
and
lower mantle
will
also increase the
lattice conductivity. Thermally
induced velocity
variations
in
deep slabs
will
be
small because
of
the
high
XL
and
low
temperature derivatives
of
velocity.
The expected increase
of
X,
with
compression,
with bulk
modulus,
with sound
speed
and
across
low
pressure
-
high
pressure
phase
changes
has been verified
by
experiment
(Fujisawa
and
others,
1968) as
has the
correlation
of
YC,
with
elastic
wave
veloci
-
ties.
We
expect about a factor
of
3
increase
in
'X,,
caused
by
the increase
in
seismic velocities, from shallow
-
mantle
to transition
-
zone pressures. The erroneous use
of
an
oli
-
vine
-
like
3CL
for the deep slab
and
a
(aVp/dT),
of
about
twice that
of
olivine (Creager
and
Jordan, 1984) are partly
responsible for the conclusion about the persistence
and
extent
of
deep slab
-
related seismic anomalies.
On
the
other
hand, changes in mineralogy associated
with
solid
-
solid
phase changes are
much more
important
than
temperature.
The
ratio
alYC,
decreases rapidly
with
depth
in
the
mantle, thereby decreasing the
Rayleigh number. Pressure
also increases the
viscosity,
an
effect that
further decreases
the Rayleigh number
of
the
lower
mantle. The
net effect
of
these pressure
-
induced changes
in
physical properties
is
to
make
convection
more
sluggish
in
the lower mantle,
to
de
-
crease thermal
-
induced
buoyancy and
to
increase the thick
-
ness
of
the thermal boundary
layer
in
D
"
,
above
the
core-
mantle boundary.
The thermal conductivity
of
crystals
is
a complex
subject
even
when
many
approximations
and
simplifying
assumptions
are
made. The mechanism for transfer
of
ther
-
mal energy
is generally
well understood
in
terms
of
lattice
vibrations, or high
-
frequency
sound waves.
This is
not
enough,
however,
since thermal conductivity
would
be
in
-
finite in
an
ideal harmonic crystal.
We
must
understand,
in
addition,
the mechanisms
for scattering thermal
energy
and
for redistributing the energy
among
the modes
and
fre
-
quencies
in
a crystal so that thermal equilibrium
can
pre
-
vail.
An understanding
of
thermal
"
resistivity,
"
therefore,
requires
an
understanding
of
higher order effects,
including
anharmonici
ty.
High
-
frequency elastic
energy
is
scattered
by
imper
-
fections
such
as
point defects, dislocations,
grain bound
-
aries
and
impurities, including isotopic differences,
and non
-
linearity or anharmonicity or interatomic forces. The latter
effects
can be
viewed as
nonlinear interactions
of
the ther
-
mal
sound
waves
themselves, a sort of self
-
scattering.
The
parameters that
enter into a
theory
of
lattice
conductivity are
fairly obvious.
One
expects
that
the
tem
-
perature,
specific heat
and
the coefficient
of
thermal expan
-
sion
will
be
involved. One
expects
that
some
measure
of
anharmonicity, such
as
y,
will
be
involved.
In addition one
needs
a measure
of
a mean
free
path
or a
mean
collision
time
or
a measure
of
the
strength
and
distribution
of
scat-
terers.
The velocities
of
the sound
waves and
the
intera
-
tomic distances are also likely
to
be
involved.
Debye
(1912) explained
the thermal conductivity
of
dielectric
or
insulating solids
in
the following
way.
The lat
-
tice
vibrations can
be
resolved
into
traveling
waves
that
carry heat.
Because
of
anharmonicities the
thermal
fluctua
-
tions
in density lead
to local fluctuations
in
the
velocity
of
lattice
waves,
which
are therefore scattered. Simple lattice
theory provides
estimates
of
specific
heat
and
sound
ve
-
locity and how
they
vary
with
temperature
and
volume. The
theory
of
attenuation
of
lattice
waves
involves
an
under
-
standing
of
how
thermal equilibrium
is
attained
and
how
momentum
is transferred
among
lattice vibrations.
The heat
flow,
Q,
associated
with
a given mode
type
(longitudinal
or
transverse) can
be
written
in
terms
of
the
energy,
wave
number
and
group
velocity
of
the
mode.
The
distinction between
group
velocity,
v,
=
dwldk,
and
phase velocity,
o/k,
is seldom
made
in the
theory
be
-
cause dispersion
is generally
ignored.
If
o
is proportional
to
the
wave
number,
k,
then
v
is
constant
and
phase
and
group velocities are
equal.
This assumption breaks down
for
high
frequencies
or
large
k.
At
high
frequencies
v,
becomes
very
small,
and high
frequencies
therefore
are
not efficient
in
transporting heat.
High
-
frequency
waves
are
those hav
-
ing
wavelengths comparable
to a lattice spacing.
Because
of
the discreteness
of
a lattice, the possible
energy levels are
quantized.
We
can
treat
the
thermal prop
-
erties
of
a lattice as a gas
of
phonons.
A
quantum
of
lattice
vibration is
called a
phonon and
acts
as a particle
of
energy
fio,
momentum
Ak
and
velocity
doldk.
There
is no limit
to
the number
of
quanta
in a normal
mode. The phonons carry
a heat
current,
which
is the sum of
the heat
currents carried
by
all normal modes:
The lattice conductivity
is a sum over all
wave
types
of
the
integral
over all
wave
numbers
where
v,
is the
group
velocity
of
the
jth
wave
type
and
T,
is
the
lifetime
or
collision time
of
this
wave
type.
The thermal resistance
is the result
of
interchange
of
energy between
lattice
waves,
that
is,
scattering. Scattering
can be
caused
by
static
imperfections
and
anharmonicity.
Static
imperfections include grain
boundaries, vacancies,
interstitials
and
dislocations
and
their
associated
strain
fields,
which
considerably broadens the
defects
cross-
section. These
"
static
"
mechanisms
generally
become
less
important
at
high
temperature. Elastic strains
in the
crystal
scatter
because
of
the
strain dependence
of
the
elastic prop
-
erties, a
nonlinear
or
anharmonic
effect.
An
elastic strain
alters
the frequencies
of
the
lattice
waves.
DIFFUSION
AND
VISCOSITY
Diffusion
and
viscosity are activated processes and depend
more strongly on
temperature
and
pressure than
the
prop
-
erties
discussed
up to
now.
The
diffusion
of
atoms, the
mo
-
bility
of
defects,
the
creep
of
the mantle and seismic
wave
attenuation
are all controlled
by
the diffusivity.
where
G*
is the Gibbs free
energy
of
activation,
5
is
a
geo
-
metric factor
and
v
is the
attempt frequency
(an
atomic
vi
-
brational frequency). The
Gibbs
free
energy is
G*
=
E*
+
PV*
-
TS*
where
E*,
V*
and
S*
are
activation
energy, volume
and
entropy, respectively.
The
diffusivity
can therefore
be
written
D
=
Do
exp
-(E*
+
PV*)IRT
Do
=
(a2
v
exp
S*IRT
where
S*
is generally
in the range
R
to
SR.
The
theory
for
the volume
dependence
of
Do
is similar
to
that
for thermal
diffusivity,
K
=
3'CL/pCv.
It increases
with
depth
but the
variation is
small,
perhaps
an
order
of
magnitude,
compared
to
the effect
of
the exponential
term.
The product
of
YCL
times viscosity is involved
in
the Rayleigh number,
and
the
above
considerations
show
that the
temperature
and
pres
-
sure dependence
of
this product depend mainly
on the ex
-
ponential
terms.
The activation
parameters are
related to
the
derivative
of
the
rigidity
(Keyes, 1963):
The
effect
of
pressure
on
D
can
be
written
For
a typical
value
of
30
for
G*/RT
we
have
V*
de
-
creasing from
4.3
to
2.3
cm
3
/mole
with
depth in
the lower
mantle,
using
elastic properties from the
PREM
Earth
model.
We
also
have
-
(d
In
D/d
In
p),
=
48
to
40
This gives a decrease
in
diffusivity, and
an
increase
in
viscosity,
of
about a factor
of
60
to
80,
due
to compression,
across the lower mantle.
In
convection calculations
and
ge-
oid modeling
it is
common
practice to assume a constant
viscosity for the
lower
mantle. This is a poor assumption.
Viscosity
A
general expression for
viscosity,
q,
is
where
n
is a constant generally
between
1
and
3
and
cr
is
the nonhydrostatic stress. This gives
an
additional increase
of
viscosity
over
that contributed
by
the
diffusivity,
D,
un
-
less
u
decreases rapidly
with
depth. The general
tendency
of
q
to increase
with
depth
may
be
reversed
in
the
D
"
zone
due
to
a high
thermal gradient
and,
possibly,
an
increase
in
o.
The decrease
in
V*
with
depth
also
means
that compres
-
sion
-
induced viscosity increases
will be milder
at the
base
of
the mantle.
A
low
-
viscosity
D
"
layer
could reduce the
ability of mantle convection
to
deform
the
core
-
mantle
boundary.
The change
in
viscosity across a chemical or phase
boundary is
not
easily determined. The pre
-
exponential
term in the diffusivity
will
increase across a boundary that
involves
an
increase in
bulk modulus
and
mean
sound
ve
-
locity,
or
Debye
frequency. Therefore, the viscosity
will
de
-
crease
due
to
this
factor.
For
the same reason lattice con
-
ductivity
and
the thermal
diffusivity will
increase. The
activation
volume
of
the low
-
density
phase will
generally
be greater
than
that
of
the high
-
density phase. This
means
that the viscosity jump across a deep boundary
will be nega
-
tive
if both
phases
had the
same viscosity at zero pressure.
The
high temperature gradient in thermal boundary layers,
such
as
D
"
, the lithosphere and, possibly, near the
650
-
km
discontinuity will cause the
diffusivity
to increase. There
-
fore, viscosities
will tend
to
decrease
across
mantle
discon
-
tinuities unless the activation energies are
sufficiently lower
for the dense
phases
that the geothermal gradient can over
-
come the above
effects
or unless pre
-
exponential, crystal
structure, defect or nonhydrostatic stress considerations
play
an
appropriate role.
The
combination
of
physical parameters that enters
into
the
Rayleigh number,
al~v
(coefficient
of expansion
a,
thermal diffusivity
K
,
and
viscosity
q),
decreases rapidly
with
compression. The decrease through
the
mantle is
of
the order
of
lo6
to
lo7.
With
parameters appropriate for the
mantle, the increase
due to
temperature is
of
the order
of
lo6.
Therefore, there is a delicate balance
between
tempera
-
ture
and
pressure. The local
Rayleigh number
in
thermal
boundary
layers
increases because
of
the dominance
of
the
thermal gradient
over
the pressure gradient.
Diffusion
Diffusion
of
atoms
is important
in
a large
number
of
geo
-
chemical
and
geophysical problems: metamorphism, ele
-
ment
partitioning, creep, attenuation
of
seismic
waves,
electrical conductivity
and
viscosity
of
the
mantle.
Diffu
-
sion means
a local nonconvective
flux
of
matter
under the
action
of
a chemical
or
electrochemical potential gradient.
The
net
flux
J
of
atoms
of
one
species
in
a solid
is
related to the gradient
of
the concentration,
N,
of
this
species
J
=
-D
grad
N
where
D
is the
diffusion
constant or
diffusivity
and has
the
same dimensions
as
the thermal
diffusivity. This
is known
as
Fick's
law
and
is
analogous
to
the heat conduction
equation.
Usually the diffusion
process requires
that an
atom,
in
changing position, surmount a potential
energy
barrier.
If
the barrier
is
of
height
G*,
the
atom
will
have sufficient
energy to pass
over
the barrier only a fraction exp
(
-
G*/
RT)
of
the time. The
frequency
of
successes
is therefore
v
=
v,
exp
(-G*/RT)
where
v,
is the attempt
frequency, usually taken
as
the
atomic vibration, or
Debye,
frequency, which
is of
the or
-
der
of
1014
Hz.
The
diffusivity
can
then
be
written
where
[
is a geometric factor
that
depends
on
crystal struc
-
ture or coordination
and
that
gives
the jump
probability
in
the desired direction
and
a
is the
jump
distance or inter
-
atomic spacing.
The
factor
v
takes
into
account the
fact that
if an
atom
can jump equally
in
m
directions
(m
=
6
in a face
-
centered
cubic
lattice,
for example),
then
the distance
moved in
the
desired
direction is
different
for each
jump
direction. Also,
the atom can only
jump
to empty sites, that
is,
vacancies.
The probability for this
is
C,,
the vacancy
concentration,
where
where
G$
is the free energy
of
formation
of
a vacancy.
Diffusion, like
most
thermally
activated
processes,
exhibits a change
in
activation energy
between
high-tem-
perature
and
low
-
temperature regimes.
At
low
temperatures
the number
of
diffusing
ions is independent
of
temperature,
and
therefore the energy
of
formation is
not
involved.
This
is the extrinsic range
of
temperature.
Diffusion proceeds via
chemical vacancies
due to
non
-
stoichiometry
or
impurities,
which outnumber
the
thermally generated vacancies.
At
high
temperature thermally generated vacancies are
pro
-
duced,
and
the energy
of
formation
is
also involved. The
transition from the extrinsic
to the
intrinsic regime
usually
occurs at about
0.8
of
the
melting
temperature. The barrier
to extrinsic diffusion,
Gz,
is the
maximum
change
in
free
energy due to
the
lattice distortion associated
with
the
mo
-
tion
of
the
ion
from
its
lattice site into a neighboring
va
-
cancy.
Do
is generally greater
than
1
cm
2
/s for intrinsic dif
-
fusion
and much
less than
1
cm
2
/s in
the extrinsic range.
Motion
of
vacancies
(Schottky defects)
and
interstitials
(Frenkel defects) are important
in
ionic conductivity. The
ionic
conductivity
is given
by
where
No
is the total
number
of
ions
of
the appropriate spe
-
cies
per
unit volume
and
p
is
the fraction
of
ions able
to
move.
The frequency factor
v
and
the activation energy
can
be
found from
diffusion
experiments.
They
can also
be
found from
dynamic experiments
such
as anelastic or at
-
tenuation
measurements involving
elastic
waves.
An
ab
-
sorption peak, for example, occurs
when
wlv
=
1 or
w~
=
1.
The
activation energy
is found from the shift in the
ab
-
sorption peak with
temperature. Generally, solids exhibit a
series
of
absorption
peaks, one for each physical mecha
-
nism
or diffusing species. The theories
of
creep, or mantle
viscosity,
and
seismic
wave
attenuation
are intimately re
-
lated to theories
of
diffusion. For a mechanism to
be
impor
-
tant at
seismic frequencies
and
mantle temperatures, the fre
-
quency
factor
v
must
be
close
to
the seismic frequency
w.
v
can
be
estimated from
vo
and
G*, given
the
temperature,
pressure,
and atomic
species. The diffusive properties
of
the
common
rock
-
forming elements (Mg, Fe,
Ca, Al, Si
and
0)
are
of
most relevance
in
studies
of
mantle
creep
and
attenu
-
ation. Experimentally determined
diffusion
parameters
in
various oxides
and
silicates are
given
in
Table
7
-
3 and
7
-
4.
E*
is generally
well
determined
if the temperature range is
extensive enough,
but
Do
requires a long extrapolation.
In
curve
-
fitting diffusion
data
there
is a trade
-
off between
Do
and
E*.
Regions
of
lattice imperfections
in
a solid are regions
of
increased mobility.
Dislocations are therefore
high-
mobility
paths for diffusing species. The rate
of
diffusion
in
these regions
can
exceed the rate
of
volume
or
lattice dif
-
fusion.
In
general,
the
activation energy for
volume diffu
-
sion
is higher
than
for other
diffusion
mechanisms.
At
high
temperature, therefore,
volume diffusion
can
be
important.
In and near
grain boundaries
and
surfaces, the jump fre
-
quencies
and
diffusivities are also high. The activation en
-
ergy
for surface
diffusion
is related
to
the enthalpy
of
vaporization.
The
effect
of
pressure
on
diffusion
is given
by
the ac
-
tivation volume,
V*:
TABLE
7
-
3
Diffusion
in Silicate
Minerals
Diffusing
T
D
Mineral
Species
(K)
(m
21s)
--
Forsterite
Mg
298
2
x
10-l8
Si
298
10-19-10-21
0
1273
2
X
Zn2Si0,
Zn
1582
3.6
x
10-l5
Zircon
0
1553
1.4
x
10-l9
Enstatite
Mg
298
10-20-
10-21
0
1553
6
x
10-l6
Si
298
6.3
X
lo-"
Diopside
Al
1513
6
x
10-l6
Ca
1573
1.5
x
10-l5
0
1553
2.4
X
10-l6
Albite
Ca
523
10
-14
Na
868
8
x
10-1'
Orthoclase
Na
1123
5
x
10-l5
0
-
1000
10
-20
Freer
(1981).
The second term can
be
estimated from lattice dynamics
and
pressure dependence
of
the lattice constant
and
elastic
moduli.
This term
is
generally small.
V*
is
usually
of
the
order
of
the atomic
volume
of
the
diffusing
species.
The
activation
volume
is also
made
up
of two
parts, the forma
-
tional part
VT
and the
migrational part
VZ.
Ordinarily the
temperature
and
pressure dependence
of
a
and
v
are small.
For
a vacancy mechanism
VT
is simply the atomic
vol
-
ume
since a
vacancy
is formed
by
removing
an
atom. This
holds
if there
is
no relaxation
of
the crystal about the
va
-
cancy.
Inevitably there
must be
some relaxation
of
neigh
-
boring atoms
inward about
a vacancy and
outward about
an
interstitial,
but
these effects are small.
In
order to move, an
atom
must
squeeze
through
the lattice,
and
V;
can also
be
expected to
about an
atomic volume.
The
work
performed
in
creating a lattice imperfection
can
be
estimated from elasticity theory
if the lattice can
be
treated
as
a continuum. In
this
case
where
G is the shear
modulus and
KT
is the isothermal
bulk
modulus
(Keyes, 1963). The magnitude
of
{G),,
from the
previous chapter, is
about
3.
The
Griineisen
assumption,
that all vibrational frequencies
of
the lattice depend on
vol
-
ume
in the same
way,
gives
(d
In
Kld
In
V),
=
-
(2y
+
113)
=
(d
In
G/d
In
V),
and
Borelius (1960) obtained a similar expression,
but
a
smaller effect,
by assuming
that the increase
in
volume
dur-
ing
the self
-
diffusion act is equal to the
volume
of
the
hard
atomic core:
V*/G*
=
-
113(K,)-'(a
In
K,ld
In
V),
=
{K,},/3K,
The activation entropy S* can also
be
estimated from
a strain energy
model
(Keyes, 1960):
S*/G*
=
-a[(d
In
Gld
In
V),
+
11
V*/S*
=
(aKT)-'
[(d
-
ln
G/d
In
V),
+
11
+
[(a
In
G/d
In
V),
+
11
With the Griineisen assumption, that
the
rigidity G depends
on temperature
and
pressure only through the volume,
and
Some
of
these approximations are
not
necessary for
upper-
mantle minerals since
(In
G/d
In
V),
=
{G},
and
(8
ln
GI
d
In
V),
=
{G),
are
both
known.
Sammis
and
others (1977) estimated
values
for
V*
of
about
8
to
12 cm
3
/mole
for oxygen self
-
diffusion
in
olivine
TABLE
7
-
4
Diffusion Parameters
in Silicate
Minerals
Diffusing
Do
Mineral
Q
Species
(m
'1s)
(kJ
mol-I)
Olivine
Garnet
Ca,Si04
CaSiO,
Albite
Orthoclase
Nepheline
Glass
Albite
Orthoclase
Basalt
Mg
Fe
0
Si
Fe
-
Mg
Sm
Fe
-
Mg
Ca
Ca
Na
0
Na
0
Na
Ca
Ca
Ca
Na
Freer
(1981).
and
about
3 to
5
cm
3
/mole
in
the
lower
mantle,
decreasing
with
depth. Another estimate for activation
volume
of
about
2
cm
3
/mole
for the
lower mantle was based
on seismic at
-
tenuation data (Anderson, 1967).
Of
course the
V*
for
at
-
tenuation
may
differ from the
V*
for creep since
different
diffusing
species
may be
involved. Nevertheless, the
effect
of
pressure on ionic volumes leads one to expect that
V*
will
decrease
with
depth
and,
therefore,
that
activated
pro
-
cesses became less sensitive
to
pressure
at high
pressure.
Indeed,
both
viscosity
and
seismic factor
Q
do
not
appear
to increase rapidly
with
depth
in
the
lower
mantle.
HOMOLOGOUS
TEMPERATURE
The ratio
E*/T,
is nearly
constant for a
variety
of
materi
-
als,
though there
is some dependence on
valency
and
crystal
structure. Thus, the factor
E*/T
in
the
exponent for acti
-
vated
processes can
be
written
AT,IT
where
X
is roughly
a
constant
and
T,
is the
"
melting temperature.
"
If
this
rela
-
tion
is assumed to
hold
at
high
pressure,
then
the
effect
of
pressure on G*, that is, the activation
volume
V*,
can
be
estimated from the
effect
of
pressure on the
melting
point:
D(P,T)
=
Do
exp
[
-
XT,(P)lRT]
and
which,
invoking
the Lindemann
law,
becomes
which
is similar to expressions given
in
the previous sec
-
tion. The temperature
T,
normalized
by
the
"
melting tem
-
perature,
"
T,,
is known
as
the
homologous
temperature.
It
is often
assumed
that activated properties depend
only
on
TIT,
and
that the
effect
of
pressure
on
these properties
can
be
estimated
from
T,(P).
The
melting point
of
a solid is related to the equilib
-
rium
between the
solid
and its melt and not
to
tb
properties
of
the solid alone.
Various
theories
of
melting
have
been
proposed that
involve
lattice instabilities, critical
vacancy
concentrations or dislocation densities, or amplitudes
of
atomic motions. These are not true theories
of
melting since
they
ignore the properties
of
the
melt phase,
which must be
in
equilibrium
with
the solid at the
melting
point.
DISLOCATIONS
Dislocations are extended imperfections
in
the
crystal lat
-
tice
and
occur
in
most
natural crystals.
They can
result from
the crystal
growth
process
itself
or
by
deformation
of
the