of 8
Chapter
21
Nonelastic
and
transport
properties
Shall
not
every
rock
be
removed
out
of
his
place?
Job
18:4
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
deform.ations.
Most
lab-
oratory
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
processes
that
are
related
to
rates
or
time.
Some
of
these
are
more
depen-
dent
on
temperature
than
those
treated
in
pre-
vious
chapters.
These
properties
give
to
geology
the
'
arrow
of
time'
and
an
irreversible
nature.
Th
er mal
conduc
t
iv
ity
There
are
three
mechanisms
contributing
to
thermal
conductivity
in
the
crust
and
mantle
.
The
lattice
part
is
produced
by
diffusion
of
ther-
mal
vibrations
in
a
crystalline
lattice
and
is
also
called
the
phonon
contribution
.
The
radiative
p a
rt
is
due
to
the
transfer
of
heat
by
infrared
elec-
tromagnetic
waves
,
if
the
mantle
is
sufficiently
tran
s
parent.
The
exciton
part,
is
due
to
the
trans-
port
of
energy
by
quasi
particles
composed
of
elec-
trons
and
positive
holes
;
this
becomes
dominant
in
intrinsic
semiconductors
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
.
Debye
theory
regards
a
solid
as
a
system
of
coupled
oscillators
transmitting
thermoelas-
tic
waves.
For
an
ideal
lattice
with
simple
har-
monic
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
conductivity
. Ther-
mal
conductivity
is
related
to
higher-order
terms
in
the
potential
and
should
be
correlated
with
thermal
expansion
.
Lattice
conductivity
can
be
viewed
as
the
exchange
of
energy
between
high-
frequency
lattice
vibrations
-
elastic
waves
.
An
approximate
theory
for
the
lattice
conductivity,
consistent
with
the
Griineisen
approximation
,
gives
K L
=
a
j
3yz
T
~
/
2Pt
;
z
where
a
is
the
lattice
parameter
,
y
is
the
Griineisen
parameter,
T
is
temperature
,
KT
is
the
isothermal
bulk
modulus
and
p
is
density.
This
is
valid
at
high
temperatures,
relative
to
the
Debye
temperature
.
This
relation
predicts
that
the
ther-
mal
conductivity
decreases
with
depth
in
the
top
part
of
the
upper
mantle.
The
thermal
conductivities
of
various
rock-
forming
minerals
are
given
in
Table
21.1.
Note
that
th
e
crust-forming
minerals
have
about
one-
half
to
one-third
of
the
conductivity
of
man-
tle
minerals
.
This
plus
the
cracks
present
at
low
crustal
pressures
means
that
a
much
higher
thermal
gradient
is
maintained
in
the
crust
,
Table
21
. 1
I
Thermal
conductivity
of
minerals
M ineral
A lbite
Anort
hi
te
M icrocli
ne
Serpe
nt i
ne
Diopside
Forsterite
Bronz
i
te
j adeite
Grossular
i
te
O l
iv
i
ne
O r thopyroxe
ne
T herma
l
Conductiv
i
ty
I
o-
3
ca
l/cm
s oc
4.7 1
3.67
5.90
7.05
1
1.79
13.32
9.99
15.92
13.49
6.7-
13.6
8.
1
6-
15.3
Horai
(1971)
, Kobay
z
shi
gy
(1974).
relative
to
the
mantle
,
to
sustain
the
same
con·
ducted
heat
flux
.
The
gradient
can
be
higher
still
in
sediments.
The
thermal
gradient
decreases
with
depth
in
the
Earth
.
If
the
crustal
radioactivity
and
mantle
heat
flow
are
constant
and
the
effects
of
temper-
ature
are
ignored,
regions
of
thick
crust
should
have
relatively
high
upper-mantle
temperatures
.
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
[100]
for
olivine
and
[001]
for
orthopyroxene.
The
most
conductive
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
orthopy-
roxene
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
inter-
mediate
P-axis
of
orthopyroxene.
These
axes,
in
turn,
tend
to
line
up
in
the
flow
direction,
which
is
in
the
horizontal
plane
in
ophiolite
sections
.
The
vertical
conductivity
in
such
situa-
tions
is
much
less
than
the
average
conductivity
computed
for
mineral
aggregates.
Conductivity
decr
e
ases
with
temperature
and
may
be
only
half
this
value
at
the
base
of
the
lithosphere
.
The
implications
of
this
anisotropy
in
thermal
con-
ductivity
and
the
lower
than
average
vertical
con-
THERMAL
CONDUCTIVITY
275
ductivity
have
not
been
investigated.
Two
obviou
s
implications
are
that
the
lithosphere
can
sup-
port
a
higher
thermal
gradient
than
g
enerally
supposed,
g
iving
higher
upper-mantle
tempera·
tures
,
and
that
the
thermal
lithosphere
grows
less
rapidly
than
previously
calculated.
For
exam-
ple,
the
thermal
lithosphere
at
80
Ma
can
be
100
km
thick
for
K
=
0.01
cal/cm
s
°C
and
only
30
km
thick
for
0 .3
cal
/
cm
s
o
c.
The
low
lattice
conductivity
of
the
oceanic
crust
is
usu-
ally
also
ig
nored
in
these
calculations,
but
this
may
be
counterbalanced
by
water
circulation
in
the
crust
.
The
lattice
(phonon)
contribution
to
the
ther-
mal
conductivity
decreases
with
temper
a
tur
e ,
but
at
high
temperature
radiative
transfer
of
heat
may
become
significant,
depending
on
the
opaci-
ties
of
mantle
rocks,
which
depend
on
grain
size
and
iron
content
.
Convection
is
probably
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
trans-
ported
across
thermal
boundary
layers
by
con-
duction
.
Thermal
boundary
layers
exist
at
the
surface
of
the
Earth,
at
the
core-mantle
bound-
ary
and,
possibly,
at
chemical
interfaces
internal
to
the
mantle
.
Conduction
is
also
the
mechanism
by
which
subducting
slabs
cool
the
mantle
,
and
become
heated
up
.
The
thicknesses
and
thermal
time
constants
of
boundary
layers
are
controlled
by
the
thermal
conductivity
,
and
these
regulate
the
rate
at
which
the
mantle
cools
and
the
rate
at
which
the
th
e
rmal
lithosphere
grows.
The
impor·
tance
of
radiative
conductivity
in
the
deep
man-
tle
is
essenti
a lly
unconstrained
.
The
thermal
conductivity
goes
throu
gh a
min-
imum
at
about
100
km
depth
in
the
upper
man-
tle.
Higher
thermal
gradients
are
then
needed
to
conduct
the
sa
me
amount
of
heat
out
,
and
this
results
in
a
further
lowering
of
the
lattice
con
-
ductivity.
The
conductivity
probably
increase
s by
at
least
a
factor
of
3
to
4
from
100
km
d e
pth
to
the
core-mantle
boundary
(CMB).
The
parameters
that
enter
int
o
a
theor
y
of
lattice
conductivity
are
fairly
obviou
s;
tempera-
ture,
specific
heat
and
the
coefficient
of
thermal
expansion,
some
measure
of
anharmonicity,
a
measure
of
a
mean
free
path
or
a
mean
276
NONELASTIC
AND
TRANSPORT
PROPERTIES
Table
21.2
I
Estimates
of
lattice
thermal
diffusivity
in
the
mantle
De pth
K
(km
)
(c
m
2
/s)
50
5
.9
X
1
o-
3
IS
O
3.0
X
10 -
3
300
2.9
X
10
-
3
400
4.
7
X
10 -
3
650
7.5
X
10
-
3
1
200
7.7
X
1
o-
3
2400
8
.1
X
1
o-
3
2900
8.4
X
1
o-
3
Hor
ai a
nd
Simmons
(1970).
collision
time
or
a
measure
of
the
strength
and
distribution
of
scatterers
, velocities
of
sound
waves
and
the
interatomic
distances
.
Both
thermal
conductivity
and
thermal
expansion
depend
on
the
anharmonicity
of
the
interatomic
potential
and
therefore
on
dimen-
sionless
measures
of
anharmonicity
such
as
y
or
a y
T.
The
lattice
or
phonon
conductivity
is
Where
V
is
the
mean
sound
speed,
l
is
the
mean
free
path,
which
depends
on
the
interatomic
dis-
tances
and
the
isothermal
bulk
modulus,
KT
.
This
gives
(8
lnKL
/8
ln
p ]
=
(8
lnKT
/8
ln
p ]
-
2(
/J
ln
yf8
ln
p ]
+
y
- 1
/3
For
lower-mantle
properties
this
expression
is
dominated
by
the
bulk
modulus
term
and
the
variation
of
KL
with
density
is
expected
to
be
sim-
ilar
to
the
variation
of
Ky.
The
lattice
conductivity
decreases
approx-
imately
linearly
with
temperature,
a
well-
known
result,
but
increases
rapidly
with
den-
sity.
The
temperature
effect
dominates
in
the
shallow
mantle,
but
pressure
dominates
in
the
lower
mantle
.
This
has
important
implications
regarding
the
properties
of
thermal
boundary
layers
,
the
ability
of
the
lower
mantle
to
con-
duct
heat
from
the
core,
and
the
convective
mode
of
the
lower
mantle
.
The
spin-pairing
and
post-perovskite
transitions
in
the
deep
mantle
may
cause
a
large
increase
in
lattice
conductiv-
ity
.
Other
pressure
effects
on
physical
properties-
viscosity,
thermal
expansion
-
all
go
in
the
direc-
tion
of
suppressing
thermal
instabilities-
narrow
plumes
-
at
the
core-mantle
boundary
.
The
ratio
a / K L
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
prop-
erties
is
to
make
convection
sluggish
in
the
lower
mantle,
to
decrease
thermally
induced
buoyancy
,
to
increase
the
likelihood
of
chemical
stratifica-
tion,
and
to
increase
the
thickness
of
the
ther-
mal
boundary
layer
in
D" ,
above
the
core-mantle
boundary.
The
mechanism
for
transfer
of
thermal
energy
is
generally
well
understood
in
terms
of
lat-
tice
vibrations
,
or
high-frequency
sound
waves.
This
is
not
enough,
however,
since
thermal
con-
ductivity
would
be
infinite
in
an
ideal
har-
monic
crystal.
We
must
understand,
in
addition
,
the
mechanisms
for
scattering
thermal
energy
and
for
redistributing
the
energy
among
the
modes
and
frequencies
in
a
crystal
so
that
ther-
mal
equilibrium
can
prevail.
An
understanding
of
thermal
'resistivity,'
therefore,
requires
an
understanding
of
higher
order
effects,
including
anharmonicity
.
Debye
theory
explains
the
thermal
conductiv-
ity
of
dielectric
or
insulating
solids
in
the
follow-
ing
way.
The
lattice
vibrations
can
be
resolved
into
traveling
waves
that
carry
heat
.
Because
of
anharmonicities
the
thermal
fluctuations
in
density
lead
to
local
fluctuations
in
the
veloc-
ity
of
lattice
waves,
which
are
therefore
scat-
tered.
Simple
lattice
theory
provides
estimates
of
specifi
c
heat
and
sound
velocity
and
how
they
vary
with
temperature
and
volume.
The
theory
of
attenuation
of
lattice
waves
involves
an
under
-
st
a
nding
of
how
thermal
equilibrium
is
attained
and
how
momentum
is
transferred
among
lattice
vibrations
.
The
thermal
resistance
is
the
result
of
interchan
ge
of
energy
between
lattice
waves
,
that
is ,
scattering.
Scattering
can
be
caused
by
static
imperfections
and
anharmonicity.
Static
imperfections
include
g
rain
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
properties,
a
nonlinear
or
anharmonic
effect.
Di
ff
usi
on
a
nd
v
isc
os
it
y
Diffusion
and
viscosity
are
activated
processes
and
depend
more
strongly
on
temperature
and
pressure
than
the
properties
discussed
up
to
now.
The
diffusion
of
atoms.
the
mobility
of
defects,
the
creep
of
the
mantle
and
seismic
wave
atten-
uation
are
all
controlled
by
the
diffusivity
.
D(P
,
T)
=
~a
2
v
exp[
- G*
(P
,
T)
I
Ril
where
G* is
the
Gibbs
free
energy
of
activation,
~
is
a
geometric
factor
and
v
is
the
attempt
frequency
(an
atomic
vibrational
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
=
D
0
exp-(E
*
+
PV
*)I
RT
Do
=
~a
2
v
exp
S *
I
RT
Typical
Do
values
are
in
Table
21.4.
The
theory
for
the
volume
dependence
of
D
0
is
similar
to
that
for
thermal
diffusivity,
K
=
K
L
1
pC
v .
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
KL
times
viscosity
is
involved
in
the
Rayleigh
number,
and
the
above
considerations
show
that
the
temper-
ature
and
pressure
dependence
of
this
product
depend
mainly
on
the
exponential
terms.
The
activation
parameters
are
related
to
the
derivative
of
the
rigidity
(Keyes,
1963)
:
v
*I
G
*
=
(I
1
K
T)
[ (
a
In
G
)
-
1
J
a
In
p
T
DIFFUSION
AND
VISCOSITY
277
The
effect
of
pressure
on
D
can
be
written
_
RT(alnG)
=V*
KT
a
ln
p
T
1
[(
a
ln
G)
J
=
K
T
a
ln
p
T
-
1
G
•
or
(
a
ln
D )
G
•
[
(a
ln
G )
J
a
ln
p
T
=
-
RT
a
ln
p
T
-
1
For
a
typical
value
of
30
for
G*
IRT
we
have
V*
decreasing
from
4.3
to
2.3
cm
3
fmole
with
depth
in
the
lower
mantle.
This
gives
a
decrease
in
dif-
fusivity,
and
an
increase
in
viscosity,
due
to
com-
pression,
across
the
lower
mantle
.
Phase
changes,
chemical
changes,
and
temperature
also
affect
these
parameters.
V isc
o
sity
There
are
also
effects
on
viscosity
that
depend
on
composition,
defects,
stress
and
grain
size.
Viscosities
will
tend
to
decrease
across
mantle
chemical
discontinuities,
because
of
the
high
thermal
gradient,
unless
the
activation
energies
are
low
for
the
dense
phases.
Even
if
the
viscosi-
ties
of
two
materials
are
the
same
at
surface
con-
ditions,
the
viscosity
contrast
at
the
boundary
depends
on
the
integrated
effects
ofT
and
P,
and
the
activation
energies
and
volumes.
The
combination
of
physical
parameters
that
enters
into
the
Rayleigh
number
decreases
rapidly
with
compression
.
The
decrease
through
the
mantle
due
to
pressure,
and
the
possibil
-
ity
of
layered
convection,
may
be
of
the
order
of
10
6
to
10
7
•
The
increase
due
to
temperature
mostly
offsets
this;
there
is
a
delicate
balance
between
temperature,
pressure
and
stable
stratifi-
cation.
All
things
considered,
the
Rayleigh
num-
ber
of
deep
mantle
layers
may
be
as
low
as
10
4
•
The
local
Rayleigh
number
in
thermal
bound-
ary
layers
increases
because
of
the
dominance
of
the
thermal
gradient
over
the
pressure
gra-
dient.
Most
convection
calculations,
and
convec-
tive
mixing
calculations,
use
Rayleigh
numbers
appropriate
for
whole-mantle
convection
and
no
pressure
effects.
These
can
be
high
by
many
orders
of
magnitude
.
The
mantle
is
unlikely
to
278
NONELASTIC
AND
TRANSPORT
PROPERTIES
be
vigorously
convecting
or
well-stirred
by
con-
vection.
Diffusion
Di
ffusion
of
atoms
is
important
in
a
large
num-
ber
of
geochemical
and
geophysical
problems:
metamorphism,
element
partitioning
,
creep,
attenuation
of
seismic
waves,
electrical
conduc-
tivity
and
viscosity
of
the
mantle.
Diffusion
means
a
local
non-convective
flux
of
matter
under
the
action
of
a
chemical
or
electrochemi-
cal
potential
gradient
.
The
net
flux]
of
atoms
of
one
species
in
a
solid
is
related
to
the
gradient
of
the
concentration,
N,
of
this
species
]
=
- D
grad
N
where
D
is
the
diffusion
constant
or
diffusivity
and
has
the
same
dimensions
as
the
thermal
diffusivity.
TI1is
is
known
as
Pick'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
the
height
G* ,
the
atom
will
have
sufficient
energy
to
pass
over
the
barrier
only
a
fraction
exp
(
- G*
j
RT)
of
the
time.
The
frequency
of
successes
is
therefore
v
=
V
0
exp(
- G
*
j
RT)
where
v
0
is
the
attempt
frequency,
usually
taken
as
the
atomic
vibration,
or
Debye,
frequency,
which
is
of
the
order
of
10
14
Hz
.
The
diffusivity
can
then
be
written
where
~
is
a
geometric
factor
that
depends
on
crystal
structure
or
coordination
and
that
gives
the
jump
probability
in
the
desired
direction
and
a is
the
jump
distance
or
interatomic
spacing.
Regions
of
lattice
imperfections
in
a
solid
are
regions
of
increased
mobility.
Dislocations
are
therefore
high-mobility
paths
for
diffusing
species.
TI1e
rate
of
diffusion
in
these
regions
can
exceed
the
rate
of
volume
or
lattice
diffusion
.
In
general,
the
activation
energy
for
volume
diffu-
sion
is
higher
than
for
other
diffusion
mecha-
nisms.
At
high
temperature
,
therefore,
volume
diffusion
can
be
important.
In
and
near
grain
boundaries
and
surfaces
,
the
jump
frequencies
Table
21.3
I
Diffusion
in
silicate
minerals
Diffus
i
ng
T
D
Mi
ner
al
Species
(K)
(m
2
/s
)
Forsterite
Mg
298
2 x
l o-
18
Si
298
I o
-19
- I o
-21
0
1273
2
X
I 0-
20
Zn
2Si0 4
Zn
1582
3.6
x
I 0-
15
Z ircon
0
1553
1
.4x
l o-
19
Enstatite
Mg
298
I o-2o
_ l o
-21
0
1553
6
X
10-
16
Si
298
6.3
x
1
o-
22
Diops
ide
AI
1513
6
X
10-
16
Ca
1573
1
.s
x
1 o-
15
0
1553
2.4
x
1 o-
16
A lbite
Ca
523
l o
-14
N a
8
68
8
X
10- l?
O rthoclase
N a
11
23
5 x
1 o-
15
0
~
100
0
l o-2o
Fre
er
(1981)
.
and
diffusivities
are
also
high.
The
activation
energy
for
surface
diffusion
is
related
to
the
enthalpy
of
vaporization.
The
effect
of
pressure
on
diffusion
is
given
by
the
activation
volume,
V*:
V *
=
RT(
a
In
D
ja
Ph-
RT
( a
ln
sa
2
v)
aP
T
The
second
term
can
be
estimated
from
lattice
dynamics
and
pressure
dependence
of
the
lattice
constant
and
elastic
moduli.
This
term
is
gener-
ally
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
,
and
the
migrational
part.
For
a
vacancy
mechanism
the
V*
of
formation
is
simply
the
atomic
volume
since
a
vacancy
is
formed
by
removing
an
atom.
This
holds
if
there
is
no
relaxation
of
the
crystal
about
the
vacancy.
Inevitably
there
must
be
some
relax-
ation
of
neighboring
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
the
migra-
tional
V*
can
also
be
expected
to
about
an
atomic
volume.