Geoscience Reference
In-Depth Information
be vigorously convecting or well-stirred by con-
vection.
Table 21.3
Diffusion in silicate minerals
Diffusing
T
D
Diffusion
Diffusion 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
J
of atoms of one species in a solid
is related to the gradient of the concentration,
N
,
of this species
(m
2
/s)
Mineral
Species
(K)
Forsterite
Mg
298
2
×
10
−
18
Si
298
10
−
19
−
10
−
21
O
1273
2
×
10
−
20
Zn
2
SiO
4
Zn
1582
3.6
×
10
−
15
Zircon
O
1553
1.4
×
10
−
19
Enstatite
Mg
298
10
−
20
−
10
−
21
O
1553
6
×
10
−
16
Si
298
6.3
×
10
−
22
10
−
16
Diopside
Al
1513
6
×
10
−
15
Ca
1573
1.5
×
10
−
16
O
1553
2.4
×
10
−
14
Albite
Ca
523
J
=−
D
grad
N
10
−
17
Na
868
8
×
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 the height
G
∗
,the
atom will have sufficient energy to pass over the
barrier only a fraction exp (
10
−
15
Orthoclase
Na
1123
5
×
10
−
20
O
1000
∼
Freer (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
∗
:
G
∗
/
RT
) of the time.
The frequency of successes is therefore
−
v
=
v
o
exp(
−
G
∗
/
RT
)
RT
∂
ln
ζ
a
2
v
V
∗
=
RT
(
∂
ln
D
/∂
P
)
T
−
where
v
o
is the attempt frequency, usually taken
as
∂
P
T
the
atomic
vibration,
or
Debye,
frequency,
which is of the order of 10
14
Hz. The diffusivity
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 then be written
va
2
D
=
ζ
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. The 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
ζ
can also be expected to about an atomic
volume.
