Geoscience Reference
In-Depth Information
Threshold
Activation
energy
E
Position
Figure 5.2
The activation energy threshold of diffusion. The maxima represent intermediate energy barriers
that the thermally activated atoms need to pass in order to reach the next stable lattice positions
represented by the minima.
where the diffusion coefficient or diffusivity
D
i
Pl
2
2 is expressed in square meters per
unit time, i.e. the frequency of vibrations
=
/
of the atom around its resting position; and (b)
the proportion of atoms with enough energy to cross the energy threshold
E
i
, known as the
activation energy, dependent on the nature of the atom and the atomic configuration of the
site. From Boltzmann's law, this proportion is equal to exp
−
ν
RT
and the temperature
E
i
/
dependence of the diffusion coefficient is therefore expressed:
D
i
0
exp
E
i
RT
D
i
=
−
(5.8)
where
D
i
0
contains the vibrational term. If we introduce the logarithm of
D
i
as a function of
the inverse 1
/
T
of absolute temperature (Arrhenius plot), we obtain a straight line of slope
E
i
−
R
. A few values of the diffusion coefficient in water, magmatic liquids, and minerals
Arrhenius lines for different elements to fan out from a common point at high temperature
reflects an inverse relationship between the activation energy and the pre-exponential factor
and is known as the Meyer-Neldel, or Winchell, “compensation” law.
The physical meaning of the diffusion coefficient can also be understood in a different
/
C
i
∂
J
i
∂
∂
=−
∂
(5.9)
t
x
where the diffusive flux
J
i
of species
i
is defined as:
C
i
∂
D
i
∂
J
i
=−
(5.10)
x
This shows that the transport of
i
by diffusion is proportional to the concentration gradi-
ent
C
i
x
and that the diffusivity
D
i
∂
/
∂
is the constant of proportionality. The minus sign
