Geoscience Reference
In-Depth Information
dissolved in olivine mainly as a neutral defect,
(2
H
)
M
(Kohlstedt
et al
., 1996). In this model, the
electrical conductivity increases linearly with
water content but independent of oxygen fugac-
ity. Also, the activation energy and anisotropy
of electrical conductivity (at a fixed hydrogen
content) are the same as those of diffusion.
As we will discuss later, the subsequent studies
demonstrated that the dissolution of hydrogen
indeed enhances the electrical conductivity in
most minerals. However, these studies also
showed that some modifications are needed to
the initial idea (the activation energy of electrical
conductivity is less than that of diffusion, the
water content exponent is often less than 1,
conductivity is sensitive to oxygen fugacity). An
important point is that Equation (5.7) must be
generalized where multiple charged species take
part in electrical conductivity, i.e.,
to ferric iron concentration, and the relation
similar to Equation (5.8) or (5.9) holds (the
same oxygen fugacity dependence) although the
activation energy for conduction is the same as
that for Mg (Fe) diffusion.
Also there is an interaction between ionic and
electronic defects in minerals. When charged
ionic defects are introduced to a mineral,
these defects create electrons or holes. Such a
reaction is well known for iron-related defects
(Reaction (5.4)). (Wang
et al
., 2011) showed
that a similar situation will arise for charged
hydrogen-related defects. Consequently, ionic
and electronic defects need to be considered
simultaneously.
These issues will be discussed later when we
review experimental and theoretical results.
5.2.3 Electrical conductivity of a multi-phase
aggregate
f
i
D
i
n
i
q
i
RT
σ
=
(5.11)
(a) Averaging scheme
Rocks are made of vari-
ousminerals with different orientations including
some fluid phases and grain-boundaries. When
different phases have largely different electrical
conductivity, then the averaging scheme becomes
an important issue. McLachlan
et al
. (1990) re-
viewed such an averaging scheme. In most of
practical purposes, the Hashin-Shtrikman's up-
per bound provide good estimates of the electrical
conductivity of a mixture. The upper (or lower)
bound is given by
i
where a quantity with a suffix
i
refers to a
quantity for the
i
-th species (e.g.,
D
i
is the (self)
diffusion coefficient of the
i
th species). Even
for a given element such as hydrogen, there are
multiple hydrogen-related species in a given ma-
terial at a given condition as shown by Nishihara
et al
. (2008) for wadsleyite. In these cases, the
hydrogen-related species that contributes mostly
to conductivity is not necessarily the most
abundant hydrogen-bearing species and therefore
the electrical conductivity is not necessarily
linearly proportional to the total hydrogen (water)
concentration.
The addition of hydrogen will also enhance
electrical conductivity through the enhancement
of diffusion of other species such as Mg (Fe)
(Hier-Majumder
et al
., 2005). Diffusion of Fe (Mg)
in olivine or other iron-bearing minerals occurs,
in most cases, by the vacancy mechanism,
i.e., through the exchange of Fe (Mg) with the
M-site (Mg or Fe-site) vacancies,
V
M
. The charge
balance in these minerals is dominated by M-site
vacancies and ferric iron, [
Fe
•
M
]
A
+
/
−
σ
+
/
−
HS
=
σ
n
+
(5.12)
A
+
−
3
σ
n
/
1
−
n
1
−
f
i
with
A
+
/
−
=
where
f
i
is the vol-
1
(
σ
i
−
1
3
σ
n
+
σ
n
)
i
=
1
ume fraction of the
i
-th component,
σ
i
is the
conductivity of the
i
th component, and
σ
n
is the
maximum (or minimum) conductivity (Hashin
& Shtrikman, 1962). When we apply the re-
sults of electrical conductivity measurements for
individual minerals to calculate the electrical
conductivity of a rock, we will use these rela-
tionships. (Simpson & Tommasi, 2005) discussed
the application of such a model to calculate
−
2[
V
M
]. In these
cases, the electrical conductivity is proportional
=
