Geoscience Reference
In-Depth Information
minerals, the conductivities are shown after nor-
malization to 0.01 wt %, i.e., the conductivity
of hydrous minerals is normalized by
σ
hyd
0.01
C
W
,
hyd
where
σ
hyd
is the conductivity of a hydrous min-
eral and
C
w
,
hyd
is the water content of the hy-
drous minerals (in wt %). Because the densities
of these minerals are similar (
any point of the material must be fixed by the
local thermodynamic equilibrium. This is much
the same way as the case of diffusion creep where
the stoichiometry is preserved during diffusion
(see Karato, 2008a, ch. 8). Consequently, among
the various hydrogen-related species, it is the
slowest moving species that controls the ''self-
diffusion coefficient'' measured by the isotopic
exchange. In contrast, when the electric charge
is transported by diffusion (ionic conductivity),
diffusion of each species occurs keeping their
concentrations unchanged and there is no direct
interaction among the diffusing species in this
case (except at the electrodes that leads to the ca-
pacitor behavior (Figure 5.3). Consequently, the
effective diffusion coefficient in ionic conductiv-
ity is simply the (weighted) arithmetic average
as seen from Equation (5.11). If one uses the
self-diffusion coefficient measured from the iso-
topic exchange, one would under-estimate the
true conductivity due to hydrogen. Also, because
the slower moving species generally has higher
activation enthalpy, one expects that the acti-
vation enthalpy for electrical conductivity due
to hydrogen is smaller than that for the ''self-
diffusion coefficient.'' These are exactly what one
finds when one compares the results of ''self-
diffusion'' measurements and hydrogen-enhanced
conductivity (Du Frane&Tyburczy, 2012). There-
fore a comparison of these two properties in fact
supports the hybrid model for hydrogen related
defects in olivine (Figure 5.9). It is misleading to
conclude, from such a comparison, that hydrogen
does not enhance conductivity enough to explain
geophysically observed conductivity. Two differ-
ent averaging schemes are involved in these two
properties, one should not compare them directly.
(Hier-Majumder
et al
. (2005) argued that (2
H
)
M
does not contribute to electrical conductivity be-
cause it is a neutral defect. This statement is
incorrect. (2
H
)
M
is a neutral defect
relative to
the perfect lattice
, but does have electric charge
relative to the vacuum
.Appliedelectricfieldex-
erts a force on each ion in an ionic crystal that
has neutral charge relative to the prefect lattice
but has electric charge relative to the vacuum.
This leads to the well-known phenomena such as
0.3 g/cm
3
)and
the charge of proton is common, a comparison
of the conductivity after this normalization is
essentially a comparison of the mobility of hy-
drogen. The results show a systematic trend in
normalized conductivity, i.e., hydrogen mobil-
ity. Hydrogen mobility in hydrous minerals is
lower than that in nominally anhydrous miner-
als. Among nominally anhydrous minerals, the
following trends can also be seen. Nominally an-
hydrous minerals with high hydrogen solubility
(e.g., wadsleyite and ringwoodite) have low hydro-
gen mobility that is consistent with the previous
discussion (see Figure 5.9). Schmidbauer
et al
.
(2000) provided clear evidence that electrical con-
ductivity in (Fe-rich) amphibole is caused by ferric
iron rather than proton, a result consistent with
our interpretation of low mobility of hydrogen
in this mineral. Also, minerals that have
r
∼
3
±
1
tend to have higher mobility than those with
r
∼
0.7. Again this supports the previous
discussion that the difference in
r
may be due to
the different hydrogen mobility.
Du Frane and Tyburczy (2012) determined the
self-diffusion coefficient (rather than the chemi-
cal diffusion coefficients), i.e., the diffusion coeffi-
cient corresponding to the exchange of H with D.
The diffusion coefficient in Equation (5.7) is the
self-diffusion coefficient and therefore the appli-
cation of their results to electrical conductivity
is more direct than those of chemical diffusion
coefficient. However, even the self-diffusion coef-
ficient corresponding to isotopic exchange cannot
be directly applied to electrical conductivity if
there are multiple hydrogen-related species and
their mobilities are different. In these cases, when
the kinetics of H-D exchange is studied to mea-
sure ''the self-diffusion coefficient of hydrogen,''
one measures the
harmonic average
of diffusion
coefficients of various species. This is due to the
fact that the concentrations of various species at
=
0.6
−
