118
3 – Transport in ionic solids
3. Propose a more precise method involving electrochemical devices to evaluate
the electronic part of the conduction for this protonic conductor.
Exercise 3.14 – Free-volume model
1. State the four hypotheses used in developing the free-volume model applied
to the variation in ionic conductivity of molten electrolytes and vitrifiable
mixtures above their glass transition temperature T G .
2. Consider an electrolyte whose cationic conductivity is due to the species M
+
(Li
+
, Na
+
, K
+
, Ag
+
, …).
a. Express the hopping probability P of this species as a function of average
free volume V f .
b. Assume that the average free volume V f is zero at the temperature T 0
and, to a first approximation, obeys the relation
V
V
(T T )
f
0
0
α
Δ
=
−
where Δα is the difference between the thermal expansion coefficients
of the liquid and the solid, V 0 is the system volume at temperature T 0 ,
and T 0 is the ideal glass transition temperature.
Express this probability as a function of temperature.
c. The relationship obtained may be written as
P e R(T T )
B
0
=
−
−
where B is a constant and R is the ideal gas constant.
Express B as a function of the parameters used in (b) and give its units.
3. By applying the free-volume model, the conductivity σ + of a cation obtained
from the hopping probability is
*
T
A
e V ( T T )
V
0
0
f
σ =
+
−
a
D
−
where A is a constant and V * f is the minimum free volume required to allow
the displacement of cation M
+
. Table 32 gives the experimental results for
the cationic conductivity of the vitreous system 2SiO 2 -K 2 O.