Solutions to exercises
159
Solution 3.14 – Free-volume model
1. The following hypotheses are put forward in developing the free-volume
model for amorphous electrolytes (glasses and polymers):
2 On a molecular scale, a volume V may be associated with each structural
unit (atom or mobile segment of a chain). This volume represents the
envelope of the displacements of the geometric center of the structural
unit, and it increases with temperature.
2 When this volume reaches and exceeds a critical volume V c , at a temperature T 0 , it becomes useful to define the notion of free volume V f by
the relation
V f = V − V c
V f is accessible to all mobile species and may be redistributed within an
amorphous solid without requiring an external energy source.
2 When the cage occupied by the structural unit attains a free volume greater
than a minimal volume V * f , the displacement of this unit becomes possible.
2 At a given temperature T and without requiring an external energy
source, the free volume oscillates about an average volume V f .
2. a. The hopping probability for cations is that for which the free volume of
the cage occupied by the cation is greater than V * f . This probability is
given by
P e V
V
*
f
f
=
−
*
b. Under these conditions, the hopping probability as a function of temperature is
P e V
T T
V
0
0
f
=
α
Δ
−
−
*
^
h
c. By identification, we have
P e
e
V
T T
V
R T T
B
0
0
f
0
=
=
α
Δ
−
−
−
−
*
^
^
h
h
which gives
B
V
R V
0
f
α
Δ
=
*
B has the same dimensions as R(T − T 0 ); namely, energy per mole. It is
not an activation energy in the classic sense because the hop requires no
thermal energy.