Page 129 - Handbook of Battery Materials

3.2 Tunnel Structures 97

4(2 − n)
y = (3.2)
2 + m
4(1 − x) − x
n = (3.3)
2(1 − x)
2x
m = (3.4)
1 − x
which allowed Ruetschi to predict the density, proton-transfer rates, electronic be-
havior, theoretical (maximum) electrochemical capacities, and electrode potentials
for a wide range of x and y.
Further improvements on the previously discussed models were proposed in
the latest model for γ -and ε-MnO 2 by Chabre and Pannetier [12, 43, 44]. Starting
from De Wolff’s model they developed a structural description of manganese
dioxides that accounts for the scattering function of all γ - and ε-MnO 2 materials
and provides a method of characterizing them quantitatively in terms of structural
defects. All γ -and ε-MnO 2 samples can be described on the basis of an ideal
ramsdellite lattice affected by two kinds of defects:
1) A stacking disorder (De Wolff disorder: intergrowth of ramsdellite- and
pyrolusite- type units, as already described above). This kind of disorder can
be quantified by two parameters:
a. the probability P r of occurrence of rutile-like slabs in the crystal structure:
the probability of the presence of ramsdellite building blocks is P R = 1 − P r .
b. the junction probability, which describes, for example, the probability P r,r
that a rutile-like layer r is followed by a similar layer. Analogous P R,R ,
P c,R , and P R,r parameters can be defined for the three other possibilities of
conjunctions.

Starting from the four general possibilities that can occur (completely ordered,
partly ordered, segregated, and completely random), Chabre and Pannetier
found that the commercially available samples are best described by a truly
random sequence of rutile and ramsdellite slabs. Furthermore, an extended
simulation study of different rutile and ramsdellite fractions in the structure
led to the findings that, for example, even a small amount P r of rutile changes
the diffraction pattern of ramsdellite significantly. The reflections with an odd
value of k are shifted and broadened significantly, while the reflections with
k/2 + l(e.g.,(021), (121), (240), (061)) arenot affected by De Wolff
disorder. Starting with P r = 1 (from a pure rutile-type structure) the typical
(1 1 0) peak of pyrolusite disappears even at low values of P r .Whenavalueof
◦
0.5 was reached, a broad peak at about 20 (in 2θ) appeared in the simulated
patterns. On the basis of these simulations Pannetier developed a method to
estimate the pyrolusite concentration in orthorhombically indexable lattices.
After a careful refinement of the lattice constants by profile refinements, the
expected (theoretical) d value for the (110) reflection of ramsdellite is calculated
from the lattice constants:
2 1/2
2
d (110),expected = [1/a + 1/b ] (3.5)