Biomedical Engineering Reference
In-Depth Information
is plotted in Figure 4.1b in terms of a Cole-Cole plot in the complex plane with the negative
imaginary parts above the real axis, as is usually used in electrochemistry.
At the peak of the semicircle, the following condition is obtained:
ω
max
R
2
C
= 1
(4.14)
and hence
1
C
=
(4.15)
2
π
max
f R
2
where Equation 4.4 has been used. Knowing the value of
R
2
and the frequency
f
max
, the
value of the capacitance can be determined. It is possible to obtain all three parameters (
R
1
,
R
2
,
and
C
) from the Cole-Cole plot as shown in Figure 4.1b, provided a sufficient frequency
range is investigated.
The application of impedance spectroscopy to the characterization of polycrystalline
materials started after Bauerle [2] showed that for zirconia with platinum electrodes, the
individual polarizations of grain interiors, grain boundaries, and electrodes could be
resolved in the admittance plane. He presented an equivalent circuit for his results, which
have now proven to be typical of most solid electrolytes. In such a circuit, the individual
elements correspond to grain interiors, grain boundaries, and electrodes connected in
series. However, estimation of the circuit parameters was made complicated by Bauerle's
choice of the admittance plane. Many subsequent researchers have therefore preferred
to work in the impedance plane, that is, Cole-Cole plot, where a more direct relationship
exists between the spectrum and the circuit [3,4]. The level of agreement between experi-
ment and simulation is quite satisfactory for the grain interior and grain boundary arcs
both in terms of shape and distribution of frequencies on the arcs, therefore supporting the
view that equivalent circuits are a meaningful way of representing the data.
Measurement of the electrical conductivity for polycrystalline materials using imped-
ance spectroscopy provides information related to the electrical behavior of the grain inte-
riors, the grain boundary regions, and electrode. This is illustrated in Figure 4.2a, which
contains the equivalent circuit for the electrical response of a polycrystalline material. The
simple
R
-
C
parallel circuit in which three components are connected in series is often
used to study the AC impedance behavior of materials. This circuit has a direct relation-
ship to the complex impedance plot (Figure 4.2b) in which
Z
→
, the imaginary part of the
complex impedance, is plotted against
Z
→
, the real part, for a wide range of frequencies
(typically 10
-1
-10
7
Hz). The frequency increases as shown by the arrow in Figure 4.2b. The
highest frequency is located at the origin. The resistances
R
gi
,
R
gb
, and
R
e
corresponding
to the grain interior, grain boundary, and electrode, respectively, can be obtained from
the intersections on the real axis of the corresponding semicircular arc. Comparing with
the other two,
R
e
can normally be ignored except for the case of imperfect ohmic contacts
which cause interfacial contact resistance. The capacitances
C
gi
,
C
gb
, and
C
e
corresponding
to the grain interior, grain boundary, and electrode, respectively, can be obtained from
Equation 4.15 [5].
The relationship between microstructural models and circuits shows its real merit when
used to correlate the equivalent circuit parameters of a material with changes in the exter-
nal conditions or the microstructure of the material. For example, in ceramics, where it is
possible to resolve the resistances due to the grain interiors and the grain boundaries, it
