Biomedical Engineering Reference
In-Depth Information
One may ask why different materials show the similar capacitance values (10
-9
F for
grain boundaries and 10
-12
F for the grain interiors) when analyzed in the Cole-Cole plot.
The reason for this may be explained from the definition of the capacitance value itself as
shown in Equations 4.20 and 4.23. The theoretical capacitance value associated with grain
interiors has been defined in Equation 4.20. In practice, all the geometric parameters in
this equation are not infinite to facilitate to carry out these experiments on a research labo-
ratory basis. Thus, these geometric parameters normally have their expected magnitude
order.
L
is the sample length within the order of millimeters (mm).
A
is the cross-sectional
area within the order of 10 mm
2
.
ε
0
is the permittivity of free space (8.854 × 10
-12
F/m).
ε
gi
is the relative dielectric constant of the grain interior of the sample. For high-purity dia-
mond, the dielectric constant is reported to be about 5.7 [21]. Thus, formula 4.20 indicates
that the grain interior capacitance from diamond is estimated about 0.5 pF.
The same
A
and
L
values can be applied to the grain boundary capacitance in Equation
4.23. In addition, the grain boundary-induced dipole polarization may cause the dielectric
constant (
ε
gb
) to be slightly higher than that of grain bulk (
ε
gi
), but probably on the same
magnitude order as grain bulk [15]. Furthermore, the grain size of a typical polycrystalline
material is usually on the order of micrometers (
µ
m), where the effective grain boundary
width is expected to be on the order of nanometers (nm) [22]. Thus, the estimated grain
boundary capacitance for a polycrystalline diamond is about 0.5 nF.
To gain further insight into these quantitative estimations, the ratio of the grain interior
capacitance and grain boundary capacitance may be considered as follows:
C
C
≈
δ
gi
(4.25)
d
gb
Equation 4.25 indicates that the ratio of the grain capacitance and grain boundary capac-
itance is proportional to the ratio of the grain boundary width (
δ
) and grain size (
d
). In
other words, the difference between the grain interior capacitance and grain boundary
capacitance mainly originates from the geometric effect of the sample. This effect may
be controlled by the following factors such as grain size, anisotropy, and porosity [1]. The
typical geometric characteristic within a polycrystalline material leads to the grain bound-
ary capacitance is normally two or three orders of magnitude higher than the grain inte-
rior capacitance.
The above discussion gives an explanation on the origin of the classic procedures and
identification criteria for the grain boundary and grain bulk contributions in an ideal con-
dition. In practice, it is difficult to calculate the theoretical capacitance of grain interior
and grain boundaries for diamond. This is because the in-plane electrode measurement
makes it unlikely to define the specific length and area of each sample. Cross-section elec-
trode measurements make the geometric parameters solved easily, but most of impedance
data cannot be detected across the samples due to extremely high resistance in diamond.
Therefore, the conduction path identification criteria used in this chapter are emphasized
on practical and empirical interpretations of materials based on the brick layer model dis-
cussed here.
Equivalent Circuits
This section presents impedance characterization on real electronic circuits made of resis-
tors and capacitors rather than materials. Two types of circuits (
R
-
C
parallel and double
