Biomedical Engineering Reference
In-Depth Information
R
gi
R
gb
R
e
(a)
C
gi
C
gb
C
e
(b)
f
e
Frequency
f
gi
f
gb
Z
'
R
gi
R
gb
R
e
FIGURE 4.2
(a) An equivalent circuit representing a polycrystalline material. (b) An idealized complex impedance plot for
a polycrystalline material showing contributions from the grain interiors (gi), grain boundaries (gb), and elec-
trode interface (e). Note that
C
e
>>
C
gb
>>
C
gi
. (From Ye, H.T., PhD thesis, University College London, 2004. With
permission.)
has facilitated the study of several important processes: sintering, grain growth, and solid-
state precipitation. In diamond, it can facilitate the investigation of nucleation, growth,
doping process, grain boundary deterioration, and nanocrystallization.
Conductivity Models
The impedance plots of polycrystalline materials can be related to their microstructure by
means of physical models of the grain interior, grain boundary, and the electrode behavior.
Three physical models used to describe the electronic materials are reviewed in details
with their respective circuit equivalents [7].
The early model used to describe the properties of two-phase mixture is the series layer
model. The model describes that the two phases are assumed to be stacked in layers paral-
lel to the measuring electrodes, with total thickness of each phase made proportional to
volume fractions
X
1
and
X
2
. This model shows a linear mixing rule for the complex resis-
tivity (
ρ
). The complex resistivity is the sum of the individual phase resistivity (
ρ
1
and
ρ
2
):
ρ
=
X
1
ρ
1
+
X
2
ρ
2
(4.16)
In the parallel circuit model, the two phases are assumed to be stacked across the elec-
trodes. For this model, the complex conductivity (
σ
) rather than resistivity follows a linear
mixing rule:
σ
= X
1
σ
1
+
X
2
σ
2
(4.17)
The widely used physical model is a more realistic one, which treats the microstructure
as an array of cubic shaped grains with flat grain boundaries of finite thickness as shown
in Figure 4.3 [8].
The volume fraction of grain boundaries is 3
δ
/
d
(
δ
is grain boundary thickness and
d
is
grain size) [7]. The current flow is assumed to be one-dimensional, and the current path
