Biomedical Engineering Reference
In-Depth Information
A
ε ε
o
gi
C
=
gi
L
(4.20)
and therefore
ε ε
σ
0 gi
=
τ
(4.21)
gi
gi
where
L
is the sample length,
A
is the cross-sectional area,
ε
0
is the permittivity of free
space (8.854 × 10
-12
F/m), and
σ
gi
and
ε
gi
are the electrical conductivity and relative dielec-
tric constant of the grain interior, respectively.
The grain boundary relaxation time constant
τ
gb
can be expressed as follows [9]:
L
δ
R
=
(4.22)
gb
A
d
σ
sgb
A
ε ε
d
0
gb
C
=
(4.23)
gb
L
δ
and therefore
ε ε
σ
0
gb
=
(4.24)
τ
gb
sgb
where
d
is the grain size,
δ
is the grain boundary width, and
σ
s
gb
and
ε
gb
are the specific electri-
cal conductivity and relative dielectric constant of the grain boundary, respectively. The resis-
tance of the grain boundary is normally associated with the presence of a second phase or a
constriction resistance, which can also result in a space charge region at the grain boundary.
The capacitance of this region is thus associated with the polarization at the interface.
By applying the brick layer model and classic identification criteria comprehensively
discussed in the literature [1,10-14], bulk (at high frequencies) and grain boundary
(at low frequencies) contributions to the total materials' impedance were in each case
distinguished.
This can be explained by the different relaxation rate of charged species within each
region or the different
R
-
C
relaxation time of the elements. The defects, non-diamond
phases, and impurities are believed to be accumulated preferentially within the grain
boundaries than grain lattices [15]. The grain boundaries may induce dipole movements,
which respond to an applied field with a delay [16]. Thus, the grain boundary contribu-
tions with a long relaxation time usually happen in the low frequencies and the grain bulk
contributions with a short relaxation time in the high frequencies. In most cases, the suf-
ficient difference in the capacitance of the
R
-
C
equivalent circuit rather than the resistance
results in the two semicircles separated on the complex Cole-Cole plane [17].
The capacitance values of the grain boundary and grain interior are reported in the order
of 10
-9
and 10
-12
F, respectively, for many polycrystalline material systems, for example, zir-
conia [7], bismuth titanate [18], ferroelectrics [19], and polycrystalline diamond [20].
