Environmental Engineering Reference
In-Depth Information
−
ε
∞
ε
s
ε
r
(
i
f
rel
f
)=
ε
∞
+
(2.34)
f
1
+
where
ε
∞
is the dielectric constant at frequency so high that molecular orientation
does not have time to contribute to polarization (i.e., at infinite frequency) [7];
ε
s
is the static dielectric permittivity;
f
rel
is the relaxation frequency (defined as the
frequency at which the permittivity is
[16]; and
f
is the frequency.
As a matter of fact, only few materials of interest behave as pure polar materials
with a single relaxation frequency; therefore, other models have been developed to
describe more accurately the dielectric behavior of materials [7]. The Cole-Cole
model, with symmetric distribution of relaxation times, is described through the
following equation:
(
ε
+
ε
∞
)
/
2
)
s
−
ε
∞
ε
s
ε
r
(
f
)=
ε
∞
+
(2.35)
i
(
1
−
β
)
f
f
rel
1
+
where
β
is a parameter that describes the spread in relaxation frequency (0
≤
β
≤
1)
[10, 17]. When
0, the Cole-Cole model reduces to the Debye model.
A further generalization of the Debye model is given by the Havriliak-Negami
formula [2, 14]:
β
=
ε
s
−
ε
∞
ε
r
(
f
)=
ε
∞
+
1
(
1
−
β
)
α
(2.36)
i
f
f
rel
+
where
describes the broadness of the permittivity spectra.
Finally, the Cole-Davidson model with asymmetric distribution of relaxation
times follows for
α
β
=
0andfor0
<
α
<
1[8]:
ε
−
ε
∞
s
ε
r
(
f
)=
ε
∞
+
1
i
α
.
(2.37)
f
f
rel
+
When the static electrical conductivity is not negligible, an additional term must
be included in the equations above; therefore, for example, the Cole-Cole model
described by (2.35) becomes
⎧
⎨
⎫
⎬
ε
s
−
ε
∞
σ
0
ε
r
(
1
(
1
−
β
)
f
)=
ε
∞
+
−
i
(2.38)
i
⎩
⎭
2
π
f
ε
0
f
f
rel
+
The five parameters are referred to as Cole-Cole parameters: they are different for
each material and they define its spectral signature. The Cole-Cole relaxation model
has proved suitable in describing relaxation phenomena of various materials, espe-
cially for lossy media [6, 9], and for a wide number of materials (e.g., concrete,
paper, petrochemicals, and even blood) [13, 15].
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