Biomedical Engineering Reference
In-Depth Information
δ
δ
d
A
δ
d
d
L
FIGURE 4.3
Brick layer model of idealized polycrystalline structure in which grains of dimensions
d
3
are separated by grain
boundaries of width
δ
. (From Maier, J.,
Prog. Solid State Chem
., 23, 171, 1995. With permission.)
at the corners of the grains is neglected. In this case, the two paths available to the current
are either through the grains and across the grain boundary, or along grain boundaries, as
depicted in Figure 4.3. Depending on the relative magnitudes of grain and grain boundary
conductivity, one of the two paths may dominate. This model has been applied to many
material systems, and according to this model, the bulk resistivity and the grain boundary
resistivity may have different response [8].
In the brick layer model, the grain interior response will be displaced from the grain
boundary response depending on the relaxation rate of charged species within each
region. It is normal to compare the relaxation rates of different processes in terms of rate
constant or relaxation time (
τ
); this can be defined as:
τ
=
RC
(4.18)
In terms of grain interior rate constant,
τ
gi
can be expressed as follows:
L
(4.19)
R
=
gi
A
σ
gi
-6 × 10
6
-5 × 10
6
-4 × 10
6
-3 × 10
6
-2 × 10
6
-1 × 10
6
0
-2 × 10
6
0
2 × 10
6
4 × 10
6
6 × 10
6
8 × 10
6
1 × 10
7
1.2 × 10
7
Re
Z
(ohms)
FIGURE 4.4
R
-
C
in parallel. (From Ye, H.T., PhD thesis, University College London, 2004. With permission.)
