Biomedical Engineering Reference
In-Depth Information
by a pure resistance. In fact, even though the lipoic acid residue is
the dielectric slab in direct contact with the metal, we cannot ex-
clude a slight ionic flux across it and a resulting high (but not an
infinitely high) resistance. Analogously, even though the capaci-
tance of the aqueous phase interposed between the lipid bilayer
and the auxiliary electrode is very low, it is not infinitely low, and
its inclusion improves the fit.
To better visualize all semicircles, we have found it conven-
ient to represent impedance spectra on a Z
Z
' vs. Z
Z
” plot.
3
Hence-
forth, this plot will be briefly referred to as an
M plot
, since Z
Z
'
and Z
Z
” are the components of the modulus function
M
. A single
RC mesh yields a semicircle even in this plot. Thus, if we multiply
both members of Eq. (3) by Z
2
and we combine the resulting equa-
tion with Eq. (2b), after simple passages we obtain:
2
2
§
1
·
§
1
·
2
2
2
2
2
Z
Z
"
Z
Z
'
Z
Z
"
/
C
0
o
Z
Z
"
Z
Z
'
(4)
¨
©
¸
¹
¨
©
¸
¹
2
C
2
C
This is the equation of a semicircle of diameter
C
-1
and center of
coordinates (2/
C
,0) on a Z
Z
' vs. Z
Z
” plot. Moreover, Z at the
maximum of the semicircle is again equal to the reciprocal of the
time constant RC of the mesh. While Z decreases along the posi-
tive direction of the abscissas on a
Z
” vs.
Z
' plot, it increases on a
0 plot. Therefore, for a series of RC meshes, the last semicircle on
the M plot is characterized by the lowest time constant. This is,
unavoidably, the semicircle simulating the solution that baths the
self-assembled film, due to its very low capacitance.
Figure 5
shows the M plot relative to the same impedance spectrum that
solid curve is the best fit of the equivalent circuit in
Fig. 3
, consist-
ing of four RC meshes in series. Proceeding along the positive
direction of the abscissas, the dashed curves express the contribu-
tion to Z
Z
' from each of the four different RC meshes, namely the
quantity Z
g
i
/(
g
i
2
+Z
2
C
i
2
), where
g
i
=1/
R
i
and
C
i
are the conductance
and capacitance of the
i
-th mesh. The deviations of the dashed
curves from a pure semicircle measure the extent of their overlap-
ping with the neighboring semicircles. The four semicircles over-
lap only to a moderate extent, thus allowing their straightforward
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