Electrochemistry of Biomimetic Membranes - Applications of Electrochemistry and Nanotechnology in Biology and Medicine - page 151

rule, an equivalent circuit is assembled from resistors and capaci-
tors, representing the electrically dominant components of biomi-
metic membranes. In general, a metal-supported self-assembled
mono- or multilayer can be regarded as consisting of a series of
slabs with different dielectric properties. When ions flow across
each slab, they give rise to an ionic current J ion = V E , where E is
the electric field and V is the conductivity. Ions may also accumu-
late at the boundary between contiguous dielectric slabs, causing a
discontinuity in the electric displacement vector D = H E , where H is
the dielectric constant. Under a.c. conditions, the accumulation of
ions at the boundary of the dielectric slabs varies in time, and so
does the electric displacement vector, giving rise to a capacitive
current J c = d D /d t . The total current is, therefore, given by the sum
of the ionic current and of the capacitive current. In this respect,
each dielectric slab can be simulated by a parallel combination of a
resistance, accounting for the ionic current, and of a capacitance,
accounting for the capacitive current, namely by an RC mesh . Ac-
cordingly, the impedance spectrum of a self-assembled layer can
be simulated by a series of RC meshes. It should be noted that lat-
eral heterogeneities, such as defects or microdomains, cannot be
accounted for by simulating them by RC meshes in parallel with
each other. In fact, in view of Kirchhoff's laws for the combination
of circuit elements, the parallel connection of RC meshes is re-
duced again to a single RC mesh, with averaged values for the
resistance and the capacitance.
Application of an a.c. voltage of amplitude v and frequency f
to a pure resistor of resistance R yields a current of equal frequen-
cy f and of amplitude v/R , in phase with the voltage. Conversely,
application of the a.c. voltage to a pure capacitor of capacitance C
yields a current of frequency f and amplitude 2S fC , out of phase by
-S/2 with respect to the voltage, i.e., in quadrature with it. This
state of affairs can be expressed by stating that the admittance Y of
a resistance element equals 1/ R , while that of a capacitance ele-
ment equals -iZ C , where Z= 2S f is the angular frequency and i is
the imaginary unit. More generally, in an equivalent circuit con-
sisting of resistances and capacitances, Y is a complex quantity,
and the impedance Z is equal to 1/ Y , by definition. Hence, Z equals
R for a resistance element, and i/Z C for a capacitance element. In
analogy with the resistance in d.c. measurements, the overall im-