Biomedical Engineering Reference
In-Depth Information
the dimensions of the electrodes are reduced, however, it is natural
to expect that some simplifications that are perfectly reasonable on
larger scales will start to break down, and that new concepts are
needed to explain experiments. Below we will discuss several dif-
ferent ways in which this can occur. We employ the term
mesoscopic to collectively describe these systems. The term refers
to an intermediate size scale between macroscopic , where the clas-
sical results apply, and microscopic , where purely molecular-scale
processes dominate.
1. Double-LayerEffects
Any charged surface in contact with an electrolyte induces a
charged region in the electrolyte that is known as an electrical
double layer. This induced charge has the same magnitude but the
opposite sign to the charge of the surface, so that it effectively
neutralizes the surface charge. The induced charge is composed of
a local increase of the concentration of counterions near the sur-
face, accompanied by a decrease of the concentration of coions. In
many cases of interest, the electrical double-layer extends only of
the order of a few nanometers away from the surface and into the
solution. For macroscopic or micron-scale electrodes, this repre-
sents a perturbation that is localized essentially at the electrode
surface. For sufficiently small electrodes, however, the lateral di-
mensions of the electrode can become comparable to the thickness
of the double layer. When this happens, it becomes necessary to
re-assess some of the simplifications on which our understanding
of the classical regime is based.
The distribution of coions and counterions near a charged sur-
face is most commonly described using a mean-field formalism
embodied by the so-called Poisson-Boltzmann equation. This is a
nonlinear differential equation that can be solved to deduce the
electrostatic potential M ( r ) as a function of position r in the vicini-
ty of the surface. Since a full discussion of the solutions would
take us too far afield, we will simply summarize some of the key
qualitative results of the Poisson-Boltzmann formalism. The reader
is referred to the literature for a more thorough treatment. 117, 118
The double layer can in many cases be described in terms of
two different, spatially separated regions: a compact layer consist-
ing of the few molecular layers next to the surface, and a diffuse
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