# Interdisciplinary Applied Mathematics

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Because of advances in nanotechnology, there is now a possibility of understanding cell behavior at a fundamental level. In particular, there is now an exciting opportunity to understand how ionic channels in cellular mem-

branes function by taking advantage of the progress in nanotechnology. Carbon nanotubes with diameters as small as 5 A to 10 A are comparable to the diameters encountered in ion channels. However, the surface characteristics of a carbon nanotube may not match closely the surface properties of an ion channel. By functionalizing carbon nanotubes, it is possible to tune the surface properties of carbon nanotubes to investigate the function    of    a    variety    of ion    channels.    To enable    such    advances,    it    is

important to understand how water, ions, and various electrolytes interact with carbon nanotubes and functionalized nanotubes. This is the subject of the second part of this chapter.

In particular, in the first section we present details on the physical mechanisms involved in self-assembly, and subsequently we give examples of patterns formed using magnetic fields for magnetorheological fluids and electrophoretic deposition for electrorheological fluids. In the second section, after a brief introduction to carbon nanotubes and ion channels in biological membranes, we present results on electrolyte transport through carbon nanotubes. Specifically, we present concepts and results showing that the transport of electrolytes can be augmented by using functionalized nanotubes and electric fields.

##### 13.1 Colloidal Particles and Self-Assembly

Construction of functional materials with feature sizes ranging from 1 nm to 1000 nm is a challenging problem and an area of active research. Due to the inherent dimensional limitations of currently available mechanical forming and lithographic techniques, it is difficult to produce structures with length scales on the order of nanometers. In addition, the following factors make it difficult for colloidal particles to assemble into ordered structures:

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