# Interdisciplinary Applied Mathematics

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FIGURE 13.7. (a) Schematic of two equal-sized particles of diameter D deposited on an electrode by EPD. The particles remain mobile due to the Brownian motion, (b) Electroosmotic flow around a colloidal particle held stationary near an electrode. When the direction of electric field is reversed, the direction of the flow is also reversed. (Courtesy of M.A. Bevan.)

electroosmotic flow brings the particles together, while the electrophoresis moves them apart. (Solomentsev et al., 2000) have shown that at r « 3D, the magnitude of the electroosmotic velocity is about seven times greater than the electrophoretic velocity, and this facilitates the aggregation process. It has to be noted that this analysis is applicable only for DC electric fields.

(Trau et al., 1996) and (Trau et al., 1997) explained the transverse migration of colloidal particles using an electrohydrodynamic mechanism wherein fluid flow transports the particles toward each other. They theorized that the particles    near    the    electrode    alter    the    local    electric    fields, and    these

perturbations can result in concentration and current density gradients at the electrode surface, resulting in fluid motion. This model is applicable for both AC and DC electric fields, and it is consistent with experimental studies.

In summary, the current theories are not elaborate enough to provide quantitative estimates of particle aggregation behavior. More experiments are needed to provide an insight into the origin of the lateral convective forces. A model that can completely describe the particle aggregation dynamics should take into account the combined effects of electrokinetics, electrohydrodynamics, and Brownian diffusion, similar to the model we presented for magnetorheological fluids.

##### 13.2 Electrolyte Transport Through Carbon Nanotubes

There is great interest in investigating fluid flow through carbon nanotubes. In Chapter 10 and Chapter 11, we presented results on transport of simple fluids and water through carbon nanotubes. The fundamental question is whether electrolytes can be transported easily through small-diameter carbon nanotubes. One of the motivations to investigate electrolyte transport through carbon nanotubes is to create nanoscale devices or design concepts to mimic biological ion channels. In this section, after an introduction to carbon nanotubes and ion channels, we address the issue of electrolyte transport through carbon nanotubes and present ideas to mimic some aspects of biological ion channels.

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