# Interdisciplinary Applied Mathematics

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To understand the flow reversal, we calculate the driving force Fe(z) for the flow using the ion concentration obtained from the MD simulation.    Specifically,    in    the    region    where    the    ions are immobilized,    Fe(z)    is

taken as zero, and in the rest of the channel, Fe(z) is computed by the expression given above, using the ion concentrations shown in Figure 12.11. Figure 12.14 (b) shows the calculated driving force. The flow Fe(z) is zero within 0.19 nm from the channel wall, since the Na+ ions adsorbed on the wall are immobilized. Notice that because of the charge inversion, Fe(z) is negative    in    the    region    0.53 nm < z <    2.96 nm.    Figure    12.14    (a)    also

shows the velocity computed by substituting the driving force obtained from the MD ion concentrations into the Stokes equation. Clearly, the new driving force can predict the flow reversal in the region 0.58 nm < z < 2.91 nm, which indicates that the major mechanism for the flow reversal is the immobilization of the adsorbed Na+ ion on the channel wall and the charge inversion. The velocity profile with the new driving force, however, still deviates    from    the    MD    velocity    profile,    especially    in    the    regions    close

to the channel wall. The reason for this deviation is not clearly understood at present,    but    is    probably caused by    the    high    local    viscosity    in    the    near

wall region.

In summary,    the    charge    inversion    and flow reversal    are    some    of    the    new

physical phenomena that have been observed in nanochannel electroosmotic flows. While these results indicate that the molecular nature of water and ions can lead to interesting new phenomena, the inclusion of the molecular nature of water and ions into continuum electrostatic and hydrodynamic theories remains an active area of research.

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### Functional Fluids and Functionalized Nanotubes

The possibility of targeting and precisely controlling the electrooptical as well as the mechanical properties of microstructures in a dynamic way using external fields has opened new horizons in microfluidics research, including new concepts and protocols for microfabrication. New functionalized ferrofluids and ferromagnetic particles have led to a range of new biomedical and diagnostic applications. Self-assembled magnetic matrices can find a large range of applications for the separation of DNA and other intermediate-size objects such as cells, proteins, organelles, and micro- or nanoparticles. Self-assembly of colloids can be used in a bottom-up approach to the fabrication of nanosystems; in particular, self-assembly offers a possible route to fabricating three-dimensional microsystems. Such selfassembly techniques are biomimetic; i.e., they are inspired by processes in biological systems that enable proteins and cells to arrange themselves organically in functionally beneficial ways. However, they need external forces to be imposed to facilitate the process and provide quality control by steering and tailoring target properties. To this end, magnetic and electric fields can be used for paramagnetic and charged particles, respectively. How exactly this is accomplished is the subject of the first part of this chapter. On a more fundamental level, systematic studies of paramagnetic particles or charged particles and their dynamics offer insight into the role of Brownian noise in microsystems as well as conceptual differences between deterministic and stochastic modeling.

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