# Interdisciplinary Applied Mathematics

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Another method to build structure and function in the nanometer scale involves the use of electric fields to assemble colloidal particles at electrode surfaces. Patterned crystalline arrays of colloidal particles can be formed by appropriately altering the surface topography of the electrode surface. Besides the disadvantage of added time and expense for altering the electrode surface (lithography and etching steps), this method cannot be used for forming patterns comprising different colloidal particles (Hayward et al., 2000). (Kim et al., 1997) exploited the presence of capillary forces in a microcontact printing mold to draw the colloidal suspension into small channels above the substrate. Evaporation of the solvent results in the colloidal particles to self-assemble into crystals. While this method eliminates the use of photolithography to form patterned surfaces, it is restricted to patterns with interconnected areas alone, since it is based on capillary flows. (Hayward et al., 2000) described a method for assembling patterned colloidal crystals using selective illumination of an optically sensitive electrode with electromagnetic radiation. This method overcomes limitations of the other methods described above.

In the following two sections we focus on the physical and modeling issues that need to be considered for understanding and simulating colloidal structures using paramagnetic beads and charged particles. We first discuss issues in magnetorheological fluids and subsequently address electrophoretic deposition.

###### 13.1.1 Magnetorheological (MR) Fluids

In order to determine the mechanics of paramagnetic micropsheres in different flow configurations, we need first to characterize the particles and the forces that act on them. Superparamagnetic beads consisting of iron oxide crystals dispersed in a styrene polymer are readily available in a range of sizes from 0.3 p,m to 5 p,m in diameter (e.g., Bang Labs). Other functionalized paramagnetic beads are available in sizes down to 20 nm that are generally spherical in shape. The particles are magnetically soft and readily acquire a dipole moment when placed in a magnetic field. They also show minimal hysteresis and quickly lose their dipole moment when the magnetic field is removed. The particles are colloidal in character and can remain in suspension for a long time, with negligible sedimentation, under the action of Brownian motion and short-range electrostatic repulsion. In MR flows for the formation of self-assembled structures the volume fraction of the particles is    typically    low,    and    the    particles    are    larger than    20    nm.

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