# Interdisciplinary Applied Mathematics

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process (Ristenpart et al., 2004).

The details of the aggregation mechanism are still not completely understood. Comprehensive theories that can completely describe the aggregation dynamics at the electrode surface are not available in the literature. As the particles and clusters were seen to interact over long ranges, (Bohmer, 1996) suggested that hydrodynamic effects resulting from electroosmotic flow around each particle were responsible for particle aggregation. This also ruled out the contribution of short-ranged van der Waals forces to particle aggregation. The motion of particles deposited on an electrode surface is governed by the relative interplay of electrokinetics, electrohydrodynamics, and Brownian diffusion. (Solomentsev et al., 1997) considered the electroosmotic flow around the charged particles near the electrode surface, and proposed an electrokinetic model for particle aggregation. This model was able to qualitatively explain the observations in (Bohmer, 1996). In addition, the particle trajectories predicted by the model were in good quantitative agreement with experimentally measured trajectories of three particles aggregating near the electrode surface. In a later work, (Solomentsev et al., 2000) studied the aggregation dynamics for two particles during electrophoretic deposition under steady electric fields. They proposed a convective-diffusive model based on electrokinetics to explain the mechanism behind the particle aggregation.

Figure 13.7(a) shows a schematic of two equal-sized colloidal particles electrophoretically deposited on an electrode. The two particles are assumed to be at the same height h above the electrode surface, and the electric field    E is    normal    to the    electrode    surface.    The    particles    attract    each

other due to the electroosmotic flow around each particle (Solomentsev et al., 2000). The authors solved Laplace’s equation to obtain the electric field about a single particle, and then they used this result to solve for the electroosmotic flow field. The streamlines around a particle are shown in Figure 13.7(b). The electrokinetic slip velocity at the surface of the colloidal particles drives the flow. The electroosmotic flow around a single particle entrains the neighboring particle, and draws it closer. Secondary electrophoretic effects also become important, since the electric field has a component that affects the relative motion of the two particles. The

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