# Interdisciplinary Applied Mathematics

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repulsion and attraction mechanisms, the particles tend to form a straight

Magnetic Field    y

FIGURE 13.5. Aggregation of eight particles inside a trianglular duct due to magnetic dipole interactions: initial seeding (top left); shortly after initial seeding (top    right);    shortly    before    the    final aggregation    (bottom    left);    final    position

(bottom right).

line to minimize the magnetic potential energy in the system. However, because    of    the    constraint imposed    by    the    geometric    boundaries,    not    all

particles can be accommodated at that height. Therefore, a five-particle chain moves down to a proper right height, while a triplet moves upward and to the right in order to join the finally formed chain parallel to the magnetic field direction.

###### 13.1.2 Electrophoretic Deposition

Electrophoretic deposition of colloidal particles at charged surfaces has been used as a manufacturing technique for coating metals, oxides, phos-phores, inorganic and organic paints, rubber, dielectrics, superconductors, and glasses (Trau et al., 1997). In his seminal paper, (Hamaker, 1940) described the formation of a colloidal deposit by electrophoresis. Electrophoretic deposition (EPD) is essentially a two-step process that involves:

1. Application of an electric field to transport the colloidal particles in the liquid suspension toward an electrode (electrophoresis), and

2. Collection of particles on the electrode to form a coherent deposit.

Building on these early ideas, flocculation at the electrode surface was explained in terms of dipole interactions (Estrelalopis et al., 1982). Hamaker’s law relates the deposit yield to the electric field strength, electrophoretic mobility, surface area of the electrode, and the particle mass concentration in the suspension, and it is given as

dY

— =l_lECA,    (13.1)

where Y is    the    yield    (in    kg),    E is the    electric    field strength    (in    V/m),    p

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