# Interdisciplinary Applied Mathematics

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These features distinguish the systems from ferrofluids (Rosenweig, 1987), in which individual    magnetite    crystals,    on    the    order    of    20    nm in size    or

smaller, are in random suspension at much higher volume fractions. In the experiments of (Hayes et al., 2001), (Furst et al., 1998), and (Doyle et al., 2002) on the formation of supraparticle structures, the particles were all about 1    p,m    to 3    p,m    in diameter,    while    in (Promislow et    al.,    1995),    they

were about 0.6 p,m in diameter.

The forces acting on the particles arise from the imposed magnetic field, Brownian motion, short-range electrostatic repulsion, van der Waals forces, gravity, and fluid forces. In particular, van der Waals forces would cause particles to bind together and aggregate due to collisions from Brownian motion or    in a shear flow, but    these    forces    act    only    over    very    short    dis

tances, while electrostatic repulsion between particles acts over a relatively larger distance, comparable to the Debye layer thickness, and will prevent aggregation; see (Russel et al., 1989; Tabor, 1991). The details of the electrostatic properties depend on the preparation of the beads, how they are stabilized, and the electrolyte content of the liquid phase. The effect of gravity diminishes rapidly with particle size, and the dominant forces to consider are due to the magnetic field, Brownian motion, and viscous fluid forces. The diffusivity of an isolated sphere is given by the Stokes-Einstein formula

6nap

for a particle of radius a, where p is the fluid viscosity, kB is Boltzmann’s constant, and T is the absolute temperature. Some characteristic features of particles of radius 1 pm and 0.1 pm are listed in Table 13.1.

A criterion for determining whether two particles that have a magnetic dipole will form a chain can be obtained by comparing the energy of thermal fluctuations kBT with the interaction energy U of the magnetic dipoles. To this end, we define the dipole strength ratio

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