Interdisciplinary Applied Mathematics

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other to form a chain of aggregated particles with the dipoles joined end to end.    The    attraction    is    quite    long-range    but    does    fall    off    quickly with

distance as    r-4.    The    uniform magnetic    field    will    produce no    force    on    the

particles, only the dipoles will create a gradient of the magnetic flux density B needed to generate a net force. At large separations the force will be below the level of Brownian thermal fluctuations or other background forces. As the particles disperse due to random motion or as longer chains form, the particles will come close enough for the magnetic forces to become dominant.

The description of the particles in terms of simple magnetic dipoles captures the    primary    dynamics,    even    though    a    more    detailed    multipole    representation may    be    warranted    for    local    variations    in    the    magnetic    field.

In (Paranjpe and Elrod, 1986), such a description has been used in terms of dipoles to effectively study the equilibrium configurations of chains of magnetic beads and to determine their stability from the overall potential U. While energy minimization principles yield information about the final states, dynamic simulations are needed to obtain the development in time and information about the time scales for chain formation and their eventual fate. This requires the solution of the coupled system of equations for the dipole    strengths,    which are    determined    by    the    external    field    and    the

fields generated by the other particles or other chains of particles.


FIGURE 13.2. Brownian motion of 2.34-pm silica particles in DI water. (a) A snapshot of monolayer of particles. (b) Particle trajectories of silica microspheres for 10 seconds.

Brownian Motion

Brownian motion and the effects of thermal fluctuations become an increasingly important feature for submicron-sized particles. Figure 13.2 (a) shows a snapshot of a monolayer of 2.34-mum-sized silica particles in DI water, taken using a conventional inverted light microscope. A sequence of digital images was taken using a monochrome CCD camera to track the Brownian diffusion of the silica particles. Image processing algorithms given in (Crocker and Grier, 1996), were employed to track the particle centers and draw their corresponding trajectories. The trajectories of the silica microspheres for 10 seconds of Brownian diffusion are shown in Figure 13.2 (b). Experiments were conducted at room temperature.

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