Interdisciplinary Applied Mathematics

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24[(1/3)1/2 — (1/3)3/2]DcA

This characteristic time is inversely proportional to D, c as before, but also to the dipole strength A. The constant prefactor is derived based on the capture volume that corresponds to the combined conditions of attraction, i.e., U < 0 and r < rc.

In the    simulations    of    (Climent    et    al.,    2004),    initially,    the    mean    cluster

size is equal to one particle diameter, since all the particles are seeded randomly throughout the domain respecting the nonoverlapping condition. Figure    13.4    shows    results    for    a    very    dilute    suspension    (c =    0.003)    that

experiences a constant magnetic field characterized by a dipole strength A ranging from    1 to    104.    We    notice    that    in    the    case    of A = 1,    no    chains are

forming and the particles always diffuse randomly. The magnetic attraction is not strong enough to join the particles together, since Brownian forcing dominates the behavior of the suspension. A clear transition is observed when A ^ 1, with {S(t)) slowly increasing during a characteristic time of order t*B. Subsequently, aggregation sets in as particles or chains gradually join together forming linear supraparticle structures. The linear increase in a log-log plot is evident for almost one decade. When the mean length is on the order of half of the domain width, the periodic conditions become invalid and the simulation is stopped.

In the previous example, the geometry was not taken into account, and chains became longer as a function of time, although they may bifurcate into smaller chains due to chain-chain interactions. Geometry effects are, however, very important, and they can be used cleverly to affect the pattern formation. The effect of geometric confinement on dynamic self-assembly has been simulated for the first time in (Liu, 2004), following the experimental studies of (Hayes et al., 2001). Figure 13.5 shows four snapshots of the dynamic aggregation process inside a trianglular duct. Initially, eight paramagnetic beads are randomly scattered inside the duct. Upon the application of a horizontal external magnetic field, the particles in close proximity form pairs first,    then triplets,    and    so    on.    Due    to    the    magetic    dipole

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