# Interdisciplinary Applied Mathematics

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paramagnetic particles seeded at random positions in a cubic domain. Figure 13.3(b) shows a typical aggregation of particles with chains of various lengths formed but with some particles still isolated. As time evolves, these particles will join together and form longer chains. Chain/chain interaction leads also to long structure formation. Most of the chains are linear, since head-to-tail aggregation of magnetic dipoles is energetically preferable. Lateral merging of chains is also possible, leading to thick clusters of particles. Such lateral merging is important for the resistance to deformation and threshold of rupture of long chains. It is directly connected to the yield stress of the suspension and has a strong impact on nanotechnology applications. In Figure 13.3(b), chain defects are observable; e.g., observe one chain divided into two branches that connect again. Careful experiments of microrheology using dual-trap optical tweezers have highlighted clearly the impact of annealing defects on the mechanical properties of chains (Furst and Gast, 2000).

Von Smoluchowki’s theory (von Smoluchowski, 1916) provides a solid basis for predicting aggregation rates in very dilute solutions. It states that

• the rate of change in the number of clusters containing n particles is connected to the reaction kernel that represents the rate of coalescence of two smaller clusters.

In diffusion-limited aggregation, the numerical simulations in (Miyazima et al.,    1987),    based    on    the    motion    of oriented    particles    on    periodic    square

FIGURE 13.4. Evolution of the mean cluster size in terms of particle diameter. The symbols denote: plus — A = 1; triangle — A = 10; squares — A = 100, and stars -A = 10,000. The dashed line indicates t07 dependence. (S(t)) is normalized with the particle diameter.

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