# Interdisciplinary Applied Mathematics

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lattices support the scaling evolution {S(t)) ж tz. The exponent z is related to the diffusion coefficient of a cluster of length s. Under the assumption of a long chain, it is commonly accepted that the diffusion coefficient scales like s-1, which in turn leads to the relation

{S(t)) Kt05.

Even though this behavior has never been strictly observed in experiments, it provides a valuable point of reference.

The determination of a characteristic time scale is a critical issue. We can expect a collapse of all the data if the time scale is properly defined,

i.e., accounting for the effect of the dipole strength A and the particle volume fraction c. A simplistic model of doublet formation is based on pure diffusion of particles in a very dilute suspension. Particles diffuse as long as collisions do not occur. When two particles are touching, a doublet is formed. The characteristic time tB in this case is inversely proportional to diffusivity D and concentration c as it corresponds to the flux of particles toward an isolated test particle, i.e.,

a

tB = 6Dc’

Based on experimental results, a correction was introduced in (Promislow et al., 1995), that improves this scaling. The identification of mechanisms leading to chain formation is of fundamental significance for the time scale determination. In particular, the motion of a paramagnetic particle can

be split into two main regimes. When particles are widely separated, the attraction force that decays quickly (r-4) is not strong enough, and particles diffuse randomly. When the separation between two beads or between a particle and a chain is smaller than a critical value rc determined by U(rc) ж kBT, the attraction force becomes dominant and aggregation occurs. Therefore, the flux of particles captured by a single sphere has to be augmented. The capture volume related to the surface U = kBT allows the determination of a new time scale based on the envelope of the anisotropic dipolar potential. Promislow et al. proposed to use a modified time scale tB given by

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