# Interdisciplinary Applied Mathematics

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X

FIGURE 13.1. Sketch and definition of geometric parameters.

U | max

квТ

so for X > 1 attraction forces overcome the Brownian motion, and aggregation is expected. If r is the center-center separation and в is the angle between the line of centers and the orientation of the dipoles (see Figure 13.1), the interaction energy (SI units) is

U (r, в)

Mo

47Г

m2(1 — 3cos2в)/г3,

where m is the dipole strength. For a magnetic field strength H, the induced dipole is

4    3

111 = -7ГО H

in terms of the effective magnetic susceptibility у of the particle. A chain will form if Umax is significantly larger than kB T. As discussed in (Hayes et al., 2001), the formation of supraparticle structures depends on particle

TABLE 13.1. Estimates for a paramagnetic bead, density 1.5 x 103 kgm 3, suspended in aqueous solution at temperature 300 K.

 d = 2/tm (a = 1/im) d = 0.2/rm (a = 0.1/im) Stokes settling velocity Ws 1.1 цт/sec 1.1 x lCD^/mi/sec Brownian diffusivity D = kT / 67Tfia 0.22 X 10~12 m2/sec 0.22 X 10-11 m2/sec Schmidt number v j D 4.5 x 106 4.5 x 105 Diffusion time scale a2 j D 4.5 sec 4.5 x 10-3 sec Scale for Brownian force kT/a 4.14 x 10—15 N 4.14 x 10~14 N Скачать в pdf «Interdisciplinary Applied Mathematics» Н. В. Камышова; С.-Петерб. национал. исслед. ун-т информ. технологий, механики и оптики, Ин-т холода и биотехнологийМетки учебные пособия