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 1015 N


4.14 x 10~14 N

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