structures formed are not truly fractal any more. However, diffusion-limited
aggregation is the exception rather than the rule in fractal aggregation.
If velocity gradients during aggregation have to be avoided, the
common situation is that the reactivity of the particles slowly increases, for
example owing to lactic acid bacteria producing acid and thereby a lower
pH and a decreased electrostatic repulsion between the particles. In such a
case, aggregation will begin and be often completed while theW valueis
quite high. In other words, it concernsreaction-limited aggregation. Both
from simulations (by Brownian dynamics) and from experiment it follows
that for largeWthe value ofDis generally 2.35–2.4 and independent ofj
(up to aboutj¼0.25). Accurate predictions cannot (yet) be made, and the
fractal dimensionality has to be experimentally determined. This can be
done by light scattering methods, especially for a dispersion of fractal
aggregates, and from gel properties as a function ofj(see Section 17.2.3).
Short-term rearrangementis another factor affecting the magnitude of
D. It concerns a change in mutual position of the particles directly after
bonding. This is illustrated in Figure 13.8a. The particles roll over each
other until they have obtained ahigher coordination number, which implies a
more stable configuration. When aggregation goes on, fractal clusters can
still be formed, as illustrated in Figure 13.8b, though the building blocks of
the aggregate now are larger. They can be characterized by an effective
radiusaeffor a number of particlesnp, wherenp&ðaeff=aÞ^3. Equation (13.12)
FIGURE13.8 Short-term rearrangement. (a) Examples of particles rolling over
each other so that a higher coordination number is attained. (b) Example of a fractal
cluster in two dimensions, where short term rearrangement has occurred (according
to Meakin). (c) Schematic example of the relations between particle numberNpand
aggregate radiusRaccording to Eqs. (13.12) (np¼1), upper curve; and (13.19), lower
curve.