Structure of Self-Assembled Globular Proteins - Food Colloids: Self-Assembly and Material Science - page 36

Figure 1 Computer simulation of irreversible aggregation of hard spheres (2 vol%) to
form aggregates (left image) and a gel (right image). (Reproduced from ref. 1.)

power law:

m ¼ aR d g : ð 1 Þ

The quantity d f is the so-called fractal dimension(ality), and it can have any value
between 1 and 3. If particles assemble into compact structures, we get d f ¼ 3; and
if they assemble to form a rod, then we get d f ¼ 1. The pre-factor a is larger if the
elementary unit of the self-similar structure is denser. The elementary unit is, of
course, larger than the individual self-assembling particles. While the value of d f
cannot be predicted theoretically, it has been determined experimentally 2 and in
computer simulations. 3,4 Irreversible diffusion-limited aggregation gives d f ¼ 1.8
as long as the aggregates are, on average, far away from each other.
It can be easily seen that the density r of aggregates with fractal dimension
smaller than 3 decreases with increasing radius of gyration: r / m = R 3 / R d f 3 .
The implication is that, as the assembly process progresses, the average distance
between the aggregates decreases until they finally fill up the whole space. The
process by which close-packed aggregates join together is different from
assembly in the dilute state, and so it results in the formation of aggregates
characterized by a larger fractal dimension on longer length-scales, i.e.,
d f ¼ 2.5. 5 This process is called as percolation and it leads to gelation. There
is thus a transition between two types of assembly process, each giving rise to
self-similar structures, but with different fractal dimensions. The complete
process has only recently been elucidated in detail using computer simulation. 1
(Here we have ignored the possibility that the aggregates sediment when they
become large, in which case a precipitate is formed instead of a gel.)
Random aggregation does not lead to monodisperse clusters, but to a
distribution of sizes. The polydispersity may be characterized by the ratio of
the weight-average molar mass M w , and the number-average molar mass M n .
Dilute aggregation leads to a relatively narrow size distribution (M w /M n ¼ 2), if
the aggregation is diffusion-controlled. But, if it is reaction-controlled, the
distribution is broad, and M w /M n increases with increasing M w . The percolation
process gives rise to such a broad distribution that the apparent fractal dimen-
sion determined by scattering techniques is lower, i.e., d f (apparent) 2.0. 5

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