Techniques for the characterization of physical gels - Physical Gels from Biological and Synthetic Polymers - page 56

long period of time, while the particles interacting over a shorter range (24:12 or 36:18
potential) show the local development of a crystalline FCC structure, but at a larger scale
they form a tenuous network. In other words, for the same reduced temperature the LJ
12:6 particles are ' kinetically trapped ' in an amorphous glassy structure.

< 0.1, the particles assemble into clusters with convex
surfaces which form strand-like structures through the box. At higher volume fractions,
ϕ

At lower volume fractions, ϕ

> 0.1, suspensions evolve into structures with many more concave surfaces. In systems
quenched deep into the two-phase region, phase separation proceeds via a spinodal
decomposition mechanism: a peak appears in the structure factor at low wave vectors
that displays an immediate increase in intensity and a decrease in position, indicating the
growing importance of longer length scales in the system.
It is clear that the structure of the aggregated phase is sensitive to small variations in
parameters such as interaction potentials, quench depth and volume fraction. In the LJ
system, cluster growth is relatively simple. On the time scale of the simulation, any
rearrangement towards a crystalline ordered phase is retarded for the 12:6 potential. For
short-range interaction potentials, particles adopt a crystalline structure more readily, an
observation which could be relevant in protein crystallization (Bonneté et al., 1996 ;
Wolde and Frenkel, 1997 ).

2.6.1 Fractal dimensions

In irreversible aggregation of spherical particles, three spatial scales of structure are
expected (Dickinson, 1987 ): (i) short-range order from packing and excluded-volume
effects, (ii) medium-range disorder associated with the rami

ed structure of the aggre-
gating clusters, and (iii) long-range uniformity for a homogeneous material. The pair
distribution function g(r) measures the probability of finding a particle at a distance r

from another particle. In the first region (shortest distances), strongly damped liquid-like

oscillations of g(r) are expected out to a few particle diameters (r < r 0 ), while in a second
region a fractal scaling regime is found with

d f 3

r
ξ

g ð r Þ ≈ r 0 ≤ r ≤ ξ: ð 2 : 37 Þ

A non-fractal, uniform structure with (g(r) = 1) appears beyond some characteristic
correlation length

. When a fractal structure appears, a power-law region in g(r)is
observed for distances over which the fractal structure holds, with df f the fractal dimen-
sion in ( 2.37 ).
For the cases investigated by Lodge and Heyes ( 1999a , 1999b ), the fractal dimension
of the clusters was almost always found close to df f ≈

ξ

3(g(r) = 1). For the systems
investigated, and on a relatively short time scale after the quench, other authors have
found lower fractal dimensions, down to df f ≈

1.4. Their fractal dimensions nevertheless
increased with time, persisting only for very slowly separating systems, i.e. at high
temperatures. It is clear that fractal dimensions are mostly associated with particles
forming permanent bonds, and then the fractal dimension depends on sticking