Techniques for the characterization of physical gels - Physical Gels from Biological and Synthetic Polymers - page 57

probability: for high sticking probability, in diffusion limited cluster aggregation
(DLCA) the fractal dimension of the clusters is df f ≈

1.7 (Weitz and Oliveria, 1984 ),
while for low bonding probability, in reaction limited cluster aggregation (RLCA) the
fractal dimension is higher, df f ≈ 2.3 (Weitz et al., 1985 ). For weak reversible interactions,

there is a substantial restructuring and condensation from the earlier stages of the phase
separation, and therefore the structure approaches a uniform structure with increasing
simulation times.
The effect of the strength of inter-particle interactions on the microstructure of
particle gels generated by Brownian dynamics simulation was also examined by
Bijsterbosch and Bos ( 1995 ). The authors simulated various potentials other than LJ,
for example where particles can form

flexible linkages when they approach within a
certain bonding distance r < d bond and where once the bond is formed it is permanent. In
addition, pairs of particle were assumed to interact with distance-dependent forces
which could be either repulsive (r < d max ) or attractive (r > d max ). The authors observed
that, under conditions of thermodynamic phase separation, with an LJ potential, the
clusters have time-dependent structures, with a process of short-range densi cation

with formation of coarser blob-like structures and large voids, as previously described
by Lodge and Heyes. On the other hand, with the second,

flexible linkage interaction
potential, they clearly observed a fractal structure with a low dimensionality (df f =1.9)
of clusters within an intermediate range of r, typically less than r <10a, with a the
particle radius.
However, for such systems the value of df f by itself does not give a full description of
the microstructure of the aggregated phase. It is interesting to observe how different the
two snapshots simulated by Bijsterbosch and Bos ( 1995 ) appear ( Figures 2.21 and 2.22 );
these exhibit the same fractal dimension in the same intermediate range but were
produced with different interaction potentials. The very different appearance of the two
network structures in terms of porosity and strand thickness indicates the danger of
adopting fractal dimensionality uncritically as an appropriate single parameter to
describe the structure of .

The fractal dimension df f of the intermediate regime is only one aspect of the gel
structure. Also important is a scaling prefactor n 0 that Bijsterbosch and Bos introduced
explicitly for the average number n(r) of particles within a range r of another particle:

' simulated particle gels '

d f

r
r 0

n ð r Þ¼ n 0 : ð 2 : 38 Þ

In this formulation, r 0 may be an arbitrary length. If r 0 is taken as the primary particle
radius, then n 0 = 1. The value of n 0 is a measure of the average number of particles in the
primary clusters from which the fractal scaling regime is built: a large value of n 0
indicates a coarse microstructure; low values of n 0 (possibly even less than unity)
imply open structures. The rate of restructuring can be measured by the rate of increase
of n 0 and depends on the attractive interactions. For instance in Figure 2.21 , n 0 = 0.5 and
in Figure 2.22 , n 0 =2.