Gel and Glass Transitions in Short-Range Attractive Colloidal Systems - Food Colloids: Self-Assembly and Material Science - page 334

Figure 3, we can argue that the final energies reached by the two different
microscopic dynamics (ND and BD) are equal within numerical error. This is
not a trivial issue, since one would expect the microscopic dynamics to play an
important role in an out-of-equilibrium situation like the one described
above. This suggests that it is possible to use ND to describe the non-
equilibrium dynamics of arrested phase separation, with great advantage in
term of computational time. This possibility can be exploited when we are
interested, as in the present case, in the final arrested structure. But if one is
primarily interested in the kinetics of the process (i.e., how the gel structure
forms), the issues concerning the microscopic dynamics should be properly
addressed.
It is interesting to note that, on the overall range of densities for this type of
quench, the final energy presents a maximum close to the critical packing
fraction. Under equilibrium conditions, one would expect the energy to grow
with the packing fraction; but, in this out-of-equilibrium situation, this is not
the case. The origin of this maximum could be related to the strength of the
critical fluctuations.
We now turn our attention to the structure of the aggregates. We simulate a
one-component system made of 10 4 particles to study large length scales. As
before we set e ¼ 0.005, and configurations equilibrated at T ¼ 1.0 were
quenched at T f ¼ 0.05. We describe the properties that are obtained when the
evolution of the system has stopped, i.e., when the energy has become constant.
Figure 4 shows snapshots of three representative configurations at different
packing fractions. In each of the illustrated situations, all the particles belong to
a single percolating cluster. This structure clearly resembles a gel with a highly
inhomogeneous percolating arrested structure. It is relevant to stress that the
percolating structure is formed during the separation process. On increasing
the density the percolating structure becomes more uniform and the degree of
inhomogeneity is reduced.
One useful concept for gels involves considering the fractal properties of the
aggregates. 36 While this concept applies strictly only to extremely low-density
aggregates, it is natural to try to understand its degree of applicability to the
gels discussed so far. In order to do this we study the integrated radial
distribution function

n ð r Þ¼ 4 pr Z r
1

x 2 g ð x Þ dx ; ð 7 Þ

where r is the number density, and g(r) the radial distribution function that can
be directly evaluated from the simulations. The quantity defined in Equation
(7) represents the average number of particles within a distance r of another
particle. It can be shown 37

that this quantity can be used to study the fractal

(r/r 0 ) d f , where r 0 is a typical length scale and d f the
fractal dimension. The fractal regime is characterized by a typical length scale x ;
for distances larger than x , we have g(r)

dimension since n(r) B

1andd f ¼ 3; the fractal scaling
regime holds for r 0 r r r x . Hence, the length scale x represents the crossover

B