Chemistry Reference
In-Depth Information
7.2.4 c
orrelaTions
, F
lucTuaTions
,
and
m
orPhology
in
a
m
odel
c
omPlex
l
liquid
wiTh
s
Pherical
i
inTeracTions
Let us now consider a model for which some microstructure can appear. It is clear
from the previous examples that microstructure cannot appear from either a purely
repulsive or a purely attractive model, when the length scale of the interaction is only
that of the size of the particles. Such models are governed by packing effects and
fluctuations. In order to make the model more complex, we need to mix both repul-
sion and attraction in order to introduce a new distance parameter within the range
of the competition between attractive and repulsive interactions. The most realistic
way to achieve this is to include an attractive interaction at short range and a repul-
sive attraction beyond. This way, we encourage particles to cluster, but not beyond
a certain range. This type of model was in fact used to mimic the experimentally
observed aggregation in globular protein solutions and colloid-polymer solutions
(Stradner et al. 2004). It was observed that varying the salt concentration in the first
solution and that of the polymers in the second, one could tune the ratio of attraction
versus repulsion, hence producing aggregation of the globular protein or the colloids.
The aggregation was monitored through the appearance of a prepeak in the neutron
scattering experiments. This work triggered intense research to find an interaction
mechanism that would produce such a prepeak. By modeling these two systems as
an effective one-component system, with particles interacting through a short-range
attraction and long-range repulsion, one can indeed reproduce the rich phase behav-
ior of the experimental systems (Broccio et al. 2006; Archer and Wilding 2007).
The experimental systems are not a one-component system and contain, in par-
ticular, water, salt, and polymers. It is the effective interaction resulting from the
contributions of the other components that provide the combination of short-range
attraction and long-range repulsion. For now, we will not worry about how long-
range repulsion can appear and we shall focus on the shape of the structure factor
when particles aggregate into domains.
Let us consider a one-component system with hard spherical particles interacting
through two additional Yukawa interactions,
+∞
r
<
σ
ur
()
=
(7.27)
r
r
exp
−
ra
−
exp
−
r
r
≥
σ
κ
κ
1
2
Such interactions are shown in Figure 7.5 in order to describe the purely attractive
case, as well as the weak and strong repulsions at large distances. In order to under-
stand how clustering can emerge from this type of interaction, we now treat the
direct correlation function associated with this model in the mean spherical approxi-
mation fashion, which amounts to setting
cr
()
=−
β
ur
()
r
≥
σ
(7.28)
