Chemistry Reference
In-Depth Information
function with distance over longer length scales than is the case for the shear
modulus. Thus, we need to generate pair-distribution functions that are precise
over long length scales and are representative of the details of the curvature of
g(r). Possibly, only computer simulations will enable this to be achieved. There
are also limitations to the applicability of a perturbation scheme as a method of
including the colloid pair potential. There are simple indications of this, for
example the maximum packing that can be achieved with spheres is 74%. When
the attraction is high between the particles the perturbation scheme can readily
predict values that exceed this. Under such circumstances where a reliable
prediction of the structure is not possible the shear modulus can be represented
in a power law form:
G
ðÞ¼
Aj
m
ð
5
:
40
Þ
where m
¼
3 to 4. This range of values is appropriate to the data in Figure 5.11.
5.5 CHARGE-REPULSION SYSTEMS
Charged colloidal spheres can be produced using both organic and inorganic
particles. In low-electrolyte conditions, the interparticle forces are dominated
by the charge-repulsion forces between the particles. Interrogation of the
microstructure using scattering techniques indicates that at a critical concen-
tration a phase change occurs. This displays a structural change akin to the
order-disorder transition seen with hard spheres. This occurs at lower volume
fractions than with hard spheres. At low particle volume fractions and low
ionic strengths the initial transition tends to a body-centred cubic order (bcc).
Further increases in concentration rapidly take the structure toward fcc order.
There can be zones of coexistence between ordered and disordered structures,
as seen with hard spheres. The transition is accompanied by a change in the
rheology of the systems, as was shown by Lindsay and Chaikin.
16
The tran-
sition between liquid-like and solid-like behaviour has been correlated to the
dimensionless energy density. As we have seen with weakly attractive systems
we can describe the internal energy, osmotic pressure and high-frequency elastic
modulus by a hierarchy of integral equations. These expressions still hold for
charge-repulsive interactions. However, once the ordered state has been
achieved we can visualise the colloidal particles constrained to a lattice site.
The nearest neighbours dominate the interactions between the particles and are
the most significant term in controlling the shear modulus. The centre-to-centre
separation for given order and packing R is given by
1
=
3
j
m
j
R
¼
2a
ð
5
:
41
Þ
where j
m
is the maximum packing fraction, which for fcc is 0.74 and for bcc is
0.68. Once the system achieves an ordered structure, the number of nearest
neighbours z remains invariant with an increase in concentration. The coordi-
nation number for fcc is z
¼
12 and for bcc is z
¼
10. This is an important
