Chemistry Reference
In-Depth Information
10
viscoelastic solid
1
viscoelastic
liquid zone
hard sphere
transitions
0.1
liquid
ϕ
freeze
= 0.494
weak attraction
(liquid like)
ϕ
melt
= 0.54
hard spheres
3kT
B
/2
0.01
0
0.1
0.2
0.3
0.4
0.5
0.6
volume fraction ϕ
Figure 5.14
The dimensionless internal energy versus volume fraction indicating
empirically defined zones of liquid-like and solid-like behaviour.
The lower volume fraction limit is set by the range of experimental data, the
upper limit is set by the hard-sphere internal energy 3k
B
T/2. In principle, a
prediction of the order-disorder transition should be possible by calculating
the free energy of the two states as a function of concentration. The system with
the lowest free energy would be the favoured one. Russel et al.
19
suggested a
simple approximation based on an effective diameter for the particles. They
argued the order-disorder transition seen in charged repulsive colloids was
analogous to the hard-sphere transition. The charged colloids would have an
effectively larger diameter and hence volume fraction. The transition would
occur when this effective volume fraction achieved j
¼
0.5. In order to establish
the effective hard-sphere diameter Russel et al. examined the integrand in the
Barker-Henderson expression (eqn (5.33)) and noted that it was a very rapidly
changing function of distance. So, for the typical form of repulsive potential at
low electrolyte concentrations and particle radii we can substitute into the
expression the repulsive pair potential:
V
ðÞ
k
B
T
¼
a
exp
kr
ð
Þ
ð
5
:
48
Þ
kr
where k is the Debye-Hu
¨
ckel length proportional to the electrolyte concentra-
tion of the dispersion, a is related to the properties of the particle and the
solvent (Section 3.5.5). The integrand changes very rapidly from zero away
from the particle surface to unity at some distance r
o
. This allows r
o
to be used
