Chemistry Reference
In-Depth Information
5.4 WEAKLY ATTRACTIVE SYSTEMS
The interparticle forces between some systems of colloidal particles can be
adjusted such that any pair of particles in the dispersion are weakly attracted
towards each other. The energy of the attraction must not be so great as to
cause a permanent contact to form. Typically, an energy of attraction of up to
20k
B
T will allow the particles to remain in close proximity but be readily
dispersed by gentle agitation. At very low levels of attraction, the interparticle
energies cause the normal Brownian collisions to become modified. The par-
ticles dwell together as a pair longer and as a consequence the equilibrium
structure in the dispersion becomes modified. This can in principle be detected
by the scattering of radiation as the mean interparticle spacing between the
particles will change. There will be a corresponding change in our structure
factor S(q). We would like to know how this change in both order and diffusive
motion away from the hard-sphere condition affects the relaxation behaviour,
the elasticity, the pressure and the viscosity. To achieve this we can take our
measured structure factor and calculate a quantity termed the pair-distribution
function g(r). The relationship between these quantities is given by:
Z
N
1
2p
2
rr
g
ðÞ¼
1
þ
ð
S
ðÞ
1
Þ
q sin q
ð
dq
ð
5
:
27
Þ
0
Z
N
S
ðÞ¼
1
þ
4pr
q
ð
g
ðÞ
1
Þ
r sin q
ð
dr
ð
5
:
28
Þ
0
In order to evaluate one from the other a numerical implementation is required.
The pair-distribution function is a tremendously useful quantity. If we consider
a dilute suspension, the particles are widely spaced with an average number
density of particles given by r. As the concentration is increased, the material
becomes more liquid-like and a short-range order develops, i.e. the particle
concentration is no longer uniform but fluctuates about r. Suppose we freeze
our system for a moment and select a test particle in our dispersion. If we
examine the local concentration a radial distance r from the test particle, we will
find regions where the concentration r(r)atr is less than the average and
regions where it is greater than the average. The further we travel away from
the test particle, the more the local order reduces and the closer r(r)istor.We
can define our pair distribution or radial distribution function as
g
ð
r
Þ¼
r
ð
r
Þ
r
ð
5
:
29
Þ
and as r
1. We do not always have the luxury of determining
g(r)orS(q) experimentally directly on our system and so computer calculations
are useful. These have been optimised against computer simulations by
-N then g(r)
-
