Chemistry Reference
In-Depth Information
allows the value of D to be related to the mechanism of aggregation. Typical
values are for:
1. Rapid aggregation, called diffusion-limited aggregation (DLA), when
species touch they stick. Particle-particle aggregation gives D
E
2.5;
aggregate-aggregate aggregation D
1.8.
2. Slow aggregation, called rate-limited aggregation (RLA), in which species
have a lower sticking probability. Some are able to rearrange and densify
the floc, D
E
2.0-2.2.
The lower the value of D the more open the packing. Suppose we now imagine
a dispersion that has been driven to instability by, for example, changing
temperature. We can visualise sites for the nucleation of flocs occurring
randomly throughout the whole volume of the dispersion. The total number
of primary particles remains unchanged so we can determine the volume
fraction of flocs:
3
D
j
f
¼
j
a
f
a
ð
6
:
61
Þ
Given that there is a relationship between the yield stress and the concentration
we might expect a power relationship between them reflecting the fractal index,
i.e. s
Y
pj
m
.
There is a concentration limit to the applicability of the fractal approach. As
simultaneous nucleation occurs the flocs will grow and as they grow they may
eventually touch. Further growth would require the interpenetration of the
flocs and when this occurs the notion of a fractal floc is lost as the structure
changes. We can roughly estimate the limits on this by assuming the flocs reach
a packing fraction before interpenetration. We need at least three particles in an
aggregate otherwise connectivity is not possible. The number in the nucleus of
the aggregate we can approximate as (a
f
/a)
¼
3. So, for a floc packing of say
62%, the maximum volume fraction without interpenetration is given by:
0
:
62
¼
3
3
D
j
ð
6
:
62
Þ
So, it is possible to determine a maximum limit where the fractal structure
begins to interpenetrate. For the various values of D we find
D
¼
1.8 j
max
¼
0.17
D
¼
2.0 j
max
¼
0.21
D
¼
2.5 j
max
¼
0.36
Once interpenetration occurs the resistance to deformation increases markedly,
so for example we would expect compaction of a sediment to become limited, as
would further concentration in a filter press. It is worth emphasising the point
that this is a simplistic approach as prior to interpenetration of the clusters,
