Chemistry Reference
In-Depth Information
where Dr is the density difference between the particles and the medium. The
sedimentation velocity v
0
can be determined by simple rearrangement:
v
0
¼
2Drga
2
9Z
o
ð
6
:
27
Þ
For a concentrated dispersion the particles feel the effects of all their
neighbours and the sedimentation velocity can be represented semiem-
pirically as
8
bj
m
ðÞ
v
v
0
¼
1
j
j
m
ðÞ
ð
6
:
28
Þ
Here b is a constant. The packing fraction, j
m
(s), as we shall see later, may
depend upon the stress applied, in this case a gravitational stress. In most
commonly encountered circumstances the stress is low enough for the packing
of the particles to be independent of the value. However, higher stresses can be
applied during centrifugation and there will be an influence on this value
relative to the g-force used. This packing fraction represents the point at which
the system becomes a viscoelastic solid and will not sediment. The above
expressions do not describe the onset of sedimentation. A long induction time is
possible before this will occur and this is very dicult to predict once a
percolation threshold has been achieved. Our model supposes that our sample
is pseudoplastic at a volume fraction below the maximum packing fraction. If it
displays plastic behaviour the low-shear viscosity will be infinite, although this
does not necessarily mean the sample will not sediment. The gravitational stress
acting on the particle depends upon its mass. For a large particle the local stress
can be relatively large. If this were to exceed the yield stress the interparticle
forces in the system would be unable to develop a structure that will support the
particle and sedimentation can occur. The yield stress required to oppose
sedimentation can be determined by projecting the gravitational force acting on
the particle over an effective area. As a first approximation we can consider the
suspension as an ''effective medium'' that is acting on the surface of the particle,
i.e. over an area 4pa
2
. We can now set up our inequality for the yield stress and
the gravitational stress:
s
Y
4
4pDrga
3
Drga
3
3
:
4pa
2
E
ð
6
:
29
Þ
This expression should be used as an approximate rather than exact formula-
tion that would require interparticle forces and tensor analysis. The difference
between a high zero-shear-rate viscosity and a significant yield stress is an
important one. A dispersion with a high zero-shear viscosity will sediment,
albeit slowly, whereas a system with a yield stress exceeding that in eqn (6.29)
will retain its integrity all the time that yield is maintained. Yielding phenomena
can provide us with important information about storage stability. One
