Chemistry Reference
In-Depth Information
fraction in turn. Then, each addition of particles is to an effective medium and
eqn (3.55) becomes the product relationship:
1
:
24
¼
Y
n
Z
ð
0
Þ
Z
o
1
j
n
j
m
ð
3
:
55
Þ
where there are n narrow modes in the distribution. The result is given in
Figure 3.12. The effect of the ratio of big to small particles by volume in a
bimodal mix is plotted in Figure 3.13. However, we must recall that these
results are for the case where the size ratios are very large. When this is not
the case the excluded volume of the larger particles will tend to increase as the
packing efficiency is reduced. There is, however, an effect of the shear rate.
Under quiescent conditions, particles in a dispersion move around with
Brownian motion. Computer simulations of hard spheres
20,21
indicate that as
the concentration is increased a liquid to solid or freezing transition occurs at
j
¼
0.495, (and that on dilution a melting transition commences at j
¼
0.54),
i.e. [Z]j
m
¼
1.24. The effect of high shear rates is to organise the particles into a
layered or string-like arrangement as the shear field dominates the Brownian
motion. This structure was first shown by Hoffman
23
and has been confirmed
by optical rheometry, neutron scattering and computer simulation. The upper
limit for j can therefore be set at the packing of monodisperse spheres in layers
with hexagonal symmetry, hence for this density of packing j
¼
0.605, i.e.
[Z]j
m
¼
1.51. eqn (3.54) may now be rewritten for monodisperse hard spheres as
Z
ð
0
Þ
Z
o
1
:
24
1
:
51
j
0
:
495
;
Z
ð
N
Þ
Z
o
j
0
:
605
¼
1
¼
1
ð
3
:
56
Þ
Low Shear Rel Vis. /1
Low Shear Rel Vis. /1:3
Low Shear Rel Vis. /1:3:6
10
3
10
2
10
1
10
0
10
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ϕ
Figure 3.12
The low-shear-limiting viscosity for unimodal, bimodal and trimodal size
distributions calculated from eqn (3.55).
