Chemistry Reference
In-Depth Information
Hence, to calculate an increase in viscosity eqn (3.51) must be integrated up to
the value of interest, so
Z
dZ
Z
ð
j
Þ
¼
2
:
5
Z
dj
ð
3
:
52
Þ
This ''effective medium'' or ''mean field'' assumption is easy to understand if
there is a very large size difference between the newly added particles and any
there previously, for example if we think of adding particles to a molecular
liquid, we merely treat the liquid as a structureless continuum. However, when
the dimensions become comparable, the finite volume of the particles present
prior to each addition must be considered, i.e. new particles can only replace
medium and not particles. The consequence of this ''crowding'' is that the
concentration change is greater than expressed in eqn (3.52) and it must be
corrected to the volume available:
Z
dZ
Z
ð
j
Þ
¼
2
:
5
Z
dj
1
j
j
m
ð
3
:
53
Þ
where j
m
is the maximum concentration at which flow is possible - above this a
solid-like behaviour will occur. j/j
m
is the volume effectively occupied by
particles in unit volume of the suspension and therefore is not just the geometric
volume but is the excluded volume. This is an important point that will have
increasing relevance later. Now, integration of eqn (3.53) with the boundary
condition that Z
-
Z
o
as j
-
0 gives
2
:
5j
m
Z
Z
o
¼
1
j
j
m
or more generally:
½
Z
j
m
Z
Z
o
¼
1
j
j
m
ð
3
:
54
Þ
There are two important issues that concern the factor that gives the
excluded volume 1/j
m
. These are (i) what is the effect of shear rate? (ii) what
is the effect of polydispersity?
Few dispersions in everyday use are monodisperse and this will mean
modification to eqn (3.54). A general expectation resulting from polydispersity
is that denser packing may be achieved.
22
The simplest case occurs for bi- or
multimodal systems with very large size differences between each mode, several
orders of magnitude for example. To calculate the zero-shear viscosity of
systems containing 1, 1:3, and 1:3:6 large to small particles as a function of
volume fraction we consider the contribution the volume fraction of each size
