whereDris the density difference between particle and continuous phase.
Especially for emulsions with droplets over 2mm in diameter, the effect of
sedimentation on aggregation rate can be significant.
Question
What shear rate would be needed in an aqueous dispersion of particles to push a
particle pair ‘‘over the repulsion barrier’’ to obtain an aggregate (doublet)? Take as
an example the situation depicted in Figure 13.3. Moreover, what shear rate would
be needed to pull the particles apart if they are aggregated in the primary minimum?
Answer
The force to be overcome is the derivative of the interaction free energyVwith
respect toh. The maximum force is where the interaction curve is steepest. For the
curve in Figure 13.3, the slope beyond the maximum is steepest wherehequals about
3.2 nm; the slope then equals 2.8kBTper nm, which corresponds to about 10^11 N
ðkBT& 4? 10 ^21 N?mÞ. The shear stress equalsZC. To obtain the force, the stress
must be multiplied by the surface area on which it is acting, roughlyd^2. Since
d¼0.5mm andZ¼1 mPa?s, the force will be about 25? 10 ^17 timesC, implying that
the latter has to be about 40,000 s^1 (to obtain 10^11 N), which is quite a large value.
The force needed for pulling the particles apart is more difficult to estimate.
We need the steepest slope to the left of the maximum, but that value is very
uncertain, since very slight surface irregularities and solvation effects can greatly
affect the interaction curve at such short distances. Taking the slope atV¼0, it is
about 10 times the one just calculated, and the force would then also be 10 times
larger, as is the shear rate needed. The latter then becomes irrealistic.
Note The particles involved are quite small. If the DLVO theory holds, the
colloidal interaction force is proportional tod, whereas the force due to the
shear stress is proportional tod^2. This then means that the shear rate needed is
proportional to 1/d. This is an example of a fairly general rule, namely that
external forces tend to become of greater importance for larger particles.
Note The calculations done here may suggest that it can simply be derived
whether the capture efficiency equals unity or zero. This is not the case,
because of the stochastic variation inamentioned above.13.2.3 Fractal Aggregation
When perikinetic aggregation occurs and the particles that become bonded
to each other stay in the same relative position as during bond formation,