particles. In a situation as in Figure 13.6, position 3, the particles are pressed
together, but a moment later, in position 4, the force is pulling at the
particles; they then may even separate, leading to position 5.
Even in the absence of colloidal repulsion, wherevan der Waals forces
will cause attraction, a will be smaller than unity. Approximately,
¼0.6(A/ZCd^3 )0.18, whereAis the Hamaker constant; this leads to
a-values between 0.1 and 0.5 for most situations. Note that it concerns an
averagea-value. This is becauseawill depend on the position at which the
particles collide, i.e., on the value ofd/a(see Figure 13.6). Note also that a
higher shear stress and larger particles cause a smaller value of.
If there is considerablecolloidal repulsionbetween the particles, the
capture efficiency can be very much smaller. On the other hand, the shear
stress may also push the particle pair ‘‘over the maximum’’ if the interaction
curve is of the type depicted in Figure 12.1, solid curve.
Other Complications. Athigh volume fractions, the encounter rate
will be more than proportional toj, because the effective volume available
for the particles is decreased owing to geometric exclusion. This also applies
to perikinetic aggregation. For orthokinetic aggregation a high volume
fraction will, moreover, affect the capture efficiency, because the stress
sensed by the particles will be greater thanZC; see Section 5.1.2.
Other flow types, especially elongational flow, give other results.
Equation (13.8) will not alter very much, but the capture efficiency may be
greatly affected. Elongational flow exerts greater stress on a particle pair
than simple shear flow (Section 5.1).
Particle shapecan also have a large effect. Anisometric particles have
an increased collision radius as compared to spheres; see the discussion in
relation to Figure 5.3. Moreover, capture efficiency is affected, but
prediction of the effect is far from easy.
Sedimentation. If particles are subject to sedimentation, this may
also lead to enhanced aggregation rate, since particles of different sizes will
move with different velocities through the liquid; see Section 13.3. This
implies that a large particle can overtake a smaller one, and a kind of
orthokinetic aggregation occurs. This will be the case if (a) there is a
substantial spread in particle size and (b) particle motion over a distance
equal to its diameter takes less time by sedimentation than by Brownian
motion. The condition for the latter is approximately
d> 2
40 kBT
pgjDrj0 : 25
& 17? 10 ^6 jDrj^0 :^25 ð 13 : 11 Þ