Physical Chemistry of Foods

Various quantities are used in relation to the viscosity of dispersions:

relative viscosity: Zrel:

Z

Zs

ð 5 : 7 Þ

specific viscosity: Zsp:

Z
Zs

Zs

¼Zrel
 1 ð 5 : 8 Þ

reduced viscosity: Zred:

1

c

Z
Zs

Zs

¼

1

c

Zsp ð 5 : 9 Þ

intrinsic viscosity: ½ZŠ:

1

Zs

dZ

dc



c¼ 0

¼lim

c? 0

Zred ð 5 : 10 Þ

wherecmeans concentration. Notice that forc¼j, insertion of Eq. (5.6)
into (5.10) yields for the intrinsic viscosity½ZŠ¼ 2 :5. For other systems,
other values for½ZŠare observed (see below);½ZŠis a measure of the capacity
of a substance to increase viscosity. Often, the concentration of the
substance is given as, for instance, kg?m
^3 , rather than volume fraction,
implying that Zred and ½ZŠ are not dimensionless but are expressed in
reciprocal concentration units.

Concentrated Dispersions. For the viscosity of not very dilute
systems, the Krieger–Dougherty equation is often useful. It reads

Z¼Zs 1 

j

jmax


½ZŠjmax

ð 5 : 11 Þ

Herejmaxis the maximum volume fraction (packing density) that the
dispersed particles can have. At that value the viscosity becomes infinite (no
flow possible). For random packing of monodisperse spheres,jmax& 0 :65.
For polydisperse systems, its value can be appreciably higher. Note that now
particle size becomes a variable, though its spread (e.g., relative standard
deviation) rather than its average is determinant.
Equation (5.11) is rigorous for hard spheres in the absence of colloidal
interaction forces, where½ZŠ¼ 2 :5; in the limit ofj?0, it equals the Einstein
equation (5.6). Some calculated results are shown in Figure 5.6. An
important aspect is that a given small increase injgives only a limited
increase in viscosity ifjis relatively small, but if it is close tojmax, the
increase inZ is large. For dispersions of other kinds of particles, the
Krieger–Dougherty equation is not quite exact, but it remains useful,
provided that½ZŠis experimentally determined (since it can generally not be
precisely predicted). However, for deformable particles Eq. (5.11) predicts
values that are markedly too high ifjis fairly close tojmax; for rigid
anisometric particlesZis underestimated at highj(see Section 17.4).