relation,
½Z 0 ¼KhMia¼KhMi^3 n^1 ð 6 : 6 ÞwhereMis expressed in daltons. The relation between the exponentsaandn
follows from the following proportionalities. Intrinsic viscosity is propor-
tional to the hydrodynamic volume of the polymer molecules, hence
½Z!Nr^3 g=nN¼r^3 g=n. From Eq. (6.4) we haverg!ðn^0 Þnand for one kind of
polymer we also have thatn!n^0 !M. Consequently,½Z!M^3 n^1.
Polymers vary widely in the value ofK, which can in general not be
rigorously derived from theory. It may further be noticed that the range
n¼ 0 :5 to 0.6 corresponds toa¼ 0 :5 to 0.8. This range applies only for very
long linear polymers in fairly good solvents. In practice,avalues of 0.5 (for
amylose or dextran) to almost unity are observed. Some very stiff and
charged polysaccharides can even exhibit an exponent >1. Branched
polymers generally havea< 0 :5; for amylopectin (highly branched) it equals
about 0.3. Some data are given in Table 6.1. Anyway,½Z, and thereby
viscosity, greatly depends on molar mass of the polymer, the more so for a
higher value ofa.
Strain Rate Dependence. The outline given above is, however, an
oversimplification, because it has been implicitly assumed that the flow
would not affect orientation or conformation of the polymer coil. This is not
true. It is always observed thatZ, and thus½Z, is affected by the shear rate
applied. As discussed in Section 5.1.1, various types of flow can occur and,
more generally, we should say strain rate or velocity gradient, rather than
shear rate. However, we will restrict the discussion here to simple shear flow.
An example is shown in Figure 6.6, lower curve. At very low shear
rate, the solution shows Newtonian behavior (no dependence ofZon shear
rate), and this is also the case at very high shear rate, but in the intermediate
range a markedstrain rate thinningis observed. The viscosity is thus an
apparent oneðZaÞ, depending on shear rate (or shear stress). It is common
practice to give the (extrapolated) intrinsic viscosity at zero shear rate, hence
the symbol½Zin Eq. (6.6). The dependence ofZon shear rate may have two
causes.
First, all molecules or particles rotate in a shear flow, but if they are
not precisely spherical, the flow causes anorientationaligned in the direction
of the flow to last for a longer time than orientation perpendicular to the
flow (Section 5.1.1). This implies that average flow disturbance, and thereby
viscosity, is smaller. The alignment depends on the rate of rotary diffusion
of the particles in relation to the magnitude of the shear rate. The rotary