Chemistry Reference
In-Depth Information
Fig. 2.7
Illustration of a
cone and plate rheometer
time as the polymer begins to relax and continues along the whole length of the chain. For short
chains this relaxation to zero takes place at a fairly constant rate. For very long chains, however, the
relaxation rate tends to be in three stages. It starts at a certain rate at stage one, but after a while,
noticeably slow down at some point, and at stage two the modulus remains relatively constant over
some period of time. After that, at stage three, the relaxation is resumed again at a rapid rate until full
equilibrium is reached. At stage two there is a period of relatively constant modulus that is not
affected by the chain architecture and the material resembles a rubber network. The length of the third
stage, however, is profoundly affected by the molecular weight, by the molecular weight distribution
and by long-chain branching of the polymer.
Koga and Tanaka [
23
] studied the behavior of normal stresses in associated networks composed
of telechelic polymers under steady shear flow. They showed numerically that the first and second
normal stress coefficients reveal thickening as a function of shear rate and that the sign of the
second normal stress coefficient changes depending on the nonlinearity in the chain tension,
the dissociation rate of the associative groups from junctions and the shear rate by analytic calculation
they showed that in the limit of small shear-rate, the sign inversion occurs by the competition
between the nonlinear stretching and dissociation of associative groups. Thus, the molecular mecha-
nism of the sign inversion is shown to be similar to that of thickening of the shear viscosity.
In the behavior of polymeric liquids two quantities are important. These are steady-state recover-
able shear compliance,
J
s
(as shown above) and steady-state viscosity at zero shear rate,
0
. These
quantities are related:
0
s
recoverable shear compliance
J
¼ g
r
=s
0
0
¼ s
0
=g
ss
zero shear viscosity
g
0
ss
is the shear rate and
where
g
r
is the total recoil strain. Both shear compliance and shear viscosity
can be obtained from creep studies. The product of the two, zero shear compliance and zero shear
viscosity is the characteristic relaxation time of the polymer:
0
s
t
0
¼ J
0
There are various techniques for determining the viscosity of molten polymers. One commonly used
piece of equipment is a cone and plate rotational viscometer. The equipment is illustrated in Fig.
2.7
:
The molten polymer is placed between the bottom plate and the cone, and the cone is rotated at
constant speed. Shear stress is obtained from the following relationship:
t ¼
3
M=
2
pR
where
is the cone radius in centimeters (or meters). The
shear rate can be obtained from the following equation
M
is the torque in dynes per centimeter, and
R
