Chemistry Reference
In-Depth Information
Fig. 2.8
Illustration
of a capillary viscometer
g
0
¼ O=a
where
is the angular velocity in radians per second
or in degrees per second. The viscosity is obtained from the following relationship [
15
,
16
]:
a
is the cone angle in radians or in degrees, and
O
¼ t=g
0
¼
3
3
aM=
2
pR
O ¼ kM=O
k
is a constant, specific for the viscometer used. It can be obtained from the relationship [
7
]:
3
k ¼
3
a=
2
pR
The cone and plate rheometers are useful at relatively low shear rates. For higher shear rates
capillary rheometers are employed. They are usually constructed from metals. The molten polymer is
forced through the capillary at a constant displacement rate. Also, they may be constructed to suit
various specific shear stresses encountered in commercial operation. Their big disadvantage is that
shear stress in the capillary tubes varies from maximum at the walls to zero at the center. On the other
hand, stable operation at much higher shear rates is possible. Determination, however, of
Z
0
is usually
not possible due to limitations of the instruments. At low shear rates. one can determine the steady-
state viscosity from measurements of the volumetric flow rates,
Q
and the pressure drop:
DP ¼ P P
0
where,
P
0
is the ambient pressure. A capillary viscometer is illustrated in Fig.
2.8
, where the diameter
of the capillary can be designated as
D
. For Newtonian liquids the viscosity can be determined from
the following equation:
4
¼ pD
DP=
128
LQ
L
where
represents the length of the capillary. The shear stress at the capillary wall can be calculated
from the pressure drop:
s
wall
¼ DDP=
4
L
Also, the shear rate at the walls of the capillary can be calculated from the flow rate [
22
]:
d log
Q
g
0
wall
¼
3
8
Q
3
þ
pD
d log
DP
Wang et al. studied the homogeneous shear, wall slip, and shear banding of entangled polymeric
liquids in simple-shear rheometry, such as in capillary viscometry, shown above [
20
]. They observed
that recent particle-tracking velocimetric observations revealed that well-entangled polymer
solutions and melts tend to either exhibit wall slip or assume an inhomogeneous state of deformation
and flow during nonlinear rheological measurements in simple-shear rheometric setups.
It is important to control the viscoelastic properties of confined polymers for many applications.
These applications are in both, microelectronics and in optics. The rheological properties of such
