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preparation may result in a small amount of unsuspected branching in
samples of ostensibly linear molecules. Such adventitious branched
molecules can have an effect on viscosity which far exceeds their
numerical abundance. It is quite possible that anomalous experimental
results may be due to such effects.
1. The following are approximate os (in dyne cm-2) versus ó datat for
three different samples of polyisoprene in tetradecane solutions of
approximately the same concentration:
tW. W. Graessley, T. Masuda, J. E. L. Rovers, and H. Hadjichristidis,
Macromolecules 9:127 (1976).
The Viscous State
sample: P-4 S-6 H-3
Mw (g mol"1) 1.61 X 106 1.95 X 106 1.45 X 106
Description Linear Four-armed star Six-armed star
Ñ (g cm-3) 0.0742 0.0773 0.0778
7 (sec -1) os X 10~3 os X 10"3 as x 10-2
0.6 0.7 --- ---
0.8 0.9 - -
1 1 0.3 ---
2 2 0.6 -
4 4 1 0.15
8 5 2 0.3
10 6 3 0.4
20 7 4 0.8
60 --- 7 2
100 - 8 4
From plots of these data, estimate the Newtonian viscosity of each
of the solutions and the approximate rate of shear at which non-Newtonian
behavior sets in. Are these two quantities better correlated with the
molecular weight of the polymer or the molecular weight of the "arms"?
2. Wagner and Dillont have described a low-shear viscometer in which
the inside diameter of the outer, stationary cylinder is 30 mm and the
outside diameter of the inner, rotating cylinder is 28 mm; the rotor is
driven by an electromagnet. The device operates at 135°C and was found to
be free of wobble and turbulence for shear rates between 3 and 8 sec-1.
The conversion of Eq. (2.7) to Eq. (2.9) shows that F/A = (T?)(dv/dr)
(instrument constant) for these instruments Evaluate the instrument
constant for this viscometer.
3. A fluid of viscosity 17 is confined within the gap between two
concentric cylinders as shown in Fig. 2.3b. Consider a cylindrical shell
of radius r, length 1, and thickness dr located within that gap.
(a) What is the torque acting on the shell if torque is the product
of force and the distance from the axis and F/A = rjr dco/dr?
(b) Under stationary-state conditions, the torques at r and at r +
dr must be equal, otherwise the shell would accelerate. This means that
the torque must be independent of r. Show that this implies the following
variation of ñî with ã: ñî = -B/2r2 + C, where Â and Ñ are constants.
(c) Evaluate the constant Â by noting that ñî = coex, the
experimental velocity, at r = R and ñî = 0 at r = fR.
(d) Combine the results of (a), (b), and (c) to obtain Eq. (2.7).
tH. L. Wagner and J. G. Dil\on,Polym. Prepr. 22:260 (1981).
4. The two-term version of equaton (2.33) contains four parameters. It
may be written as
F / sinh"1 (0i 7) sinh-1 (/327) \ .
= a,ft------------:----- + a202 ---------------- 17 = r)7
A ' 0i7 P27 '
It is sometimes observed that 0X is sufficiently small that sinh-1
1 for all 7 values, in other words, an exclusively Newtonian
contribution. In such a case 17 = t?n + a2j32 sinh_1(027)/02 7. Consider
both limiting values of (1/07) sinh_1(07) from Table 2.1 to suggest a
procedure whereby t?n, a2 > and 02 could be evaluated from experimental
data describing F/A versus 7, assuming that this model applies and that
data are available over a sufficiently wide range of 7 values._
5. The bulk viscosity of polystyrene (Mw = 371,000) at 200°C was
measured by Graessley and Segalt at different rates of shear. At low
rates of shear t?n = 330,000 P and drops off with 7 approximately as
7 (sec-1) 0.03 0.1 0.3 1 3
V/VN °-89 °-76 °-56 °-35 °-22
Does a 0 value of 1, 10, or 100 sec work best in Eq. (2.28) to
describe the variation of 17 with 7 for this sample? Note that no single-
term version of the Eyring theory gives a totally acceptable fit.
6. A slightly different but useful way of defining the viscosity
average molecular weight is the following:
-- Ss fs M; M;a M,
2: f: M:
where fj Mj is the weighting factor used to average Mja. A
satisfactory way of treating many polymer distributions is to define
f. = _L e"M^"
My3 = -----------------