JOURNAL OF COLLOID AND INTERFACE SCIENCE 2~, 5 1 7  5 3 0
(1966)
The Dependence of NonNewtonian Viscosity on Molecular Weight for "Monodisperse" Polystyrene R O B E R T A. S T R A T T O N Mobil Chemical Company, Metuchen, New Jersey
Received February 3, 1966 The apparent viscosities as a function of shear rate of five narrowdistribution polystyrenes were obtained using a CIL capillary rheometer. Relationships were derived between the shape of the flow curve and its dependence on molecular weight. Shift factors derived from these relationships were employed to reduce all the data (and some literature data) to a single master flow curve. It is shown that shift factors predicted by current theory do not satisfactorily reduce the data. A mechanism for nonNewtonian flow is postulated which ascribes the decrease in viscosity with increasing shear rate to an increase in the spacing of coupling entanglements. INTRODUCTION The determination of molecular weight and molecular weight distribution from the dependence of apparent viscosity on shear rate has been the subject of a great number of investigations (15) dating back to the early work of Philippoff (1). Only empirical relationships and qualitative interpretations, however, have been obtained. What is needed for a quantitative study of the effects of molecular weight and molecular weight distribution on the flow properties is a series of samples with known molecular weight and very narrow molecular weight distribution. These may then be blended to synthesize broader distributions. Until recently it was impossible to obtain sizable quantities of wellcharacterized materials. However, with the advent of the "living polymer" technique (6) and, alternatively, large sealefraetionation (7, 8), such samples became available. In 1960 It~udd (9) reported flow data on some anionieally polymerized polystyrenes, and during the course of the present investigation, Balhnan and Simon (10) presented a paper en a similar series of polystyrenes. (The latter's data will be reinterpreted below.)
Once the dependence of the flow Froperties of a polymer melt on molecular weight and molecular weight distribution has been ascertained, it is desirable to interpret this behavior in terms of a molecular theory. Several theories have been proposed (1113) but, as will be seen subsequently, all are inadequate. The present study was undertaken to provide data on a wellcharacterized series of polystyrenes with the goal of making quantitative interpretations and obtaining a better understanding of the mechanism of flow. EXPERIMENTAL E~/IATERIALS
A series of polystyrene samples (S103, S109, $111, S108) prepared by the "living polymer" technique (9) was generously furnished by Dr. H. McCormick of the Dow Chemical Company. To supplement this series, a sample of lower molecular weight (GDSB1) was synthesized by Dr. C. Geacintov of these laboratories using the techniques developed by Szwarc and his school. The polymerization was carried out in tetrahydrofuran with sodium a517
518
STRATTON TABLE I Sample
(M)n ~K 10Sb
GDSB1 $103 S109 Slll S108
0.48~ 1.17 1.79 2.17 2.42
0.45 1.09 1.67 2.00 2.36
[~]c
0.276 0. 522 0.697 0.786 0.869
~0 X i04
(pOise)d
0.153 2.88 12.4 21.9 33.5
Light scattering (14) in eyelohexane at 50°C. Osmometry (14) in toluene at 30°C. ° In toluene at 25°C. At 183°C. methyl styrene tetramer as the initiator; water was used to kill the living ends. The samples were characterized by light scattering and osmometry (14). The results are given in Table I; the breadth of the molecular weight distribution as mirrored by the ratio (M}~/(M},~is seen to be quite narrow ( < 1 . 1 ) . The values of (M}~, though lower thart those reported by Rudd (9), are believed to be more accurate. For comparison the present result for S109 is in excellent agreement with that for the NBS sample 705 (identical with S109) obtained by light scattering (15) and with a recent determination (16) using a magnetically suspended equilibrium ultracentrifuge. This concurrence lends confidence to the values in Table I. Intrinsic viscosities were obtained in toluene at 25.0°C. and were in excellent agreement with those quoted by Rudd (9) for the same samples. The present data are also shown in Table I. MELT YISCOSITIES Shear rateshear stress data were obtained with a compressed nitrogen gasdriven capillary rheometer ~ (similar to that described by Bagley) (17) over a range of shear stresses between 1 X 105 and 2 X 10 ~ dynes/era. 2. The die had a diameter of 0.0430 inch, a length to diameter ratio of 19.49, and an included angle at the entrance of 80 °. Shear rates at the wall, calculated assuming a Newtonian fluid, were 1F. J. Mullowney Co., Yardley, Pennsylvania.
corrected for nonparabolic velocity profile using the Rabinowitsch equation (18) # = D[ a~ + ~ ( d log
D/d log ~)],
[1]
where # is the true shear rate at the wall, % is the shear stress at the wall, and D is (4/~rR ~) times the volumetric flow rate, R being the die radius. Because the nonNewtonian behavior is quite pronounced at higher shear stresses, the correction factor in brackets in Eq. [1] amounted to as much as 2.2. The shear rates were not corrected for the increased density due to compressibility of the melt (less than 2 % change at the highest pressures used) (19). Kinetic energy corrections were negligible. Shear stressshear rate data were also obtained on $111 using a die with a length to diameter ratio of 3.80. The difference in shear stress at the wall at the same shear rate (with Rabinowitsch correction) for the two dies averaged 2 % and was randomly positive or negative over the entire range of shear rates. This finding for S l l l , in agreement with that of others (20, 21) for polymers with narrow molecular weight distributions, coupled with the limited quantity of the samples caused us to assume that the end correction (22, 23) was negligible in all our measurements. The accuracy of the Newtonian shear rates D and shear stresses is estimated to be 5% and 3%, respectively. The necessity of measuring a slope graphically for use in Eq. [1] increases the maximum error in the true shear rate to perhaps 10 %. The shear stresses at the wall and shear rates at the wall (with Rabinowitsch correction) are given in Table II for the various samples. The temperature was 1 8 3 ° + I°C. except for sample GDSB1, for which 1 5 9 ° + 1°C. was used to obtain comparable shear stresses. Since no antioxidant was added to the polymers, checks for the occurrence of degradation during the measurements were made by determining the intrinsic viscosity of pieces of the extrudate. The results agreed with the initial values to within 2 %,
NONNEWTONIAN VISCOSITY AND " M O N O D I S P E R S E " POLYSTYRENE TABLE II
in linear viscoelastic measurements (24, 25).
CAPILLARY EXTRUSION DATA (RABINOWITSCtt CORRECTION INCLUDED) Sample
log
(dynes/
log ~, (sec.D Sample
log (dynes/ c ¢,t~fl )
log
5. 030  0 . 2 6 5 5.150 0.130 5. 270 0.005 5.395 0.145 5. 530 0.300 5. 645 0.470 5. 775 O.680 5. 900 1.010 6,030 1.710 6.150 2.450
$103b
5.030 0.615 S108b 5.150 0. 740 5. 280 0.880 5.405 1.010 5.530 1.165 5. 650 1. 320 5.775 1.510 5.900 1. 775 6.030 2.155
5.030 0.445 5.150 0.315 5. 280 0.160 5.405  0.020 5. 530 0.145 5. 650 0.350 5. 775 0.615 5.900 1.035 6.030 1.735
S109b
5.125 5.250 5. 505 5. 625 5.745 5. 875 5. 990 6.125 6.250
?~r = ~ p o T o / p T ;
[2]
% = ~a~ ;
[3]
~ = ~poTo/pTa~.
[4]
(sec.~)
$111b GDSB1~ 5. 290 1 . 0 0 5 5.405 1.130 5. 530 1.260 5. 645 1. 385 5. 775 1. 545 5.900 1. 685 6. O3O 1.870 6.095 1.965 6.150 2.080 6.200 2.150 6. 270 2.285
5.380
519
0.085 0.220 0. 380 0. 530 O.700 O.890 1.190 1.615 2.425 2.920
At 159°C. At 183°C
Attempts to use the extrudate for a second melt viscosity determination showed that degradation did occur upon repeating the thermal history (evidenced by the decrease
Here po and p are the sample densities at the reference temperature To and the ternperature of measurement T (°K.). The function ar was derived from the work of H6gberg et al. (26) with To = 456°K.: log ar =  6 . 0 6 ( T 
To)/
[5]
(148.6 [ T  To). All results discussed subsequently will have been reduced to 183°C. The apparent viscosities are plotted doubly logarithmically against shear rates for the five samples in Fig. 1. Because there is still a nonzero slope at the lowest obtainable shear rates, no zero rate of shear viscosity could be extracted from these graphs. An extrapolation procedure, therefore, was used. The data for Sample S109 1 were plotted as v against shear stress as proposed by Ferry (27). The data at low shear stresses can be fitted very well with a straight line which, when extrapolated to zero shear stress, yields the reciprocal of the zero rate of shear viscosity. Sample $109 was then used as the reference, and the log viscosity against log shear rate plots for the other samples were superposed on it by horizontal and vertical shifts. Superposition was excellent over the entire range of shear rates. The vertical shift factors a, were converted into zero rate of shear viscosities n0 using 70 = a~.n0 ($109),
in apparent viscosity, ~/~). RESULTS After the Rabinowitsch correction was performed, the data for sample GDSB1 were reduced to 183°C. using the reduced variables defined by analogy to the corresponding quantities found to be applicable
where n0 (S109) is the extrapolated value. The Newtonian viscosities thus obtained are listed in Table I and are plotted ]ogarithmieally against log (M)~ in Fig. 2. The slope of the line obtained by the method of least squares (3.34) is in good agreement with the usual value (3.4) and at variance
520
STRATTON
\\ \ \
"~
5
S
4 _o
3 2
I
o
5
log ~" (sec I)
FIe. 1. Logarithmic plots of apparent viscosity against shear rate for samples with different molecular weight. Pip up, 48,500, successive 45° rotations clockwise correspond to 117,000, 179,000, 217,000, and 242,000; all da%a reduced to 183°C. The dashed line has a slope of 0.82, I'
'
f.
'[
.
.
.
.
I .....
5.5
5.0
g 4.5 o
4.0
3.5
4.7
I
,
1
4.9
5.1
5.3
log W
Fie. 2. Logarithmic plot of zero rate of shear viscosity (see text) against weightaverage molecular weight at 183°C.
with the recent finding of Tobolskg et aI. (4.0) (28) on the same samples. The latter's results are indirect, however, being deter
mined from stress relaxation data by a graphical integration. Our result also differs from that of Rudd (3.14) (9) on the same samples. Because the latter's zero rate of shear viscosities were extrapolated by eye on viscosityshear stress plots from a region of pronounced nonNewtonian behavior to zero shear stress, the present findings are believed to be the more accurate. It is recommended that the procedure which has been described be used to obtain zero rate of shear viscosities when the Newtonian range is experimentally inaccessible. The values obtained will then be internally consistent although the absolute magnitude may be in error to the extent that the one extrapolation is in error. This procedure, of course, is valid only if the log viscositylog shear rate plots for the various samples will superpose over the entire range of shear rates, which fact presupposes that all the samples have nearly the same molecular weight distribution. The logarithms of the horizon£al shift factors ah are plotted in Fig. 3 against Iog (M}~ giving a stra.ight line with a slope of 4.10.
NONNEWTONIAN VISCOSITY AND "MONODISPERSE" POLYSTYRENE i
i
olog
f'°l
= _ ( log) /log (Oolog
2.0
[6]
log ~ )~1
with log
=
log fV/ .~
[7]
2
log
0~
4.7
521
419 Ioq (M>~
From Fig. 1 it is seen that in the limit of large ~ the first term on the righthand side of Eq. [7] is zero; the second term is just the slope in Fig. 2 which will be denoted by a. In the limit of large
Flo. 3. Logarithmic plot of the horizontal shift factor a~ against weightaverage molecular weight.
DISCUSSION D E P ~ ' n ~ c ~ oF TI~n APPARE~ VISCOSITY ON MOLECULAR WEIGHT The most remarkable feature of Fig. 1 is the coincidence of all the results at high shear rates. This fact has been recently reported also for a series of polydimethylsiloxanes (29) and for some linear and t e t r a c h a i n  branched narrow molecular weight distribution polystyrene samples (30). This behavior has profound implications with regard to the mechanism of flow and the dependence of the apparent viscosity on molecular weight. For a monodisperse polymer at constant temperature, the reduced viscosity (n/n0) is a function only of the two independent parameters   shear rate and molecular weight. (That this must be the ease is apparent from the superposition of the flow curves described above.) Here v is the apparent viscosity and v0 the zero rate of shear viscosity. For brevity we shall denote this reduced viscosity by the letter H in the following derivations. From the total differential of H as a function of the two variables we obtain
where ;~ is the limiting slope in Fig. 1 (dashed line) and
(01og 3 0 log M ] ~ ~ ~"
[9]
However, since plots of log n/n0 against log q at various molecular weights will superpose by a simple horizontal shift, Eq. [9] must hold over the whole range of variables covered in these experiments. It follows upon integration of Eq. [9] that for two samples with molecular weights M1 and 2k% ('fl/%)~r = (M~/MI) ~1~,
[10]
or since Fig. 2 yields no = K M ~ we can rewrite Eq. [10] ('Yl/5~2)H = (no2/nol) (M2/M1) ~(1+~)1~. [11] Hence the product ~noM~(~+~)~ should be a constant at a given level of reduced viscosity. This expression was cast into this form for comparison with the predictions of Bueche's theory (2, 11) which defines a terminal relaxation time ~ : 2
rl = 12 noMR/~r p R T .
[12]
Here M~ is defined as a rheologicM molecular weight with no indication of the
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[911
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q
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,oI~ )
'oIe]
[gI]
\ :~ Sot e /
\ .z. So~ e / .I0
\~, sot e /
\:~sote / =
\~soto/
oqa.aat u'go o~a 'X~.t.soast.A qmoaqsdd~ oq~t jo uo.~#t.utJop oq:~ ~m.s£1 "uoDonpoa o~sa a~oqs oqc~ moaj puno t osos[~ o~ poc~s[oa g[dm~.s oa~ pou.t~c~qo snq~ sao~a~j ~j.~qs Is~uoz!aoq oq,L uo~3t.sodaodns oA~.qo~ o~ £iW~uoz~.aoq dq paonpaa oq oslv u~a vc~p ~aog aq~g "I "~.X m. om.i paqssp aq~ moaj gg'0 = g pros g "~.t~ moa~ W'8 = a~ q~.~a~ g/~a m Isnba s.~ g "~.~d m. odois oq& "[II] ao [0I] "sba u~. ~(ag/L,c) q:~.~ po}vnbo s! (8 "~.
~O£&VkLLS
~8~
NONNEWTONIAN VISCOSITY AND "MONODISPERSE" POLYSTYRENE
{secb 5.6
0
5.2
/
5
IO .~ 4,4 o
~. 4,0
ii
3.2
2.81 I
r
I
i
f
4.6 4.8 5.0 5.2 5.4 log {M>w ]~IG, 4. Logarithmic plots of apparent viscosity against molecular weight with shear rates as indicated.
although the data for sample GDSB1 certainly suggest that the viscosity decreases, if any, would be very small at moderate shear rates. The nonlinearity of the curves at constant shear rates is readily apparent even at fairly small values of this parameter. A plot analogous to Fig. 4 but using shear stress rather than shear rate as the variable also clearly yields nonlinear results for the present data as would be predicted by Eq. [16]. Collins and Bauer (38) have shown that (0 log v/0 log M)~ cannot be a constant for polymer systems which obey the BueehetIarding (2) empirical flow equation. It is of interest to compare the predictions of Eq. [11] with those of the Buechettarding semiempirical equation (2) which has a limiting slope at high shear rates of 0 log n/0 log ~ equal to  0 . 7 5 and a terminal relaxation time given by Eq. [12]. Equating, as before, MR with {M}~~°+~)/~ and letting a be 3.40, we obtain Mn = (M}~/13. Brodnyan et al. (39) showed that MR was smaller
523
than (M}w in disagreement with the present postulates. They state, however, that the various samples they compared did not in general fit the BuecheHarding curve very well. The limiting slope ~ which determines the exponent in Eq. [11] is very difficult to measure for polymers with a broad molecular weight distribution. Large shear rates must be attained, and in this range shear heating and melt fracture can become troublesome. ]~XPERIMENTAL VERIFICATION OF THE I:~EDUCED VARIABLES
In Figs. 5 and 6 the present data and the data of BMlman and Simon (10) are plotted logarithmically using the reduced variables ~/~0 against "i,~o(M}w/pRT. (The RabinowRsch correction has been applied to the latter's data taken st 205°C.) These are the reduced variables predicted by the BuecheHarding method using the terminal relaxation time from Bueche's molecular theory (11). It is readily apparent that these coordinates do not satisfactorily reduce the data which in these plots tend to fan out at the higher shear rates. Lines have been drawn through data points of the same molecular weight showing that the deviations are not random but are systematic with respect to this parameter. The data are replotted in Figs. 7 and 8 using log w0(M}~" 0.7~/pRT as the horizontal coordinate as suggested by Eq. [11] with a = 3.4 and ~ = 0.82. The reduction of all the data is seen to be excellent. THE NATURE OF NONNEWTONIAN
FLow The molecular mechanism accounting for the deviation from Newton's law has long been a subject of much debate. Bueche (11) and Takemura (12) obtain a decrease in viscosity arising from the periodic deformation of a molecule in dilute solution resulting from its rotation induced by the solvent flow. The phase angle between the shearing force and the deformation is a
I
I
I
I o
2 I
0
t
I
I log ~ " ~ M ) w//oRT
2
Fro. 5. Logarithmic plots of reduced apparent viscosity against shear rate reduced according to Bueehe's theory. Molecular weights identified as ia Fig. 1; lines are drawn through data points of constant molecular weight. I
I
i
I
I
I
I
I
0
I
2
5
o
to 9 ~""~o(M)wIp RT FIa. 6. Logarithmic plots of reduced apparent viscosity against shear rate reduced according to Bueche's theory. Data of Ballman and Simon (10). Pip up, molecular weight is 43,000, successive 45° rotations clockwise correspond to 62,000, 107,000, 127,000, and 180,000. Temperature is 204.5°C. 524
I
t
I
i
F F" I.0 o
2.0 E
r
I.0
I
0
1.0
log "~'~o(M)O" 7 5/p R T FIo. 7. Logarithmic plot of reduced apparent viscosity against reduced shear rate according to Eq. [11]. Molecular weights identified as in Pig. 1. I
I
r
0
O
%
~L.o o
2.0 I
2.0
I.0
I
I
0
1.0
0.75 Io g "~"~o(M)w /toRT
FIG. 8. Logarithmm plot of reduced apparent viscosity against reduced shear rate according to Eq. [11]. Data of Ballman and Simon; molecular weights identified as in Fig. 6. 525
526
STRATTON
function of the shear rate and hence the apparent viscosity is also. The resulting equations, however, are not in quantitative agreement with available data on polymer melts. The predicted dependence of n/n0 on molecular weight is incorrect (see preceding section). Both theories exhibit (40) a limiting slope, Eq. [8], of ~/~; experimentally the value is invariably larger in absolute magnitude. The reason for the failure would appear to be the same as that accounting for the rapid rise of viscosity with increasing molecular weight of concentrated solutions and polymer melts above a certain critical molecular weight Me, namely, entanglement coupling. Neither of the theories mentioned contains this feature of polymer chain interaction in its model (the theories were derived for isolated molecules), and the failure to reproduce the experimental data is, therefore, not unexpected. Lodge (41) has presented a molecular theory which views a concentrated polymer solution as a homogeneous network composed of temporary physical entanglements. The result is a constant viscosity independent of shear rate. This follows, he states, from his assumption that the "junction age distribution fmlction" is independent of flow history or in other words that the entanglement junction density is not a function of shear stress. Indeed ¥amamoto (42) showed in very general terms that this should be the case. The representation of a polymer melt as a dynamic entanglement network with a shear rate dependent degree of coupling has been elaborated upon recently by Graessley (13). Although the model he uses to calculate the nonNewtonian viscosity apparently is oversimplified (it predicts a limiting slope, Eq. [8], of _3/~ in disagreement with the present and other data), the qualitative picture is appealing and will be used in the following interpretation of our data. Since in this representation the shape of the retardation spectrum would change with shear rate,
it would not be possible to calculate the apparent viscosity and normal stresses from linear viscoelastic measurements as Pao (43) has theorized except at very low shear rates, where the viscosity is Newtonian. For the same reason plots of n' (e)/n0 or I n(¢0)* l/n0 and 7(3;)/n0 against w and 3% respectiveIy, would not be expected to have the same shape. Osald et a/. (44) have recently shown experimentally the change in shape of the relaxation spectrum with increasing shear rate. Two alternative analyses of the data in terms of a shear rate dependent degree of entanglement are now presented. For a monodisperse undiluted polymer the molecules in steady flow with apparent viscosity y(~) may be viewed as being in the same state of entanglement as that of another network of entangled molecules of lower molecular weight with a zero rate of shear viscosity of magnitude equal to 7(3)). If we consider again Fig. 4, the unusual feature is that at high molecular weight and high, constant shear rate the apparent viscosity is independent of molecular weight. Because the flow behavior in this molecular weight range is dominated by the effect of entanglements, one interpretation of the independence is that the various samples all have the same number of effective entanglemeats. This number can be calculated by extrapolating the horizontal lines in Fig. 4 to intersect the log 70 vs. log (M>. line. The ratio of the molecular weight at this intersection to M~ ( = 3.3 X 10~ for polystyrene) (32) equals the number of effective entanglements per molecule. Here we follow the recommended convention (37) of interpreting M~ as the molecular weight at which there is an average of one entanglement per molecule by analogy to the theory of gelarich. Then, from the dashed line in Fig. I, the zero rate of shear viscositymolecular weight relation (Fig. 2), and the viscosity m at the critical molecular weight obtained by extrapolatilfg the line in Fig. 2 to Me as in Fig. 4, the dependence of the average
NONNEWTONIAN VISCOSITY AND "MONODISPERSE" POLYSTYRENE I
l
I
f
527
I
I0~t ~"~o(M>0"75//~ RT cO
5.6
0.5 0.0 4.8
0.5 m o
1.0 ~'4.0 o
3.2
f I
f
J I 4.4
I
I
I
4.8
I
I
5.2 log
(M)' w
FIG. 9. Logarithmic plots of apparent viscosity against weightaverage molecular weight at the indicated reduced shear rates; data from Fig. 1. number e of effective entanglements per molecule on the shear rate can be calculated. 

3,
[17]
k~/ Here nl is the apparent viscosity given by the dashed line in Fig. 1 at a shear rate of 1 see.  1 B y substituting the present data into Eq. [17] we obtain e =
8 . 4 ~;0.2~.
Equation [17], of course, is valid only in the region where the apparent viscosity as a function of shear rate has reached its limiting slope B. An alternative analysis of the data which gives a better physical picture of the flow phenomenon can be performed by plotting log n against log (M)~ at constant values of • 0.72 the reduced shear rate "/no(M)~ /pRT. This
leads to a series of parallel lines of slope 3.4 as is shown in Fig. 9. It has been shown (32, 33) that the Newtonian viscosity increases linearly with molecular weight below Mc if the viscosity is measured under or corrected to conditions of isofree volume. Allen and Fox (32) have obtained an accurate value for ]4c for polystyrene of 3.3 X 104. Using this information we constructed the heavy dashed line in Fig. 9 which represents the variation of viscosity with molecular weight accounting for chain length but not for the effect of entanglements. The light dashed lines are extrapolations of the data beyond experimentally accessible shear rates to the line of slope one. The intersections of the heavy and light dashed lines represent critical molecular weights for entanglement coupling at the respective reduced shear rates. The
528
STRATTON
5.ot
I
1
I
L
~
I t
2.2
2
< 1.4
0.6

I
I.5
I
I
1
0.5
log
I
0.5

I
0.75
/pRT
Fro. 10. Plot of entanglemen£ spacing parameter A against logarithm of reduced shear rate. The dashed line is the v~lue of £ under zero shear rate conditions.
family of lines with slopes of 3.4 denote lines of constant entanglement spacing (37) A (defined as the average number of chain bonds between coupling entanglements) determined by the intersection points. From this set of intersections the increase in apparent entanglement spacing with increasing reduced shear rate can be obtained gnd is shown in Fig. 10. The quantity 0.75/ pRT may be taken as proportional to a relaxation time r~ characteristic of the time necessary for reentanglement upon cessation of steady flow. At low shear rates, ~, is much greater than r~ and the entanglement spacing is the same as that obtained under zero flow conditions; the viscosity is Newtonian. As the shear rate is increased, the disentanglement produced by the shearing field is not able to recover during the experimental time scale (#~) and the entanglement spacing increases until an equi
librium between the entangling and disentangling processes is reached. It is to be expected that these processes proceed via segmental motion as does diffusion, but their exact nature as indeed that of the entanglements themselves must be left unspecified at the present. It is easily shown that in the region where the terminal slope ~ has been attained, A is proportional to the reduced shear rate raised to the fl/(1  a) power. It must be pointed out that these two analyses do not give equivalent results; the apparent number of entanglements per molecule for a given sample at a given shear rate is different when calculated by the two methods. This is a consequence of the fact that the first method takes no account of the molecular weight but only of the apparent number of entanglements. The second method, because it accounts for both these effects, is thought to be the more consistent
NONNEWTON1AN VISCOSITY AND "MONODISPEP~SE" POLYSTYI~ENE
but no choice between the two is possible at the present. The absolute magnitude of the limiting slope ~ is seen to be related to the ease of disentanglement and hence would be expected to be a function of the chemical structure of the polymer. Equation [11] predicts that for a polymer with I2 1 larger than that for polystyrene, a larger value of the quantity "~no/pT would have to be used to obtain an equivalent viscosity loss at equal levels of molecular weight. Indeed, this is just the result obtained by Gruver and Kraus (21) for ardonically polymerized polybutadiene, which because of its lack of bulky side groups might be expected to disentangle rather easily under a shearing stress. This discussion has been limited to polymers with very narrow molecular weight distributions. For broader distributions, the equations used here will have to be modified to account for the heterogeneity of sizes and hence of relaxation times. Experiments to attempt to determine the contributions of the various sizes to the measured apparent viscosity of polydisperse materials are now in progress in these laboratories. ACKNOWLEDGMENT The author wishes to thank Drs. R. L. Ballman add R. It. M. Simon of Monsanto Chemical Company for making their data available to him in tabular form, Dr. C. Geacintov for synthesizing the low molecular weight sample, and Mr. A. F. Butcher for assistance in the experimental work. REFERENCES 1. PHILIPPOFF, W., "Viskosit~it der Kolloide." Steinkopff, Dresden and Leipzig, 1942. 2. BUECHE, F., AND HARDING,S. W., J. Polymer SeA. 32, 177 (1958). 3. ScHuRz, J., J. Colloid Sci. 14, 492 (1959). 4. CHII'qAI,S. N., AND SCHNEIDER,W. C., Rheol. Acta 3, 148 (1964). 5. PORTER, ~:~. S., AND JOHNSON, J. F., Proc. 4th Intern. Contr. Rheol., Part 2, p. 479 (1965). 6. SZWARC, M., LEVY, M., AND MILKOVICH, i~., J. Am. Chem. Soc. 78, 2656 (1956). 7. CANTOW,M. J. R., PORTER, 1:~. S., AND JOHNSON, J. F., J. Polymer Sci. C1,187 (1963).
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