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Solution viscosities of linear flexible high polymers
JOURNAL
OF COLLOID
SCIENCE
SOLUTION
Robert
17,
270-287
(1962)
VISCOSITIES OF LINEAR HIGH POLYMERS’
FLEXIBLE
Simha2 and Jacques L. i&kin3
Department of Chemical Engineering, Received
September
New York University,
New
York
6, 1961
INTRODUCTION The viscosities of macromolecular systems have been studied frequently from two alternative points of view and, hence, under two extreme conditions of concentration: (1) to deduce information about the size and shape of the solute unit in dilute solution, and (2) to elucidate the mechanism of transport phenomena in essentially bulk systems of large molecules. There also exist several investigations in intermediate ranges of concentration which have led to moderately successful empirical viscosity-concentration relations. For infinitely dilute solutions, the well-known and successful hydrodynamic theories of the intrinsic viscosity for solutes of various shapes (1) have been developed. The extension of these theories to finite concentrations, i.e., the introduction of hydrodynamic interactions, has, for the most part, been restricted to first-order perturbations (1) or very dilute solutions in which the average solute-solute distances are large compared with the solute dimensions. Hydrodynamic theories for concentrated systems have been developed for compact spherical particles only (1). Some time ago, Weissberg, Simha, and Rothman (2) (WSR) set out to investigate in detail the solution viscosities of flexible polymers and the effect of the solvent environment in the intermediate region of concentration, where the pervaded volumes of the polymer coils, as measured at infinite dilution, start to overlap. Depending on the molecular weight and the solvent used, this occurred at concentrations varying from about 0.5 to 3 g./dl. The following observations were made by these authors: In good solvents, the curves of the quantities T~,./([v]c) vs. concentration for different molecular weights are approximately superimposable, provided the 1 Part of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Engineering Science. 2 Present address: Department of Chemistry, University of Southern California, Los Angeles, California. 3 Present address: Socony Mobil Oil Company, Research Department, Brooklyn Laboratory, Brooklyn 22, New York. 270
SOLUTION
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POLYMERS
271
concentrations are expressed in units of CO . Here co is the concentration of incipient overlap, as derived from molecular dimensions at infinite dilution. corresponding Thus the reduced variable, C/CO., defines approximately states and this applies over a range of c’s extending beyond the usual linear region of the Q,./c vs. c curves. As the solvent becomes poorer, this approximation increasingly fails. The first derivatives of these reduced functions are considerably smaller in good than in poor solvents. In the latter, they approach the curves which result for rigid solutes, such as globular proteins or resin particles in suspension. Finally, older data suggested that the corresponding states representation increasingly fails also in good solvents when the molecular weight is reduced below a number average of about 50,000. There the curves also rise more steeply than in the conventional range. Their data illustrated that the sharp increase in the viscosities of suspensions of compact solutes which is observed when C/CO approaches unity, is absent for flexible polymer solutions. This is to be expected because of the low density of the polymer coils. On the other hand, the observed differences in good and poor solvents can not be due to the differences in the average coil densities for a given molecular weight. The latter are, for a given polymer type, approximately inversely proportional to the intrinsic viscosity. Therefore, the same coil densities can be generated in good and in poor solvents by the proper selection of molecular weight in each solvent. A factor reducing the concentration dependence of the viscosity is the osmotic compression of the polymer coils in good solvents which results from the interchain segment repulsions. This effect is absent or minimized in poor solvents. In order to relate the quantitative differences between good and poor solvents to this compression effect, it is necessary to obtain independent experimental or theoretical information about the latter. WSR were concerned with the region C/COm 1 and gave an approximate treatment in terms of the osmotic pressure which has been recently modified (3) and adjusted for application to the concentrated systems involved here. In order to obtain an estimate of the magnitude of the compression to be expected, and in the absence of quantitative experimental results, WSR assumed that it can be obtained directly from the apparent intrinsic viscosity at finite concentrations, viz.,
This, in turn, amounted to the assumption that the viscosity in a solvent in which the osmotic pressure assumes the ideal value, &conditions, is a universal function of c/co and, in particular, an exponential function. It became clear from the limited experimental information that this was at best only approximately true. Nevertheless, better than the right orders of
272
SIMHA AND ZAKIN
magnitude for the expansion factors of the coils, which result from intramolecular segment repulsions, could be derived in this way from the viscosities at C/COin the neighborhood of unity. The above discussion suggests investigations along the following lines: first, an experimental verification of the behavior in &solvents; secondly, an investigation of relatively small in comparison with high molecular weights in good and poor solvents. WSR examined polystyrene solutions only with number-average molecular weights varying by a factor of ten. The range has now been extended considerably and other polymers and copolymers also included. EXPERIMENTAL
Viscosity-concentration measurements in the Newtonian region were made with three different kinds of polymers-polystyrene, polymethyl methacrylate, and copolymers close to the azeotropic composition (51.4 mole % styrene (4))-in a good solvent, toluene at 3O”C., and in their respective theta solvents. Theta solvents were chosen close to 30°C. For polystyrene, cyclohexane at 34.O”C. was used (5) and for polymethyl methacrylate 4-heptanone at 31.8”C. (6). Theta solvents for the copolymers were mixtures of methyl isopropyl ketone and n-heptane at 25.O”C. the compositions of which were estimated from data obtained by Dr. B. N. Epstein (4) Table I summarizes the characteristics of the fractions investigated. A. Polymers
1. Polystyrene (PS). A thermally polymerized (1SO’C.) polymer obtained from Dr. Leo A. Wall of the National Bureau of Standards was fractionTABLE Characteristics
Polymer Polystyrene
Sample number
Copolymer Styrene)
(48.8%
PSl PS2 PS3 PS4 PS5 PS6 CPl
Copolymer Styrene)
(47.8%
CP2
Polymethyl methacrylate
PMMl PMM2
I of Polymers
Method of polymerization
Tok!ene 30” c.
T#ta
Estimated molecular weight
Anionic Anionic Thermal 180°C. Anionic Thermal 180°C. Thermal 0.11 mole To beneoyl peroxide @ 60°C. 0.11 mole y0 benzoyl peroxide @ 60°C. Free radical Free radical
0.142 0.278 0.308 0.732 0.782 2.026 0.594
0.112 0.186 0.198 0.362 0.365 0.701 0.321
15,000 40,000 45,000 150,000 165,000 600,000 215,000
1.333
0.555
630,000
0.373 0.802
0.192 0.298
135,000 470 ) 000
SOLUTION
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FLEXIBLE
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POLYMERS
273
ated. Seven broad fractions were precipitated from a l%Yo solution in MEK at 25°C. by addition of isopropyl alcohol. Two of these were refraetionated from 1 Yo solutions. Narrow middle fractions were taken from each and dried by the frozen benzene technique. Four narrow distribution polymers were obtained from Dr. H. W. McCormick of the Dow Chemical Company. Three were anionic polymers prepared with a sodium naphthalene complex. The fourth was a high molecular weight fraction from a thermal polymerization. One of the former, PS 2, is identical with S-3 described by McCormick (7) as having a ratio of Mm/ill, = 1.35. Another, PS 4, is believed to be similar to his S-9, which has a ratio of about 1.1. Molecular weights were estimat,ed from log-log plots of [q] in toluene at 30°C. vs. &!, (2). 2. Polymethyl Methacrylate (PMM). A sample of Lucite Molding Powder Composition 41 was fractionated. This polymer was furnished by Dr. F. W. Billmeyer, Jr., of the Polychemicals Department of DuPont and was prepared by a free radical polymerization following bulk kinetics (8). Five broad fractions were precipitated from a 2% solution in MEK at 25°C. with methanol as the precipitant. The first and last were refractionated from 0.9% solutions. Middle fractions from each of these were separated and dried from frozen benzene solution. Molecular weights were estimated from recalculated M, vs. [v] data of Kapur in toluene at 30°C. (9). S. Copolymer (CP). A copolymer of styrene and methyl methacrylate near the azeotropic composition (51.4 YOstyrene) was fractionated. The copolymer was also furnished by Dr. Wall and was prepared at 60°C. with 0.11 mole % benzoyl peroxide as catalyst and contained 49.7% styrene. Although the difference in chemical composition of individual copolymer chains prepared from a ratio of monomers close to the azeotropic ratio is small, the system MEK-diisopropyl ether at 25°C. suggested by Stockmayer et al. (10) was used to minimize fractionation by composition rather than by molecular weight. Five broad fractions were precipitat,ed from a 0.5% solution of MEK at 25°C. by addition of diisopropyl ether. Two of these were refractionated from 0.5 % solutions. Middle fractions from each of these were separated and dried from frozen benzene solutions. Molecular weights were estimated from recalculated M, vs. [r] data of Kapur in toluene at 30°C. (9). B. Solvents Toluene (Fisher’s Reagent Grade) and cyclohexane (Phillips Pure Grade 99+ mole Yo) were distilled over sodium wire and middle cuts were taken. Methyl isopropyl ketone (Eastman’s Analytical Reagent Grade), n-heptane (Phillips ASTM Grade 99.5+ mole %), and 4-heptanone (Sealey) were fractionated in a 42-theoretical plate adiabatic packed column and constant boiling middle fractions were collected. Physical properties of the solvents are given in Table II.
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SIMHA
AND
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TABLE Physical
II
of Solvents
Properties
Viscosity (Cmtistokes&) At subscript temp., (“C).
Solvent Toluene Cyclohexane Methyl isopropyl ketone (MIPK) n-Heptane di-n-Propyl ketone Mixed solvent 0.435 ml. n-heptane/l .OOO ml. MIPK 0.415ml.n-heptane/l.OOOml.Mt[PK 8 Based
on
1.002
centipoises
for
Copolymer
Theta
water TABLE
Mole
70 styrene 24.5 51.4 74.2
in copolymer
(azeotrope)
at
Density (g./mZ.) at subscript temperature PC).
Refractive index at 200c’
0.604530 .o 1.00234.0 0.774831.8
0.8567ao.o 0.7646ad.o 0.800225.s 0.679923.0 0.804531.~
1.4958 1.4256 1.3881 1.3871 1.4065
0.553&&L? 0.556Gr,.o
0.7612zs.a 0.763926.0
-
2O.O”C. III
Compositions
at 15%‘.
(4)
Theta solvent composition at 25” C. ml. n-heptanelml. MIPK 0.290 0.435 0.620
Epstein’s data (4) showing the variation of mixed solvent composition at the theta point at 25°C. with copolymer composition are shown in Table III. It was assumed at the start of these viscosity studies that the copolymer compositions were close enough to the azeotrope to use that equivalent ratio of solvents, i.e., 0.435 ml. n-heptanelml. MIPK. This solvent for the low molecular weight sample, CPl, was used for its measurements. The theta solvent composition for CP2 was estimated by interpolating the above data and was used for its viscosity measurements. C. Polymer Solution Preparation,
Viscosjty, and Density Measurements
Master solutions were made up by weight to concentrations of at least C = 2/[~] where sample size permitted and their densities and those of the solvents were measured with a Lipkin pycnometer at the appropriate temperatures. Densities at intermediate concentrations were linearly interpolated. The master solutions were filtered through a medium porosity sintered glass filter into a Friedman La Mer weight buret using a setup similar to that suggested by Rothman (11). The quantity of solution drained into a viscometer was measured by weight difference. Ubbelohde viscometers with dilution bulbs were used, employing in most cases a closed system similar to that described by McElwain (12). The sys-
SOLUTION
VISCOSITIES
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TABLE
FLEXIBLE
HIGH
275
POLYMERS
IV
Viscometer Characteristics
viscometer A B-l
in
Capillary diameter (mnt.)
0.400 0.600
Capillary length (cm.)
10 10
Efflux bulb volume (CC.)
5.0 4.2
Viscometer A*
0.00202 0.01123
constants B*
1.28 0.39
* A and B are defined by the equation t/p = At - B/t, where t is in seconds, p is gm./ml., and q is in centipoises.
tern was modified to avoid applying vacuum to any portion of the solution. Solution viscosities were measured down to concentrations corresponding to relative viscosities of 1.05. Least squares computations were made on calibration measurements with standard oils and purified solvents to calculate the viscometer constants. The viscometer characteristics are shown in Table IV. Flow times in Viscometer A generally exceeded 300 sec. and in Viscometer B-l, which has a very low kinetic energy correction constant, they generally exceeded 150 sec. Kinetic energy corrections were applied to all readings. The temperature of the bath was controlled to within fO.Ol”C. The average error in the timing operation is estimated to be of the order of 0.1 sec. At the maximum flow rates, the average shear rate at the capillary walls in Viscometer A was about 1800 sec.-l and in Viscometer B-l about 880 sec.-l. These shear rates are relatively low and significant non-Newtonian effects would not ordinarily be expected under these conditions even for the higher molecular weight polymers in good solvents. Fox and Flory (13) state that the shear correction is small for intrinsic viscosities under 3 dl./g. Sharmon, Sones, and Cragg (14) found no effect in polystyrene in toluene up to intrinsic viscosities of 2 dl./g. and in cyclohexane up to intrinsic viscosities of about 1.3 dl./g. The maximum intrinsic viscosity in this work was about 2 dl./g. for polystyrene in toluene. For polystyrene in cyclohexane, the maximum intrinsic viscosity was 0.7 dl./g. WSR were also unable to detect any appreciable shear rate effect in their studies of polystyrene in toluene. The possibility that the viscosities of the high concentration solutions have some shear rate dependence cannot be completely discounted. However, it should be noted that the most concentrated solutions have the highest viscosities and hence the lowest shear rates. Furthermore, in many of the runs, sections of the viscosity concentration curves were measured in both Viscometers A and B-l, and no apparent shear dependence was found, although the shear rate at equal viscosities was about 35% higher in Viscometer B-l than it was in A.
276
SIMHA AND ZAKIN RESULTS
To compare the data for the different types of polymers and widely varying molecular weights and concentrations in different solvents, viscosityconcentration curves are presented in the reduced form suggested by WSR, that is ~,p./(c[~]) as a function of C/Q . Here COis the concentration at which polymer molecules, or rather the equivalent spheres closely packed, would begin to overlap if they retained the same volume in concentrated solution that they pervade at infinite dilution. Thus, c/co would represent relative volumes if no compression of molecules occurred as the concentration increases. The region of incipient overlap corresponds to c/co values of unity. Most of the data presented extend to c/a values of about 2 or c x 2/[7]. The intrinsic viscosities were computed by least squares analyses of the linear portions of the plots of r]8P./c vs. c (15) and the co’s were obtained from the relation CO= 1.08/h] (3). In representing the reduced viscosity-concentration data (Figs. l-4), a family of Baker equations in the reduced form: TSP. -MC
_
(1+[&“-, > Fit
has also been plotted for the selection of appropriate values of n. These can be used as a convenient device for comparing the relative positions of the experimental curves. The parameter n may assume both positive and negative values. As n goes from 0 to 00, the slopes increase monotonically. The curves steepen still further for negative n’s, increasing in slope with smaller negative values of n. Toluene at 30°C. is a good solvent for all the polymers used by us, as may be inferred from the respective intrinsic viscosities in toluene and theta solvents shown in Table I. The viscosity-concentration curves for the six polystyrene samples are shown in Fig. 1. The Baker equation is represented by the dotted lines. In this good solvent, the steepness of the curves increases as molecular weight decreases. At the higher molecular weights, curves for the 600,000, 165,000, and 150,000 samples are close to each other at n X 3. This agrees with the results of WSR, but the spread between curves was found to be slightly larger than they reported. In the 40,000 molecular weight range, the curves are displaced upwards and n is between 3 and 4. The parameter increases rapidly between 40,000 and 15,000 molecular weight where n reaches a value larger than 6. Turning to the results for the copolymers and polymethyl methacrylate (Fig. 2), we note somewhat larger slopes in the range of molecular weights exceeding 100,000. The n values now vary from less than 4 to over 5. None of these curves is as steep as that for the polystyrene fraction of molecular weight 15,000.
SOLUTION
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FLEXIBLE
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POLYMERS
277
n=3
2.40
/
2.30
!! /
2.20 /
/
P
2.10
2.00
1.90
1.80 rlsp c [T]
I70 1.6C
I.50
I.40
130
1.20
0
FIG. toluene
040
0.80
1. Experimental at 30°C.
reduced
0 0
Overlapping sent Baker
points equation
600,000 165,000
near the origin with n-values
2.00 1.60 c/co
1.20
2.40
Viscosity-Concentration
a A
150,000 45,000
have been indicated.
omitted
2.00 curves
for
3.20
3.60
polystyrene
in
q 40,000 15,000
n
for
clarity.
Dotted
lines
repre-
278
SIMHA
AND
ZAKIN
2.00
2.60
I-
n=4
n=3
/
2.40
Tsp. cPll I .80
I .“”
0
0.80
1.60
240
3.20
4.00
4.80
G/Co 2. Experimental reduced Viscosity-Concentration methacrylate and copolymers in toluene at 30°C.
curves
FIG.
0 CP 630,000 0 CP 215,000 Overlapping sent Baker
points equation
near with
the origin n-values
n PMM A PMM
have been indicated.
omitted
for
polymethyl
470,000 135,000 for
clarity.
Dotted
lines
repre-
Thus, for all the polymers investigated here, in good solvents we observe a systematic increase in slope with decreasing molecular weight. However, for molecular weights above 100,000, the effect is small. Considering the differences in solution viscosities between polymers of different molecular weights when conventional variables are used, these results demonstrate an approximate and useful corresponding states representation for a given
SOLUTION
VISCOSITIES
9.00
6.20
OF
LINEAR
FLEXIBLE
HIGH
2.40
3.20
POLYMERS
279
I
7.40
6.60
5.80
Lc fyj]
5.00
4.20
3.40
2.60
I .oo
FIG.
theta
3. Experimental solvent.
0
0.60
reduced 0 600,000 o 165,000
Overlapping sent Baker
points equation
near the origin with n-values
1.60
Viscosity-Concentration A 150,000 A 45,000 have been indicated.
omitted
curves
4.00
for
polystyrene
in
0 40,000 15,000
n
for
clarity.
Dotted
lines
repre-
polymer type. Moreover, on the basis of our present results, we can compare different polymer types. Next we consider the same polymers in their respective theta solvents. Without exception, the curves are c.onsiderably steeper than in the good
280
SIMHA
AND
ZAKIN
z 5.40
2 1.80 1 20.20
18.60
I700 1 1540 t 13.80 1 rjSP qq’22o t
n=-3
1060 i I
9.00 1
I
7.40
5.80
4.20 I 2.60 t I .oo 0
FIG.
4. Experimental
methacrylate
Overlapping sent Baker
and copolymers
points equation
040
0.80
120
160
reduced Viscosity-Concentration in theta solvents.
curves
0 CP 630,000
n
PMM
470,000
0 CP 215,000
A PMM
135,000
near the origin with n-values
have been indicated.
omitted
200
for
clarity.
2.40
for
Dotted
polymethyl
lines
repre-
SOLUTIOr;
VISCOSITIES
OF
LIKEAR
FLEXIBLE
HIGH
POLYMERS
281
solvent and the n values are all negative. Specifically, for polystyrene (Fig. 3) n varies between -4 and co. As in the good solvent, the lowest molecular weight fraction has the steepest slope with n near -4. The slopes decrease with increasing molecular weight up to 150,000, where n E w. However, on increasing the molecular weight further, this trend is reversed for the 165,000 and 600,000 fractions. For the copolymers in or near their t.heta solvents (Fig. 4), the viscosity variables rise more rapidly than for polystyrene of equal molecular weight in its theta solvent. The n values range from -4 to about -7 and therefore approach the value for the lowest molecular weight polystyrene. Finally, for polymethyl methacrylate (Fig. 4)) the viscosities increase most rapidly, corresponding to n-values between -1 and -2. For the copolymers as well as for polymethyl methacrylate, the slopes increase with molecular weight, as was found for the high molecular weight polystyrene in the theta solvent. Comparison of Figs. 1 with 3 and 2 with 4 shows larger spreads between different molecular weights in the theta systems. As stated earlier, the influence of shear gradient on viscosity was not detected in these experiments. If such an effect were present, it would tend to decrease the slopes of the curves, particularly for the high molecular weights.
DISCUSSION The relation between viscosity and concentration in the concentration range considered here is determined by: (a) long-range hydrodynamic interactions of single solute molecules; (b) osmotic compression of the polymer coils; (c) formation of aggregates and their hydrodynamic interaction and, possibly, entanglements. We have noted in the Introduction that current hydrodynamic theory is not adequately developed for our present systems in the concentration ranges studied. Recent work by Debye (16) suggests that compression effects should be amenable to independent experimental investigation by light scattering. We have earlier attempted to calculate the magnitude of the effect (2, 3, 15). In a theta solvent factors (a) and (c) should predominate, since (b) is negligible, when B, and higher virial coefficients are close to zero. In what follows we shall consider, in the light of this compression effect, two sets of observations, viz., the corresponding states representation in good solvents alid the differences between good and poor solvents. Values of the compression ratio, VJV, , have been given previously (see Table II, reference 3) using two alternative equations. Here the V’s refer to the encompassed coil volumes at concentration c and infinite dilution, respectively. One is based on the exact relation between instantaneous values of the radius of gyration, R, and the end-to-end distance, r, for Gaussian
282
SIMHA
AND
TABLE Effect
of Concentration
on Compression Polusturene
ZAKIN
V Ratio for in Toluene
Three
Molecular
Weights
of
VclVo M
M
M
= 600,000 1.00 1.41 2.07 = 150,000 0.48 0.95 1.63 = 40,000 0.49 0.98 1.47
Gaussian chain (Eq. 4’, Ref. 3)
Gaussian distribution of radius of gyration (Eq. 4a’, Ref. 3)
0.80 0.65 0.40
0.78 0.64 0.47
0.94 0.77 0.49
0.93 0.76 0.53
0.92 0.73 0.52
0.90 0.71 0.53
coils (Eq. 4’ reference 3), the other on a simplification valid only for averages (Eq. 4a’ reference 3). These data are included in Table V here, together with results for a series of concentrations using Eq. 4’ not reported previously. The two equations give very nearly the same results. For a given concentration, c/co , the magnitude of the compression varies but slightly with molecular weight. Thus, c/c0 again appears to represent a corresponding state variable but, as in viscosity, this is only approximately true. The calculated compression ratios decrease with increasing molecular weight; this is in the opposite direction to the slope variations of the viscosity curves. However, this seems hardly significant in view of the approximations inherent in the theory (3) and of the uncertainties in the osmotic data used in the preparation of Table V. The highest rate of change in viscosity slopes occurs in the region between 40,000 and 15,000 molecular weight. If, in view of results for high molecular weights, we adopt the hypothesis that the reduced concentration variable, c/co, should indeed be a corresponding state variable, then this suggests the possibility of a change in the relationship between [q] and co as low molecular weights are approached. Measurements at still lower molecular weights would be desirable (as well as at higher). The proportionality factor between CO and l/[v] would have to increase with molecular weight in order to account for the observations. It would require a factor varying with a very low power of molecular weight, about 0.1 on the basis of Fig. 1, to cover the whole range of 600,000 to 15,000 or 40/l. Actually the Debye-Bueche (17) and Kirkwood-Riseman (18) theories of intrinsic viscosity lead to a molecular weight dependence of the above factor. If for very low molecular weights
SOLUTIOP;
VISCOSITIES
OF
LINEAR
FLEXIBLE
HIGH
POLYMERS
283
3.007 v/c b-j,1
FIG. toluene
5. Reduced Viscosity-Concentration at 30°C. corrected for compression.
curves
for
polystyrene
(600,000)
in
the extreme of rodlike configurat’ions is approached, the proportionality factor would vary with the 0.7 power of the axis ratio. Next we examine to what extent the compression can account for the differences in slopes between good and poor solvents. In order to attain truly corresponding stat’es for good and poor solvents, the values of c/co in the former must be corrected for compression, that is, multiplied by the factor V,/V, . This is shown for polystyrene of molecular weight 600,000 in Fig. 5, which is based on the results of Table V (Eq. 4~‘). The theory from which the compression values were calculated is most applicable above c/c, w 1 (uncorrected) (3). In this region, the correction reduces the spread bet,ween the two original curves considerably. However, in view of the uncertainties in the numerical values of the compression ratios, we can merely stat’e that at c/c, z 1, we account for a significant, portion of the difference between the two viscosity curves. EssenCally the same pictlire results for lower molecular weight polyst>yrenes. A direct’ experimental determination of the average coil configurations in concentrated solutions will be necessary to corroborate these results. If the present compression calculations are assumed to be adequate, the remaining difference must be accounted for by effects (a) and (c). In theta solvents, the n-values or slopes have been shown to reach a maximum as a function of molecular weight. Here the higher molecular weight polymers are only a few degrees above their precipitat’ion points and
284
SIMHA
AND
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clustering of polymer molecules is most pronounced. Actually in the case of the highest molecular weight polystyrene, we observed a blue Tyndall effect. This may account for the reversal in the general trend toward decreasing slope with increasing molecular weight. At some lower molecular weight, the precipitation temperature, and hence the clustering, is sufficiently reduced and the trend with molecular weight should be the same as in the good solvent. In the case of polystyrene, where sufficiently low molecular weights were studied, this reversal was actually observed. Whereas, in general, the differences between theta and good solvents are pronounced and in the same direction for each polymer type, there exist also large differences between different polymers in their respective theta solvents (compare Figs. 3 and 4). The variations from polymethyl methacrylate to the other polymers are particularly striking. The polymethyl methacrylate coil in the theta solvent has the smallest [q] and hence the highest density for a given molecular weight. However, density alone can not explain the steep slopes, as the lower molecular weight copolymer has
4.00 -
565,000
-
1?5,000
3.00
rlsp/ c[ril 2.00
C,g/dl FIG.
mers
6. Comparison of reduced in their theta solvents.
Viscosity
vs.
Concentration
curves
for three
poly-
SOLUTION
VISCOSITIES
OF
LINEAR
FLEXIBLE
HIGH
POLYMERS
285
about the same [v] and hence the same density within the pervaded coil volume in its theta solvent as the higher molecular weight polymethyl methacrylate. Nevertheless, the curve for the copolymer is nowhere near as steep. 31.8%. was reported by Schultz and Flory (5) to be the theta temperature for polymethyl methacrylate in 4-heptanone. In view of the extremely steep slopes observed, the possibility that hhis temperature is actually below the theta point can not be overlooked. However, the curve for CPl refers to a solution poorer than its theta solvent and here there is no striking increase in slope. The differences between the theta systems may reflect differences in the extent of clustering. A useful representation of viscosity data in theta solutions is illustrated in Fig. 6. Here the reduced ordinate has been retained and the actual concentration used. For approximately equal molecular weights a single curve accounts for all polymers up to an ordinate value of about 2. This implies definite relations between the coefficients of the conventional power series representation of Q~./c vs. c for different polymers in their respective theta solvents, but further confirming data are obviously required. A f-ma1 observation refers to the parameter k’ defined by the equation:
y = [q]+ k’[?#c as a function of molecular weight for polystyrene in toluene at 30°C. The results are summarized in Table VI. A significant increase of k’ for the lowest molecular weights is apparent. This could have been ant.icipated from the n-values of Fig. 1. These results were later confirmed by McCormick for low molecular weight polystyrene in toluene (19). The measured M,/M, ratio for PS2 (40,000) is greater than the estimated value for PS4 (150,000). However, it would appear that even a general trend to greater heterogeneity with decreasing molecular weight for the fractions used in this work could not, account for the difference between 0.34 and 0.46 in Table VI (20). An increase in k’ at low molecular weights where the chain becomes more rigid was suggested by Eirich and Riseman (21). Variation
of k’ with Molecular G00,000 165,000 150,000 45,000 40,000 15,000
Molecular
TABLE Weight
VI of Polystyrene
in
weight
Toluene k’
0.34 0.37 0.31 0.40 0.41 0.46
f i f * f f
.Ol .07 .Ol .Ol .02 .02
at 30°C.
286
SIMHA AND ZAKIN
In the theta solvent, k’ is much larger as expected. However, the data do not permit a definite conclusion. SUMMARY
The Newtonian viscosities of polymer solutions have been measured for three solute systems, namely, polystyrene, polymethyl methacrylate, and copolymers of these. The molecular weights of the fractions used ranged from 15,000 to 630,000. The upper limits of concentration were of the order of 2/[7]; this includes therefore the concentration co of incipient overlap of the pervaded coil volumes, as they exist at infinite dilution. Toluene, a good solvent for all three polymers, and individual theta solvents for each were investigated. Provided the dimensionless coordinates ~sp./(c[~]) and c/c, are used, one obtains for a given polymer type an approximate “corresponding states” representation in good solvents and over a wide range of molecular weights for the whole range of concentrations used here. The latter extends beyond the linear portion of the usual plots. As the molecular weights are reduced below 50,000, however, the viscosity functions exhibit considerably higher slopes than for larger molecular weights. This results in an increase of the parameter lc' with decreasing molecular weight. The corresponding states representation is a poorer approximation at all molecular weight levels in theta solvents, where the slopes are furthermore universally larger than in good solvents. This is in accord with previous measurements of Weissberg, Simha, and Rothman. Nearly universal curves are, however, obtained for the three theta systems when ~~./(c[rl]), for approximately equal molecular weights, is treated as a function of c. In good solvents an osmotic compression of the polymer coils must take place as the concentration increases. When a theory of this effect previously developed, is applied to polystyrene in toluene and the concentration C/CO is suitably corrected for compression, it accounts for a significant portion of the differences observed between good and poor solvents in the neighborhood of c = co . On the other hand, we suggest that the increased slopes of our reduced plots for low molecular weights in good solvents result from a molecular weight dependent relation between COand l/[q]. The trend of decreasing slopes with increasing molecular weights is reversed in theta solvents at high molecular weights. This may be due to clustering of polymer molecules near their precipitation points. Differences in the degree of clustering for different polymers, but approximately equal molecular weights, may account for the differences in the slopes of our reduced viscosity curves in theta solvents. ACKNOWLEDGMENT Grateful acknowledgment is offered to the Socony Mobil Oil Company, Inc., for the grant of an Employee Incentive Fellow-ship to one of us (J.L.Z.). The authors also
SOLUTION wish to thank discussions.
VISCOSITIES Dr.
S. G. Weissberg
OF
LINEAR
of the
National
FLEXIBLE Bureau
HIGH
POLYMERS
of Standards
for
287 his helpful
REFERENCES I. 2.
3. 4. 5. 6.
7. 8.
9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20. 21.
H. L., AND SIIMHA, R., in F. R. Eirich, ed., ‘LRheology,” Vol. 1. Academic Press, New York, 1956. WEISSBERG, S. G., SIMHA, R., AND ROTHMAN, S., J. Research Natl. Bur. Standards 47, 298 (1951); referred to as WSR. SIMHA, R., AND ZAKIN, J, L., J. Chem. Phys. 33,179l (1960). EPSTEIN, B. N., Ph.D. Thesis, Massachusetts Institute of Technology, Boston, 1956. SCHULTZ, A. R., AND FLORY, P. J., J. Am. Chem. Sot. 74,476O (1952). SCHULTZ, A. R., AND FLORY, P. J., J. Am. Chem. Sot. 75,3888 (1953). MCCORMICK, H. W., J. Polymer Sci. 39,87 (1959). BILLMEYER, F. W., JR., Private communication. KAPUR, S. L., “Solution Properties of Styrene and Methyl Methacrylate Copolymer,” Ph.D. Thesis, Polytechnic Institute of Brooklyn, Brooklyn, 1948. STOCKIVIAYER, W. H., MOORE, L. D., JR., FIXMAN, M., AND EPSTEIN, B. hi., J. Polymer Sci. 16,517 (1955). ROTHMAN, S., Anal. Chem. 22,367 (1950). MCELWAIN, J. W., Anal. Chem. 21,194 (1949). Fox, T. G., JR., AND FLORY, P., J. Am. Chem. Sot. 73,1915 (1951). SHARMON, L. J., SONES, R. H., AND CRAGG, L. H., J. Appl. Phys. 24,703 (1953). ZA~IN, J. L., “The Effect of Concentration on the Solution Viscosities of Linear Flexible High Polymers at Moderate Concentrations.” Doct. Eng. Sci. Thesis, New York University, New York, 1959. DEBYE, P., COLL, H., AND WOERMANN, D., J. Chem. Phys. 33,1746 (1960). DEBYE, P., AND BUECHE, A. M., J. Chem. Phys. 16,573 (1948). KIRKWOOD, J. G., AND RISEMAN, J. J., J. Chem. Phys. 16,565 (1948). MCCORMICK, H. W., J. Colloid Sci. 16, 635 (1961). TONIPA, IT., “Polymer Solutions,” p. 269. Academic Press, New York, 1956. EIRICH, F., AND RISEMAN, J. J., J. Polymer Sci. 4,417 (1949). FRISCH,
OF COLLOID
SCIENCE
SOLUTION
Robert
17,
270-287
(1962)
VISCOSITIES OF LINEAR HIGH POLYMERS’
FLEXIBLE
Simha2 and Jacques L. i&kin3
Department of Chemical Engineering, Received
September
New York University,
New
York
6, 1961
INTRODUCTION The viscosities of macromolecular systems have been studied frequently from two alternative points of view and, hence, under two extreme conditions of concentration: (1) to deduce information about the size and shape of the solute unit in dilute solution, and (2) to elucidate the mechanism of transport phenomena in essentially bulk systems of large molecules. There also exist several investigations in intermediate ranges of concentration which have led to moderately successful empirical viscosity-concentration relations. For infinitely dilute solutions, the well-known and successful hydrodynamic theories of the intrinsic viscosity for solutes of various shapes (1) have been developed. The extension of these theories to finite concentrations, i.e., the introduction of hydrodynamic interactions, has, for the most part, been restricted to first-order perturbations (1) or very dilute solutions in which the average solute-solute distances are large compared with the solute dimensions. Hydrodynamic theories for concentrated systems have been developed for compact spherical particles only (1). Some time ago, Weissberg, Simha, and Rothman (2) (WSR) set out to investigate in detail the solution viscosities of flexible polymers and the effect of the solvent environment in the intermediate region of concentration, where the pervaded volumes of the polymer coils, as measured at infinite dilution, start to overlap. Depending on the molecular weight and the solvent used, this occurred at concentrations varying from about 0.5 to 3 g./dl. The following observations were made by these authors: In good solvents, the curves of the quantities T~,./([v]c) vs. concentration for different molecular weights are approximately superimposable, provided the 1 Part of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Engineering Science. 2 Present address: Department of Chemistry, University of Southern California, Los Angeles, California. 3 Present address: Socony Mobil Oil Company, Research Department, Brooklyn Laboratory, Brooklyn 22, New York. 270
SOLUTION
VISCOSITIES
OF
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FLEXIBLE
HIGH
POLYMERS
271
concentrations are expressed in units of CO . Here co is the concentration of incipient overlap, as derived from molecular dimensions at infinite dilution. corresponding Thus the reduced variable, C/CO., defines approximately states and this applies over a range of c’s extending beyond the usual linear region of the Q,./c vs. c curves. As the solvent becomes poorer, this approximation increasingly fails. The first derivatives of these reduced functions are considerably smaller in good than in poor solvents. In the latter, they approach the curves which result for rigid solutes, such as globular proteins or resin particles in suspension. Finally, older data suggested that the corresponding states representation increasingly fails also in good solvents when the molecular weight is reduced below a number average of about 50,000. There the curves also rise more steeply than in the conventional range. Their data illustrated that the sharp increase in the viscosities of suspensions of compact solutes which is observed when C/CO approaches unity, is absent for flexible polymer solutions. This is to be expected because of the low density of the polymer coils. On the other hand, the observed differences in good and poor solvents can not be due to the differences in the average coil densities for a given molecular weight. The latter are, for a given polymer type, approximately inversely proportional to the intrinsic viscosity. Therefore, the same coil densities can be generated in good and in poor solvents by the proper selection of molecular weight in each solvent. A factor reducing the concentration dependence of the viscosity is the osmotic compression of the polymer coils in good solvents which results from the interchain segment repulsions. This effect is absent or minimized in poor solvents. In order to relate the quantitative differences between good and poor solvents to this compression effect, it is necessary to obtain independent experimental or theoretical information about the latter. WSR were concerned with the region C/COm 1 and gave an approximate treatment in terms of the osmotic pressure which has been recently modified (3) and adjusted for application to the concentrated systems involved here. In order to obtain an estimate of the magnitude of the compression to be expected, and in the absence of quantitative experimental results, WSR assumed that it can be obtained directly from the apparent intrinsic viscosity at finite concentrations, viz.,
This, in turn, amounted to the assumption that the viscosity in a solvent in which the osmotic pressure assumes the ideal value, &conditions, is a universal function of c/co and, in particular, an exponential function. It became clear from the limited experimental information that this was at best only approximately true. Nevertheless, better than the right orders of
272
SIMHA AND ZAKIN
magnitude for the expansion factors of the coils, which result from intramolecular segment repulsions, could be derived in this way from the viscosities at C/COin the neighborhood of unity. The above discussion suggests investigations along the following lines: first, an experimental verification of the behavior in &solvents; secondly, an investigation of relatively small in comparison with high molecular weights in good and poor solvents. WSR examined polystyrene solutions only with number-average molecular weights varying by a factor of ten. The range has now been extended considerably and other polymers and copolymers also included. EXPERIMENTAL
Viscosity-concentration measurements in the Newtonian region were made with three different kinds of polymers-polystyrene, polymethyl methacrylate, and copolymers close to the azeotropic composition (51.4 mole % styrene (4))-in a good solvent, toluene at 3O”C., and in their respective theta solvents. Theta solvents were chosen close to 30°C. For polystyrene, cyclohexane at 34.O”C. was used (5) and for polymethyl methacrylate 4-heptanone at 31.8”C. (6). Theta solvents for the copolymers were mixtures of methyl isopropyl ketone and n-heptane at 25.O”C. the compositions of which were estimated from data obtained by Dr. B. N. Epstein (4) Table I summarizes the characteristics of the fractions investigated. A. Polymers
1. Polystyrene (PS). A thermally polymerized (1SO’C.) polymer obtained from Dr. Leo A. Wall of the National Bureau of Standards was fractionTABLE Characteristics
Polymer Polystyrene
Sample number
Copolymer Styrene)
(48.8%
PSl PS2 PS3 PS4 PS5 PS6 CPl
Copolymer Styrene)
(47.8%
CP2
Polymethyl methacrylate
PMMl PMM2
I of Polymers
Method of polymerization
Tok!ene 30” c.
T#ta
Estimated molecular weight
Anionic Anionic Thermal 180°C. Anionic Thermal 180°C. Thermal 0.11 mole To beneoyl peroxide @ 60°C. 0.11 mole y0 benzoyl peroxide @ 60°C. Free radical Free radical
0.142 0.278 0.308 0.732 0.782 2.026 0.594
0.112 0.186 0.198 0.362 0.365 0.701 0.321
15,000 40,000 45,000 150,000 165,000 600,000 215,000
1.333
0.555
630,000
0.373 0.802
0.192 0.298
135,000 470 ) 000
SOLUTION
VISCOSITIES
OF
LINEAR
FLEXIBLE
HIGH
POLYMERS
273
ated. Seven broad fractions were precipitated from a l%Yo solution in MEK at 25°C. by addition of isopropyl alcohol. Two of these were refraetionated from 1 Yo solutions. Narrow middle fractions were taken from each and dried by the frozen benzene technique. Four narrow distribution polymers were obtained from Dr. H. W. McCormick of the Dow Chemical Company. Three were anionic polymers prepared with a sodium naphthalene complex. The fourth was a high molecular weight fraction from a thermal polymerization. One of the former, PS 2, is identical with S-3 described by McCormick (7) as having a ratio of Mm/ill, = 1.35. Another, PS 4, is believed to be similar to his S-9, which has a ratio of about 1.1. Molecular weights were estimat,ed from log-log plots of [q] in toluene at 30°C. vs. &!, (2). 2. Polymethyl Methacrylate (PMM). A sample of Lucite Molding Powder Composition 41 was fractionated. This polymer was furnished by Dr. F. W. Billmeyer, Jr., of the Polychemicals Department of DuPont and was prepared by a free radical polymerization following bulk kinetics (8). Five broad fractions were precipitated from a 2% solution in MEK at 25°C. with methanol as the precipitant. The first and last were refractionated from 0.9% solutions. Middle fractions from each of these were separated and dried from frozen benzene solution. Molecular weights were estimated from recalculated M, vs. [v] data of Kapur in toluene at 30°C. (9). S. Copolymer (CP). A copolymer of styrene and methyl methacrylate near the azeotropic composition (51.4 YOstyrene) was fractionated. The copolymer was also furnished by Dr. Wall and was prepared at 60°C. with 0.11 mole % benzoyl peroxide as catalyst and contained 49.7% styrene. Although the difference in chemical composition of individual copolymer chains prepared from a ratio of monomers close to the azeotropic ratio is small, the system MEK-diisopropyl ether at 25°C. suggested by Stockmayer et al. (10) was used to minimize fractionation by composition rather than by molecular weight. Five broad fractions were precipitat,ed from a 0.5% solution of MEK at 25°C. by addition of diisopropyl ether. Two of these were refractionated from 0.5 % solutions. Middle fractions from each of these were separated and dried from frozen benzene solutions. Molecular weights were estimated from recalculated M, vs. [r] data of Kapur in toluene at 30°C. (9). B. Solvents Toluene (Fisher’s Reagent Grade) and cyclohexane (Phillips Pure Grade 99+ mole Yo) were distilled over sodium wire and middle cuts were taken. Methyl isopropyl ketone (Eastman’s Analytical Reagent Grade), n-heptane (Phillips ASTM Grade 99.5+ mole %), and 4-heptanone (Sealey) were fractionated in a 42-theoretical plate adiabatic packed column and constant boiling middle fractions were collected. Physical properties of the solvents are given in Table II.
274
SIMHA
AND
ZAKIN
TABLE Physical
II
of Solvents
Properties
Viscosity (Cmtistokes&) At subscript temp., (“C).
Solvent Toluene Cyclohexane Methyl isopropyl ketone (MIPK) n-Heptane di-n-Propyl ketone Mixed solvent 0.435 ml. n-heptane/l .OOO ml. MIPK 0.415ml.n-heptane/l.OOOml.Mt[PK 8 Based
on
1.002
centipoises
for
Copolymer
Theta
water TABLE
Mole
70 styrene 24.5 51.4 74.2
in copolymer
(azeotrope)
at
Density (g./mZ.) at subscript temperature PC).
Refractive index at 200c’
0.604530 .o 1.00234.0 0.774831.8
0.8567ao.o 0.7646ad.o 0.800225.s 0.679923.0 0.804531.~
1.4958 1.4256 1.3881 1.3871 1.4065
0.553&&L? 0.556Gr,.o
0.7612zs.a 0.763926.0
-
2O.O”C. III
Compositions
at 15%‘.
(4)
Theta solvent composition at 25” C. ml. n-heptanelml. MIPK 0.290 0.435 0.620
Epstein’s data (4) showing the variation of mixed solvent composition at the theta point at 25°C. with copolymer composition are shown in Table III. It was assumed at the start of these viscosity studies that the copolymer compositions were close enough to the azeotrope to use that equivalent ratio of solvents, i.e., 0.435 ml. n-heptanelml. MIPK. This solvent for the low molecular weight sample, CPl, was used for its measurements. The theta solvent composition for CP2 was estimated by interpolating the above data and was used for its viscosity measurements. C. Polymer Solution Preparation,
Viscosjty, and Density Measurements
Master solutions were made up by weight to concentrations of at least C = 2/[~] where sample size permitted and their densities and those of the solvents were measured with a Lipkin pycnometer at the appropriate temperatures. Densities at intermediate concentrations were linearly interpolated. The master solutions were filtered through a medium porosity sintered glass filter into a Friedman La Mer weight buret using a setup similar to that suggested by Rothman (11). The quantity of solution drained into a viscometer was measured by weight difference. Ubbelohde viscometers with dilution bulbs were used, employing in most cases a closed system similar to that described by McElwain (12). The sys-
SOLUTION
VISCOSITIES
OF
LINEAR
TABLE
FLEXIBLE
HIGH
275
POLYMERS
IV
Viscometer Characteristics
viscometer A B-l
in
Capillary diameter (mnt.)
0.400 0.600
Capillary length (cm.)
10 10
Efflux bulb volume (CC.)
5.0 4.2
Viscometer A*
0.00202 0.01123
constants B*
1.28 0.39
* A and B are defined by the equation t/p = At - B/t, where t is in seconds, p is gm./ml., and q is in centipoises.
tern was modified to avoid applying vacuum to any portion of the solution. Solution viscosities were measured down to concentrations corresponding to relative viscosities of 1.05. Least squares computations were made on calibration measurements with standard oils and purified solvents to calculate the viscometer constants. The viscometer characteristics are shown in Table IV. Flow times in Viscometer A generally exceeded 300 sec. and in Viscometer B-l, which has a very low kinetic energy correction constant, they generally exceeded 150 sec. Kinetic energy corrections were applied to all readings. The temperature of the bath was controlled to within fO.Ol”C. The average error in the timing operation is estimated to be of the order of 0.1 sec. At the maximum flow rates, the average shear rate at the capillary walls in Viscometer A was about 1800 sec.-l and in Viscometer B-l about 880 sec.-l. These shear rates are relatively low and significant non-Newtonian effects would not ordinarily be expected under these conditions even for the higher molecular weight polymers in good solvents. Fox and Flory (13) state that the shear correction is small for intrinsic viscosities under 3 dl./g. Sharmon, Sones, and Cragg (14) found no effect in polystyrene in toluene up to intrinsic viscosities of 2 dl./g. and in cyclohexane up to intrinsic viscosities of about 1.3 dl./g. The maximum intrinsic viscosity in this work was about 2 dl./g. for polystyrene in toluene. For polystyrene in cyclohexane, the maximum intrinsic viscosity was 0.7 dl./g. WSR were also unable to detect any appreciable shear rate effect in their studies of polystyrene in toluene. The possibility that the viscosities of the high concentration solutions have some shear rate dependence cannot be completely discounted. However, it should be noted that the most concentrated solutions have the highest viscosities and hence the lowest shear rates. Furthermore, in many of the runs, sections of the viscosity concentration curves were measured in both Viscometers A and B-l, and no apparent shear dependence was found, although the shear rate at equal viscosities was about 35% higher in Viscometer B-l than it was in A.
276
SIMHA AND ZAKIN RESULTS
To compare the data for the different types of polymers and widely varying molecular weights and concentrations in different solvents, viscosityconcentration curves are presented in the reduced form suggested by WSR, that is ~,p./(c[~]) as a function of C/Q . Here COis the concentration at which polymer molecules, or rather the equivalent spheres closely packed, would begin to overlap if they retained the same volume in concentrated solution that they pervade at infinite dilution. Thus, c/co would represent relative volumes if no compression of molecules occurred as the concentration increases. The region of incipient overlap corresponds to c/co values of unity. Most of the data presented extend to c/a values of about 2 or c x 2/[7]. The intrinsic viscosities were computed by least squares analyses of the linear portions of the plots of r]8P./c vs. c (15) and the co’s were obtained from the relation CO= 1.08/h] (3). In representing the reduced viscosity-concentration data (Figs. l-4), a family of Baker equations in the reduced form: TSP. -MC
_
(1+[&“-, > Fit
has also been plotted for the selection of appropriate values of n. These can be used as a convenient device for comparing the relative positions of the experimental curves. The parameter n may assume both positive and negative values. As n goes from 0 to 00, the slopes increase monotonically. The curves steepen still further for negative n’s, increasing in slope with smaller negative values of n. Toluene at 30°C. is a good solvent for all the polymers used by us, as may be inferred from the respective intrinsic viscosities in toluene and theta solvents shown in Table I. The viscosity-concentration curves for the six polystyrene samples are shown in Fig. 1. The Baker equation is represented by the dotted lines. In this good solvent, the steepness of the curves increases as molecular weight decreases. At the higher molecular weights, curves for the 600,000, 165,000, and 150,000 samples are close to each other at n X 3. This agrees with the results of WSR, but the spread between curves was found to be slightly larger than they reported. In the 40,000 molecular weight range, the curves are displaced upwards and n is between 3 and 4. The parameter increases rapidly between 40,000 and 15,000 molecular weight where n reaches a value larger than 6. Turning to the results for the copolymers and polymethyl methacrylate (Fig. 2), we note somewhat larger slopes in the range of molecular weights exceeding 100,000. The n values now vary from less than 4 to over 5. None of these curves is as steep as that for the polystyrene fraction of molecular weight 15,000.
SOLUTION
VISCOSITIES
OF
LINEAR
FLEXIBLE
HIGH
POLYMERS
277
n=3
2.40
/
2.30
!! /
2.20 /
/
P
2.10
2.00
1.90
1.80 rlsp c [T]
I70 1.6C
I.50
I.40
130
1.20
0
FIG. toluene
040
0.80
1. Experimental at 30°C.
reduced
0 0
Overlapping sent Baker
points equation
600,000 165,000
near the origin with n-values
2.00 1.60 c/co
1.20
2.40
Viscosity-Concentration
a A
150,000 45,000
have been indicated.
omitted
2.00 curves
for
3.20
3.60
polystyrene
in
q 40,000 15,000
n
for
clarity.
Dotted
lines
repre-
278
SIMHA
AND
ZAKIN
2.00
2.60
I-
n=4
n=3
/
2.40
Tsp. cPll I .80
I .“”
0
0.80
1.60
240
3.20
4.00
4.80
G/Co 2. Experimental reduced Viscosity-Concentration methacrylate and copolymers in toluene at 30°C.
curves
FIG.
0 CP 630,000 0 CP 215,000 Overlapping sent Baker
points equation
near with
the origin n-values
n PMM A PMM
have been indicated.
omitted
for
polymethyl
470,000 135,000 for
clarity.
Dotted
lines
repre-
Thus, for all the polymers investigated here, in good solvents we observe a systematic increase in slope with decreasing molecular weight. However, for molecular weights above 100,000, the effect is small. Considering the differences in solution viscosities between polymers of different molecular weights when conventional variables are used, these results demonstrate an approximate and useful corresponding states representation for a given
SOLUTION
VISCOSITIES
9.00
6.20
OF
LINEAR
FLEXIBLE
HIGH
2.40
3.20
POLYMERS
279
I
7.40
6.60
5.80
Lc fyj]
5.00
4.20
3.40
2.60
I .oo
FIG.
theta
3. Experimental solvent.
0
0.60
reduced 0 600,000 o 165,000
Overlapping sent Baker
points equation
near the origin with n-values
1.60
Viscosity-Concentration A 150,000 A 45,000 have been indicated.
omitted
curves
4.00
for
polystyrene
in
0 40,000 15,000
n
for
clarity.
Dotted
lines
repre-
polymer type. Moreover, on the basis of our present results, we can compare different polymer types. Next we consider the same polymers in their respective theta solvents. Without exception, the curves are c.onsiderably steeper than in the good
280
SIMHA
AND
ZAKIN
z 5.40
2 1.80 1 20.20
18.60
I700 1 1540 t 13.80 1 rjSP qq’22o t
n=-3
1060 i I
9.00 1
I
7.40
5.80
4.20 I 2.60 t I .oo 0
FIG.
4. Experimental
methacrylate
Overlapping sent Baker
and copolymers
points equation
040
0.80
120
160
reduced Viscosity-Concentration in theta solvents.
curves
0 CP 630,000
n
PMM
470,000
0 CP 215,000
A PMM
135,000
near the origin with n-values
have been indicated.
omitted
200
for
clarity.
2.40
for
Dotted
polymethyl
lines
repre-
SOLUTIOr;
VISCOSITIES
OF
LIKEAR
FLEXIBLE
HIGH
POLYMERS
281
solvent and the n values are all negative. Specifically, for polystyrene (Fig. 3) n varies between -4 and co. As in the good solvent, the lowest molecular weight fraction has the steepest slope with n near -4. The slopes decrease with increasing molecular weight up to 150,000, where n E w. However, on increasing the molecular weight further, this trend is reversed for the 165,000 and 600,000 fractions. For the copolymers in or near their t.heta solvents (Fig. 4), the viscosity variables rise more rapidly than for polystyrene of equal molecular weight in its theta solvent. The n values range from -4 to about -7 and therefore approach the value for the lowest molecular weight polystyrene. Finally, for polymethyl methacrylate (Fig. 4)) the viscosities increase most rapidly, corresponding to n-values between -1 and -2. For the copolymers as well as for polymethyl methacrylate, the slopes increase with molecular weight, as was found for the high molecular weight polystyrene in the theta solvent. Comparison of Figs. 1 with 3 and 2 with 4 shows larger spreads between different molecular weights in the theta systems. As stated earlier, the influence of shear gradient on viscosity was not detected in these experiments. If such an effect were present, it would tend to decrease the slopes of the curves, particularly for the high molecular weights.
DISCUSSION The relation between viscosity and concentration in the concentration range considered here is determined by: (a) long-range hydrodynamic interactions of single solute molecules; (b) osmotic compression of the polymer coils; (c) formation of aggregates and their hydrodynamic interaction and, possibly, entanglements. We have noted in the Introduction that current hydrodynamic theory is not adequately developed for our present systems in the concentration ranges studied. Recent work by Debye (16) suggests that compression effects should be amenable to independent experimental investigation by light scattering. We have earlier attempted to calculate the magnitude of the effect (2, 3, 15). In a theta solvent factors (a) and (c) should predominate, since (b) is negligible, when B, and higher virial coefficients are close to zero. In what follows we shall consider, in the light of this compression effect, two sets of observations, viz., the corresponding states representation in good solvents alid the differences between good and poor solvents. Values of the compression ratio, VJV, , have been given previously (see Table II, reference 3) using two alternative equations. Here the V’s refer to the encompassed coil volumes at concentration c and infinite dilution, respectively. One is based on the exact relation between instantaneous values of the radius of gyration, R, and the end-to-end distance, r, for Gaussian
282
SIMHA
AND
TABLE Effect
of Concentration
on Compression Polusturene
ZAKIN
V Ratio for in Toluene
Three
Molecular
Weights
of
VclVo M
M
M
= 600,000 1.00 1.41 2.07 = 150,000 0.48 0.95 1.63 = 40,000 0.49 0.98 1.47
Gaussian chain (Eq. 4’, Ref. 3)
Gaussian distribution of radius of gyration (Eq. 4a’, Ref. 3)
0.80 0.65 0.40
0.78 0.64 0.47
0.94 0.77 0.49
0.93 0.76 0.53
0.92 0.73 0.52
0.90 0.71 0.53
coils (Eq. 4’ reference 3), the other on a simplification valid only for averages (Eq. 4a’ reference 3). These data are included in Table V here, together with results for a series of concentrations using Eq. 4’ not reported previously. The two equations give very nearly the same results. For a given concentration, c/co , the magnitude of the compression varies but slightly with molecular weight. Thus, c/c0 again appears to represent a corresponding state variable but, as in viscosity, this is only approximately true. The calculated compression ratios decrease with increasing molecular weight; this is in the opposite direction to the slope variations of the viscosity curves. However, this seems hardly significant in view of the approximations inherent in the theory (3) and of the uncertainties in the osmotic data used in the preparation of Table V. The highest rate of change in viscosity slopes occurs in the region between 40,000 and 15,000 molecular weight. If, in view of results for high molecular weights, we adopt the hypothesis that the reduced concentration variable, c/co, should indeed be a corresponding state variable, then this suggests the possibility of a change in the relationship between [q] and co as low molecular weights are approached. Measurements at still lower molecular weights would be desirable (as well as at higher). The proportionality factor between CO and l/[v] would have to increase with molecular weight in order to account for the observations. It would require a factor varying with a very low power of molecular weight, about 0.1 on the basis of Fig. 1, to cover the whole range of 600,000 to 15,000 or 40/l. Actually the Debye-Bueche (17) and Kirkwood-Riseman (18) theories of intrinsic viscosity lead to a molecular weight dependence of the above factor. If for very low molecular weights
SOLUTIOP;
VISCOSITIES
OF
LINEAR
FLEXIBLE
HIGH
POLYMERS
283
3.007 v/c b-j,1
FIG. toluene
5. Reduced Viscosity-Concentration at 30°C. corrected for compression.
curves
for
polystyrene
(600,000)
in
the extreme of rodlike configurat’ions is approached, the proportionality factor would vary with the 0.7 power of the axis ratio. Next we examine to what extent the compression can account for the differences in slopes between good and poor solvents. In order to attain truly corresponding stat’es for good and poor solvents, the values of c/co in the former must be corrected for compression, that is, multiplied by the factor V,/V, . This is shown for polystyrene of molecular weight 600,000 in Fig. 5, which is based on the results of Table V (Eq. 4~‘). The theory from which the compression values were calculated is most applicable above c/c, w 1 (uncorrected) (3). In this region, the correction reduces the spread bet,ween the two original curves considerably. However, in view of the uncertainties in the numerical values of the compression ratios, we can merely stat’e that at c/c, z 1, we account for a significant, portion of the difference between the two viscosity curves. EssenCally the same pictlire results for lower molecular weight polyst>yrenes. A direct’ experimental determination of the average coil configurations in concentrated solutions will be necessary to corroborate these results. If the present compression calculations are assumed to be adequate, the remaining difference must be accounted for by effects (a) and (c). In theta solvents, the n-values or slopes have been shown to reach a maximum as a function of molecular weight. Here the higher molecular weight polymers are only a few degrees above their precipitat’ion points and
284
SIMHA
AND
ZAKIN
clustering of polymer molecules is most pronounced. Actually in the case of the highest molecular weight polystyrene, we observed a blue Tyndall effect. This may account for the reversal in the general trend toward decreasing slope with increasing molecular weight. At some lower molecular weight, the precipitation temperature, and hence the clustering, is sufficiently reduced and the trend with molecular weight should be the same as in the good solvent. In the case of polystyrene, where sufficiently low molecular weights were studied, this reversal was actually observed. Whereas, in general, the differences between theta and good solvents are pronounced and in the same direction for each polymer type, there exist also large differences between different polymers in their respective theta solvents (compare Figs. 3 and 4). The variations from polymethyl methacrylate to the other polymers are particularly striking. The polymethyl methacrylate coil in the theta solvent has the smallest [q] and hence the highest density for a given molecular weight. However, density alone can not explain the steep slopes, as the lower molecular weight copolymer has
4.00 -
565,000
-
1?5,000
3.00
rlsp/ c[ril 2.00
C,g/dl FIG.
mers
6. Comparison of reduced in their theta solvents.
Viscosity
vs.
Concentration
curves
for three
poly-
SOLUTION
VISCOSITIES
OF
LINEAR
FLEXIBLE
HIGH
POLYMERS
285
about the same [v] and hence the same density within the pervaded coil volume in its theta solvent as the higher molecular weight polymethyl methacrylate. Nevertheless, the curve for the copolymer is nowhere near as steep. 31.8%. was reported by Schultz and Flory (5) to be the theta temperature for polymethyl methacrylate in 4-heptanone. In view of the extremely steep slopes observed, the possibility that hhis temperature is actually below the theta point can not be overlooked. However, the curve for CPl refers to a solution poorer than its theta solvent and here there is no striking increase in slope. The differences between the theta systems may reflect differences in the extent of clustering. A useful representation of viscosity data in theta solutions is illustrated in Fig. 6. Here the reduced ordinate has been retained and the actual concentration used. For approximately equal molecular weights a single curve accounts for all polymers up to an ordinate value of about 2. This implies definite relations between the coefficients of the conventional power series representation of Q~./c vs. c for different polymers in their respective theta solvents, but further confirming data are obviously required. A f-ma1 observation refers to the parameter k’ defined by the equation:
y = [q]+ k’[?#c as a function of molecular weight for polystyrene in toluene at 30°C. The results are summarized in Table VI. A significant increase of k’ for the lowest molecular weights is apparent. This could have been ant.icipated from the n-values of Fig. 1. These results were later confirmed by McCormick for low molecular weight polystyrene in toluene (19). The measured M,/M, ratio for PS2 (40,000) is greater than the estimated value for PS4 (150,000). However, it would appear that even a general trend to greater heterogeneity with decreasing molecular weight for the fractions used in this work could not, account for the difference between 0.34 and 0.46 in Table VI (20). An increase in k’ at low molecular weights where the chain becomes more rigid was suggested by Eirich and Riseman (21). Variation
of k’ with Molecular G00,000 165,000 150,000 45,000 40,000 15,000
Molecular
TABLE Weight
VI of Polystyrene
in
weight
Toluene k’
0.34 0.37 0.31 0.40 0.41 0.46
f i f * f f
.Ol .07 .Ol .Ol .02 .02
at 30°C.
286
SIMHA AND ZAKIN
In the theta solvent, k’ is much larger as expected. However, the data do not permit a definite conclusion. SUMMARY
The Newtonian viscosities of polymer solutions have been measured for three solute systems, namely, polystyrene, polymethyl methacrylate, and copolymers of these. The molecular weights of the fractions used ranged from 15,000 to 630,000. The upper limits of concentration were of the order of 2/[7]; this includes therefore the concentration co of incipient overlap of the pervaded coil volumes, as they exist at infinite dilution. Toluene, a good solvent for all three polymers, and individual theta solvents for each were investigated. Provided the dimensionless coordinates ~sp./(c[~]) and c/c, are used, one obtains for a given polymer type an approximate “corresponding states” representation in good solvents and over a wide range of molecular weights for the whole range of concentrations used here. The latter extends beyond the linear portion of the usual plots. As the molecular weights are reduced below 50,000, however, the viscosity functions exhibit considerably higher slopes than for larger molecular weights. This results in an increase of the parameter lc' with decreasing molecular weight. The corresponding states representation is a poorer approximation at all molecular weight levels in theta solvents, where the slopes are furthermore universally larger than in good solvents. This is in accord with previous measurements of Weissberg, Simha, and Rothman. Nearly universal curves are, however, obtained for the three theta systems when ~~./(c[rl]), for approximately equal molecular weights, is treated as a function of c. In good solvents an osmotic compression of the polymer coils must take place as the concentration increases. When a theory of this effect previously developed, is applied to polystyrene in toluene and the concentration C/CO is suitably corrected for compression, it accounts for a significant portion of the differences observed between good and poor solvents in the neighborhood of c = co . On the other hand, we suggest that the increased slopes of our reduced plots for low molecular weights in good solvents result from a molecular weight dependent relation between COand l/[q]. The trend of decreasing slopes with increasing molecular weights is reversed in theta solvents at high molecular weights. This may be due to clustering of polymer molecules near their precipitation points. Differences in the degree of clustering for different polymers, but approximately equal molecular weights, may account for the differences in the slopes of our reduced viscosity curves in theta solvents. ACKNOWLEDGMENT Grateful acknowledgment is offered to the Socony Mobil Oil Company, Inc., for the grant of an Employee Incentive Fellow-ship to one of us (J.L.Z.). The authors also
SOLUTION wish to thank discussions.
VISCOSITIES Dr.
S. G. Weissberg
OF
LINEAR
of the
National
FLEXIBLE Bureau
HIGH
POLYMERS
of Standards
for
287 his helpful
REFERENCES I. 2.
3. 4. 5. 6.
7. 8.
9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20. 21.
H. L., AND SIIMHA, R., in F. R. Eirich, ed., ‘LRheology,” Vol. 1. Academic Press, New York, 1956. WEISSBERG, S. G., SIMHA, R., AND ROTHMAN, S., J. Research Natl. Bur. Standards 47, 298 (1951); referred to as WSR. SIMHA, R., AND ZAKIN, J, L., J. Chem. Phys. 33,179l (1960). EPSTEIN, B. N., Ph.D. Thesis, Massachusetts Institute of Technology, Boston, 1956. SCHULTZ, A. R., AND FLORY, P. J., J. Am. Chem. Sot. 74,476O (1952). SCHULTZ, A. R., AND FLORY, P. J., J. Am. Chem. Sot. 75,3888 (1953). MCCORMICK, H. W., J. Polymer Sci. 39,87 (1959). BILLMEYER, F. W., JR., Private communication. KAPUR, S. L., “Solution Properties of Styrene and Methyl Methacrylate Copolymer,” Ph.D. Thesis, Polytechnic Institute of Brooklyn, Brooklyn, 1948. STOCKIVIAYER, W. H., MOORE, L. D., JR., FIXMAN, M., AND EPSTEIN, B. hi., J. Polymer Sci. 16,517 (1955). ROTHMAN, S., Anal. Chem. 22,367 (1950). MCELWAIN, J. W., Anal. Chem. 21,194 (1949). Fox, T. G., JR., AND FLORY, P., J. Am. Chem. Sot. 73,1915 (1951). SHARMON, L. J., SONES, R. H., AND CRAGG, L. H., J. Appl. Phys. 24,703 (1953). ZA~IN, J. L., “The Effect of Concentration on the Solution Viscosities of Linear Flexible High Polymers at Moderate Concentrations.” Doct. Eng. Sci. Thesis, New York University, New York, 1959. DEBYE, P., COLL, H., AND WOERMANN, D., J. Chem. Phys. 33,1746 (1960). DEBYE, P., AND BUECHE, A. M., J. Chem. Phys. 16,573 (1948). KIRKWOOD, J. G., AND RISEMAN, J. J., J. Chem. Phys. 16,565 (1948). MCCORMICK, H. W., J. Colloid Sci. 16, 635 (1961). TONIPA, IT., “Polymer Solutions,” p. 269. Academic Press, New York, 1956. EIRICH, F., AND RISEMAN, J. J., J. Polymer Sci. 4,417 (1949). FRISCH,