NORMAL STRESS EFFECT IN DILUTE POLYMER SOLUTIONS
Normal Stress Effect in Dilute Polymer Solutions. 11.
2271
Polystyrene
in Chlorinated Diphenyl
by Kunihiro Osaki, Katsuhisa Tanaka, Michio Kurata, and Mikio Tamura Department of Industrial Chemistry and Institute for Chemical Research, Kyoto University, Kyoto, Japan (Received January 25, 1966)
Normal stresses of a series of 0.3-2.0 wt % polystyrene solutions in chlorinated diphenyl (good solvent) were measured with a parallel plate rheogoniometer. The shear stresses and the complex moduli were also measured with a coaxial cylinder rheometer. The normal stress ( - b P / b In r ) and shear stress ( 6 1 2 - K ~ J were linear in K~ and K , respectively, in the lowest range of shear rate K observed. The ratios ( - b P / b In T ) / K ~and (u12 - K ~ ~ ) / decreased K with increasing rate of shear more rapidly than did those of polystyrene solutions in dioctyl phthalate (poor solvent). It was also found that both the normal stress and shear stress were proportional to the number of polymer molecules per unit volume, n. For a wide variety of molecular weights and concentrations, the reduced (or intrinsic) normal and shear stresses, (- bP/b In r)/n and ( 6 1 2 - K V ~ ) / ~were , found to be universal functions of the reduced rate of shear (qo - qs)K/n. The effect of heterogeneity in molecular weight was also investigated through a comparison of the data obtained for ordinary fractionated samples and an anionically polymerized sample.
Introduction Measurement of normal stress effect of dilute solutions of fractionated polystyrenes in dioctyl phthalate (a poor solvent) was reported in part I of this series.' The incremental normal stress attributable to the polymer was found to be exerted primarily in the flow direction and to be proportional to the square of the rate of shear K over a wide range of K . As for the effect of molecular weight and concentration, the data obtained with various combinations of the variables were superimposed to give a composite curve in the plot of (- bP/b In rjM/c vs. K ( T ~ - q S ) M / c . Here (- bP/ b In r ) is the normal stress as measured with a parallel plate rheogoniometer, qo is the zero shear viscosity of the solution, qs is the viscosity of the solvent, M is the molecular weight, and c is the concentration in grams per cubic centimeter. All of the findings mentioned above were compatible with the predictions of the spring-beads model theory for the normal stress eff e ~ t , ~ which corresponds to the Rouse4 or Zimm6 theory for the dynamic mechanical properties. Not only qualitative but also quantitative agreement was found between the observed and the theoretical values of the steady-
state compliance after an appropriate correction was made for the effect of molecular weight distribution (Mw/M,, = 1.8). However, we could not assess the degree of the hydrodynamic draining effect, because the steady-state compliance was rather insensitive to the hydrodynamic interaction for the material used. In this paper, we report the results of normal stress measurements carried out in a polymer-good solvent system, polystyrene solutions in chlorinated diphenyl. The effect of molecular weight distribution is also studied.
Experimental Section Materials. The fractionated polystyrene used in this study was prepared by thermal polymerization at 95" without solvent or catalyst. Fractionation was performed at 30" from a 0.5% benzene solution using (1) , ~ M. Tamura, M. Kurata, K. Osaki, and K. Tanaka, J . Phys. Chem., 7 0 , 516 (1966). (2) Y.Ikeda, Kobunshi, 5 , 635 (1957). (3) M.C.Williams, J . Chem. Phys., 42, 2988 (1965). (4) P. E. Rouse, Jr., ibid., 21, 1272 (1953). (5) B.H.Zimm, ibid., 24, 266 (1956).
Volume 70, Number 7 July 1966
K. OSAKI,K. TANAKA, 111. KURATA, AND M. TAMURA
2272
methanol as precipitant. The viscosity-average molecular weight of each fraction was evaluated by [q] =
1.13 X lO6MV0J3 (benzene, 25°)6
(1)
and tabulated in Table I. Among these samples, F1, F3, and F4 are the same samples as used in the previous study.l Table I Nomenclature
(I
co.
M"
loE 108 lo8 lo6 lo*
F1 F3 F4 F5 F6
5.0 X 1.2 x 0.66 X 0.54 X 0.27 X
Slll
2.39 X l o b a
ter, 1 is the gap between the two plates, and Q is the angular velocity of the lower (rotating) disk. The shear stress a12 was measured with a coaxial cylinder rheometer. The single-bob method of Krieger and Marong was applied to determine the shear stress as a function of the rate of shear. The complex modulus was measured with the same coaxial cylinder rheometer. A detailed description of the apparatus was also published earlier. lo Separate determination of the components of the normal stress was based on the phenomenological theory of Coleman and No1l.l' As was shown by Coleman and Markovitz,12 this theory leads to the following relationships for the so-called second-order fluid lim G'/w2 = l.W.0
lim (ull X-0
M , evaluated by Dr. H. W. McCormick, Dow Chemical "2
Another polystyrene, Sill*, prepared by anionic polymerization was also used in this study. (This sample was kindly given us by Dr. J. E. Frederick, University of Wisconsin.) The weight-average molecular weight M , of this sample was 2.39 X 105 and M,/M, was 1.08 as reported by Dr. H. W. McCormick, Dow Chemical Co. Aroclor 1248 (mixture of chlorinated diphenyl and polyphenyl, Monsanto Chemicals Ltd.) was used as the solvent. This was a good solvent for polystyrene, and the viscosity was as high as 2.70 poises at 25". Polymer solutions were prepared by storing weighed amounts of polymer and solvent a t 50" with occasional stirring with a spatula. Several weeks was necessary for obtaining homogeneous solutions. Evaporation of the solvent was negligible. Apparatus and Method. Measurement of the normal stress was performed with a parallel plate rheogoniometer. A precise description of the apparatus has been published previously? In this apparatus, the pressure exerted on the upper (stationary) plate by the test fluid is related to the components of the normal stress b p
- b P / d In r =
lim
-0
a11
K
433
+
b(a22
- m ) / b In K
(2)
= fi/1
where uIl, 422,and u33 are the normal stress components in the flow direction, in the shear plane, and in the plane perpendicular to both the flow direction and the shear plane, respectively, r is the distance from the cenThe Journal of Physical Chemistry
- U22)/K2
(
~
-2 U 3 d / K 2
(3)
(4)
The two components, (all - 422) and ( 6 2 2 - 4,of the normal stress can be determined separately from each other by using these relationships. A detailed account of the method may be found in our previous publication.'*
Results and Discussion Figure 1 shows comparison of the results of steadyshear measurement with those of dynamic measurement. The small and large black circles represent the incremental shear stress (uI2 - K ~ , J and the normal stress - l / ~ ( b P /In b r ) plotted against the rate of shear K , respectively. The factor '/2 is introduced in front of - ( b P / b In r ) for the sake of convenience (see eq 4). Small and large white circles represent the incremental loss modulus (G" - wqs) and the storage modulus G' plotted against the angular frequency w, respectively. The incremental shear stress (a12 - KT/J and loss modulus (G" - wq8) are proportional to K and w, respectively, only in the lowest region of t'he variables ( K or o < 10). (6) W. R. Krigboum and P. J. Flory, J . Polymer Sci., 11, 37 (1953). (7) M. Tamura, M. Kurata, and T. Kotaka, Bull. Chem. Soc. Japan, 3 2 , 471 (1959); T. Kotaka, M. Kurata, and M. Tamura, J . Appl. Phys., 30, 1705 (1959). (8) H. Markovitz, Trans. Soc. Rheology, 1 , 37 (1957). (9) I. M.Krieger and S. H. ,Maron, J . Appl. Phys., 23, 147 (1952). (10) T. Kotaka and K. Osaki, Bull. Inst. Chem. Res. Kvoto Univ., 39, 331 (1961); K. Osaki, M. Tamura, M. Kurata, and T . Kotaka, t o be published. (11) B. D. Coleman and W. Noll, Arch. Rational Mech. Anal., 6, 355 (1960); Ann. N . Y . Acad. Sci., 89, 672 (1961). (12) B. D. Coleman and H. Markovitz, J . Appl. Phys., 3 5 , 1 (1964). (13) K. Osaki, M. Tamura, T. Kotaka, and M. Kurata, J . Phys. Chem., 69, 3642 (1965).
NORMAL STRESS EFFECTIN DILUTEPOLYMER SOLUTIONS
2273
I
101
I
I
I
101
10,
101 w, K , seC-l.
Figure 1. Comparison between stresses in steady shear and dynamic moduli of a 1 wt yo solution of polystyrene (F3) in Aroclor 1248 a t 25": small black circles, incremental shear stress (alz- qa)plotted against the rate of shear K ; large black circles, normal stress measured as -1/2(BP/d In T ) and plotted against K ; small open circles, loss modulus (G" - UT). plotted against the angular frequency o ; large white circles, storage modulus G' plotted against w .
Figure 2. Effect of concentration on normal and shear stresses in solutions of polystyrene (F3) in Aroclor 1248 a t 25". Concentrations are 1, 0.5, and 0.3% in weight from left to right, respectively. Small circles represent (u12 - K V ~ )and large circles ( - bP/B In r).
The normal stress ( - b P / b In r) and storage modulus G' are not proportional to K~ and w2, respectively, over almost the whole range of variables, except at the lowest values, say K , o < 5 sec-I. In other words, the system behaves as a second-order fluid only in a very narrow range of long time scale (small K or a). This is in sharp contrast to the previous observation for polystyrene in dioctyl phthalate (poor solvent), to which the secondorder fluid approximation was applicable over a fairly wide range of K or w . Now, in the range of K or w lower than 5 sec-', it may be natural to expect from the figure that - 1/2(bP/b In r) and G' come to coincidence with each other. This, in turn, implies that the (az2- ua3)component of normal stress, if it exists, should be quite small compared with the (all - az2)component a t least in this range of small K. Accordingly, we may assume ( - d P / b In T ) to be equal to the (UII - a22) component of the normal stress. Although this conclusion seems inescapable so far as our data are concerned, this is not based on the direct evaluation of the normal stress components. Therefore, we do not insist that a22 is equal to ~ 7 3 ~ . Final solution of this problem is still open to future study. For a higher rate of shear, at which the solution does not behave 8s a second-order fluid, we cannot determine two components of the normal stress separately from each other by the parallel-plate data alone.
Figure 2 shows the log-log plots of the incremental ~ ) the normal stress ( - b P / shear stress (u12 - ~ 7 and b In r) against rate of shear K , which were obtained for a series of solutions of polystyrene (F3) in Aroclor 1248 with various concentrations. The large circles represent (- bP/b In r ) and the small circles, (au ~ 7 ~ )Figures . 3 and 4 show the effect of molecular weight on the normal stress ( - b P / b In r ) and the incre. 3 gives the mental shear stress (al2 - ~ 7 ~ )Figure results obtained for 0.5% solutions of fractionated polystyrenes (Fl, F3, and F4) and Figure 4 gives the results for 2% solutions of fractionated polystyrenes (F5 and F6) and the living polystyrene (Slll). Viscosities of relatively high molecular weight polystyrenes depend markedly on the rate of shear in the measured range of the rate of shear (Figure 3). This effect is observed even in the lowest concentration tested, 0.3% (Figure 2). On the other hand, viscosities of relatively low molecular weight polystyrenes are practically independent of the rate of shear (Figure 4) The log-log plot of ( - b P / b In r ) against K has a slope lower than the second-order fluid slope, 2, at any value of K . This deviation of the observed slope from 2 exists to some extent even in the case of low molecular weight samples whose viscosities are practically independent of K .
100 K,
8%C-'.
l ' O ~ U m 6 70,Number
7 July 1066
K. OSAKI,K. TANAKA, M. KURATA, AND M. TAMURA
2274
I
I
J
V
1
I
I
1lTJ
10'
10%
1 100
I, sec-1.
10%
101 x , sec-1.
Figure 3. Effect of molecular weight on normal and shear stresses in 0.5 wt % solut'ions of polystyrene F1, F3, and F4 (from left to right) in Aroclor 1248 at 25". Molecular weight of polystyrene is 5.0 X 106 for F1, 1.2 x 106 for F3, and 0.66 X 106 for F4. Small circles represent ((112 - K ? ~ ) and large circles represent ( -dP/d In r).
Figure 4. Effect of molecular weight on normal and shear stresses in 2.0 wt yo solutions of polystyrene F5, F6, and S l l l (from left to right) in Aroclor 1248 at 25". Molecular weight of polystyrene is 5.4 X 105 for F5, 2.7 X 105 for F6, and 2.39 X 105 for 5111. Small circles represent (012 ~ 7 and ~ )large circles represent ( - dP/d In 7).
Finally, it is to be noted that in Figure 4, the normal stress of the solution of the living polystyrene (5111) is considerably lower in magnitude than that of the fractionated polystyrene (F6) which has an approximately equal molecular weight. This difference may be due to the difference in molecular weight distribution and will be discussed later. Now, it has been found by Harrison and collaborators'* that dynamic mechanical data obtained for dilute polystyrene solutions with various molecular weight polystyrenes and concentrations can be superimposed on a composite curve by using reduced variables. These are: G'M/c for the storage modulus, (G" wq,)M/c for the loss modulus, and w(qo - v a ) M / cfor the angular frequency, where c is the polymer concentration in grams per cubic centimeter and hence c/M is the number of moles in a unit volume. We may expect in analogy that the reduced plots of - ( d P / b In r ) M / c and ( q 2 - K ~ ~ ) M against / c ~ ( 7 0- q,)M/c also give composite curves in the case of steady-shear data (see eq 3 and 4 and Figure 1). This expectation was tested in Figure 5, where the data given in Figures 2 to 4 all were replotted using the reduced scales. The data of (w K ~ J and ( - b P / b In r ) obtained for various combinations of molecular weights and concentrations except for S l l l were superimposable on two composite
curves, one for the shear stress and the other for the normal stress. Excellent superposition was obtained when the procedure was applied to the data obtained for different concentrations of the single sample F3. They are represented in Figure 5 by various types of circles. A closer inspection of the figure reveals that the reduced normal stress for F3 is slightly larger than those for other fractions. The same tendency was also observed in dioctyl phthalate.' (The parameter y defined in eq 5 was about 25% higher for the solution of F3 than for the solution of F1 or F4 in dioctyl phthalate. See ref 1.) Therefore, the difference may be attributable to possible difference in molecular weight distribution among these samples. I n conclusion, we may say that the stresses in steadyshear flow of dilute polymer solutions are essentially proportional to the number of polymer molecules in a unit volume. In Figure 5, we show by the dashed lines the reduced curves obtained in dioctyl phthalate for the sake of comparison.' At high reduced rates of shear, ( - b P / b In r ) M / c and (alz - K ~ , ) M / Cin Aroclor (good solvent) are both smaller t~hanthose in dioctyl phthalate (poor solvent). At lower reduced rates of shear, no solvent effect is observed on the reduced
-
The Journal of Physical Chemistry
-
~~
~
~~
(14) G . Harrison, J. Lamb, and A. J. Matheson, J . Phys. Chem., 68, 1072 (1964).
NORMAL STRESS EFFECT IN DILUTE POLYMER SOLUTIONS
2275
1.06
2.39
3.72
M x 10-6.
Figure 6. Triangular distribution of molecular weight.
X(?
- qa)M/c, dyne cm-9 mole-1 cma.
Figwe 5. ltediiced plot of the normal and shear stresses obtained for polystyrene in Aroclor 1248 with various combinations of M and c: A, F1 with c = 7.25 X g/cc (or 0.5 wt c/c); 0, 0, F3 with c = 1.45 X lo-* (or 1.0 wt %); c = 7 25 X 1 0 - 3 (or 0.5 wt %) and c = 4.35 x 10-s (or 0.3 wt %) respectively; V, F4 with c = 7.25 X (or 0.5 w t 70); 0, F5 with c = 2.90 X 14, F6 with c = 2.90 X 10-2; S l l l with c = 2.90 X 10-8.
a,
ity of our samples is possibly in this region, we cannot derive any definite conclusion on the solvent effect on y from the above observation. The normal and shear stresses of S l l l solution give 0.35 for y. A theoretical value of y for the heterogeneous polymer with M,/M, = 1.08 is also calculated according to the theory of Lovell, et aZ.,lEfor the cases of zero and infinite hydrodynamic interaction. Two types of molecular weight distribution are assumed. One is the Schulz distribution and the other is the triangular distribution illustrated in Figure 6, where the weight distribution function of molecular weight g(M) is shown. The calculated values of y are given in Table 11. The triangular distribution gives a little larger value of y than does the Schulz distribution.
a,
Table 11
normal stress as well as on the reduced shear stress. This leads to the parameter y which is independent of the nature of the solvent. y =
cRT '/2
- Iim
M
-0
( - d P / b In r ) (412
- d2
Hydrodynamic interaction
0
(5)
According to Frederick and co-workers, the dynamic mechanical properties of dilute polymer solutions change from Zimm-like to Rouse-like behavior with increasing solvent power.16 If this is the case, the parameter 7 which is affected by the degree Of hydrodynamic interaction should be a function of the solvent power. Our results apparently conflict with this conjecture. However, as was mentioned in the previous paper, the Darameter Y is also affected by the molecular weight insensitive to the 'istribution' and it becomes degree of hydrodynamic interaction if the heterogeneity is such that Mw/Mn = 1.8-2.0. Since the heterogene
W
Calcd value of Triangular distn
0.526 0.308
y
Schulz distn
0.500 0.270
The observed value 0.35 does not agree with the value for the limit of nor hydrodynamic interac-
tion when the Schulz distribution is used, On the other hand, if the triangular distribution is adopted, the theoretical value of for the limit of infinite hydrodynamic interaction cOmeS quite close to the (15) J. E. Frederick, N.W. Tschoegl, and J. D. Ferry, J. Phys. C h m . , 68. 1974 (1964). . . (16) s. E. Love11 and J. D. Ferry, ibid., 65,2274 (1961).
Volume 70,Number 7 July 1966
WILLIAM G. STEVENS AND IRVING SHAIN
2276
served value. Remembering that the triangular distribution, rather than the Schulz distribution is more appropriate for this type of polymer, we may conclude that the polymer S l l l displays Zimm-like, instead of Rouse-like, behavior in Aroclor.
Acknowledgment. Discussions with Professor T. Kotaka of this university have been indispensable during the course of this study. We are also indebted to Professors J. D. Ferry and J. E. Frederick for their kind supply of the solvent and the living polystyrene.
Electrolysis with Constant Potential. Diffusion Currents of Metal Species Dissolved in Spherical Mercury Electrodes
by William G. Stevens1 and Irving Shain Department of Chemistry, University of Wisconsin, Madison, Wisconsin (Received January 26,19fi6)
~
~~
The hanging mercury drop electrode was used in a potentiostatic method for the determination of the diffusion coefficient of metal species in mercury. The amalgam was prepared in a constant-potential preelectrolysis step, during which the quantity of electricity involved in the electrodeposition was measured to determine the amalgam concentration. Then, current-time curves were obtained for the diff usion-controlled dissolution of the amalgam. Diffusion coefficients calculated from the time-dependent current decay were compared with those calculated from the “spherical correction” term in a critical evaluation of the method. As a demonstration system, the diffusion coefficient of cadmium in cm2/sec, relative standard deviation mercury was measured. The value was 1.61 X 3.0%.
The use of stationary spherical mercury electrodes in potentiostatic experiments has been suggested prev i ~ u s l for y ~ the ~ ~unambiguous determination of the diffusion coefficients of electroactive species. By evaluating the diffusion coefficients from both the slope and the intercept of potentiostatic current-time curves, it was shown that the uncertainties involving other experimental parameters, particularly the electrode area, could be minimized. This method has been applied previously only for the determination of reducible ions in the solution. It is obvious, however, that the same approach could be used to determine diffusion coefficients of metal atoms in a hanging mercury drop electrode, by analyzing the current-time curves obtained from potentiostatic experiments involving the oxidation (dissolution) of the amalThe Journal of Phyaical Chemistry
gams. Developing the method appeared to be of considerable importance as a result of the possibility of studying intermetallic compound reactions in the merc ~ r y . For ~ any such studies, accurate values of the diffusion coefficients are required. The use of the hanging mercury drop electrode in stripping analysis had indicated that it would be possible to prepare amalgams of known concentration by electrodeposition in a carefully controlled preelectrolysis step.5 This work was carried out to determine exactly (1) National Science Foundation Predoctoral Fellow, 1961-1965. (2) I. S h a h and K. J. Martin, J . Phys. Chem., 65, 254 (1961). (3) I. Shain and D. S. Polcyn, ibid., 65, 1649 (1961). (4) W. Kemula, Z. Galus, and Z. Kublik, Bull. Acad. Polon. Sei. Ser. Sei. Chim., ffeol. Geograph., 6 , 661 (1958); W.Kemula and Z. Galus, ibid., 7 , 553, 607, 729 (1959).
