51 6
34. TAMURA, M. KURATA, K. OSAKI,AND K. TANAKA
Normal Stress Effect in Dilute Polymer Solutions. I. Polystyrene in Dioctyl Phthalate
by Mikio Tamura, Michio Kurata, Kunihiro Osaki, and Katsuhisa Tanaka Department of Industrial Chemistry and Institute for Chemical Research, Kyoto University, Kyoto, Japan (Received August 50,1966)
Xormal stresses of a series of polystyrene solutions in dioctyl phthalate (poor solvent) were measured with a parallel-plate rheogoniometer in the range of concentration, 0.82.0 wt %. The shear stress and dynamic moduli were also measured in the same range of concentration with a coaxial cylinder rheometer. It was found that the incremental normal stress attributable to the polymer was primarily exerted in the flow direction and that u11 - (r33 was proportional to K~ and c/M over a wide range of variables. Here ull is the normal stress in the flow direction, q3is the normal stress in the direction perpendicular both to the flow line and the sheared plane, K is the rate of shear, c is the polymer concentration in grams per cubic centimeter, and k! is the molecular weight.
1. Introduction Measurements of the dynamic mechanical properties of dilute polymer solutions have recently been reported by several groups of investigators. Ferry and coworkers measured the storage (G’) and loss (G”) shear moduli of dilute solutions of polystyrene and other polymers in the range of relatively low frequencies (0.016400 cps) by using very viscous liquids as solvents, and they presented a map of the so-called “hydrodynamic interaction parameter” h as a function of polymer concentration and molecular The result shows that the value of h changes progressively from (Zimm behavior6) to 0 (Rouse behavior’) with increasing concentration, increasing molecular weight, and increasing solvent power. Lamb and coworkers measured G’ and G” of dilute solutions of polystyrene in various ordinary solvents (actually in toluene, butanone, and cyclohexane) using torsional quartz crystals resonant a t 40 and 73 kc/sec, and they found a very similar dependence of h on molecular weight and solvent power as a b o ~ e .A~ similar ~ ~ investigation has also been carried out by Furuichi and co-morkers.lOL1l The success of these studies in obtaining detailed information on the hydrodynamic interaction between polymer segments has awakened our interest in measuring the normal stress components in steady shearing flow in dilute polymer solutions. The Journal ofphysical Chemistry
The normal stress effect in concentrated polymer solutions has been extensively studied by many investigators with use of the cone-plate or parallel-plate type of apparatus.12-15 I n dilute solutions, however, the effect generally becomes too small in magnitude to (1) R. B. DeMallie, M. H. Birnboim, J. E. Frederick, N. W. Tschoegl, and J. D. Ferry, J . Phys. Chem., 66, 537 (1962). (2) N. W. Tschoegl and J. D. Ferry, ibid., 68, 867 (1964). (3) N. W. Tschoegl and J. D. Ferry, Kolloicl-Z., 189, 37 (1963). (4) J. E. Frederick, N. W. Tschoegl, and J. D. Ferry, J . Phys. Chem., 68, 1974 (1964).
(5) J. E. Frederick and J. D. Ferry, ibid., 69, 346 (1965). (6) B. H. Zimm, J . Chem. Phys., 24, 266 (1956). (7) P. E. Rouse, ibid., 21, 1272 (1953). (8) G. Harrison, J. Lamb, and A. J. Matheson, J. Phys. Chem., 68, 1072 (1964). (9) J. Lamb and A. J. Matheson, Proc. Roy. SOC.(London), A281, 207 (1964). (10) H. Tanaka, A. Sakanishi, and J. Furuichi, Rept. Progr. Polymer Phys. Japan, 7, 145 (1964). (11) A. Sakanishi, H. Tanaka, M. Kaneko, and J. Furuichi, ibid., 8, 191 (1965). (12) H. Markovits and R. B. Williamson, Trans. SOC.Rheol., 1, 25 (1957); H. Markovitz, ibid., 1, 37 (1957). (13) W. Philippoff, ibid., 1, 95 (1957); J . G. Brodnyan, F. H. Gaskins, and W. Philippoff, ibid., 1, 105 (1957). (14) Tamura, M. Kurata, and T. Kotaka, Bull. Chem. SOC. Japan, 32, 471 (1959); T. Kotaka, M. Kurata, and M.Tamura, J . A p p l . Phys., 30, 1705 (1959). (15) T. Kotaka, M. Kurata, and M. Tamura, Rheol. Acta, 2, 179 (1962).
NORMAL STRESS EFFECT IN DILUTEPOLYMER SOLUTIONS
be detected by an ordinary apparatus, and no measurement has been reported for solutions with concentration less than about 2 wt yo so far as we are aware. This difficulty may be solved if we adopt a very viscous liquid as solvent such as those used in the recent work by Ferry, et Such a possibility is tested here. In the present paper, we report the normal stress data as well as the steady shear viscosity and the complex modulus data which were obtained for polystyrene in dioctyl phthalate in the range of polymer concentration between 0.8 and 2.0 wt %. This combination of polymer and solvent was chosen as an example of poor-solvent systems; actually, the 0 temperature was , ~ the measuredesignated as 12' by Ferry, et ~ l . while ment was performed at 20'. Effects of polymer concentration and molecular weight on the normal and shear stresses were investigated in detail, and they were compared with the predictions of the existing molecular theories.
2. Theory We first summarize the results of the theory of normal stress effect for dilute polymer solutions, which were obtained by Ikeda16 and Williams." The polymer model used in their theories is the so-called RouseZimm chain which consists of N 1 identical beads connected by Gaussian springs. The hydrodynamic interaction between beads is taken into account in the form of the isotropic average of the Oseen tensor (Kirkwood-Riseman approximation'*). With the model and approximation, the increments of stress components caused by the presence of polymer molecules are given as
+
-
~ 1 2
~
a13
(r13'
a22 Ull
1
=2 nkT(Z,r,)K ~ =
a23
- (r23O
=
0
2nkT(ZpTp2)K2
U1io
=
- u33O
=
0
(1) where n is the number of polymer molecules per unit volume, k is the Boltzmann constant, T is the absolute temperature, and K is the rate of shear. The suffixes 1, 2, and 3 attached to u and uo denote the directions parallel to the flow line, perpendicular to the plane of shear, and perpendicular to both 1 and 2, respectively. r p represents the relaxation time relating the pth mode of cooperative segmental motion. The distribution of r, depends on the strength of the hydrodynamic interaction which is usually represented by the parameter h h = {N'/2/6'/2 n- '/2 aqs (2) Here { is the friction constant of a bead, a is the rootu22O
u33
517
mean-square length of a spring, and ?le is the solvent viscosity. In the case of h > 1,they are given as6 7,
=
1.71(qo - qs)/XpnkT
(4)
with
X,
=
...
4.04, 12.79, 24.2, 37.9,
(for p
=
1, 2, 3, 4,
. . .)
(5)
The former case is often called the free-draining limit, and the latter is the nonfree-draining limit. For intermediate values of h, Tschoegl has recently tabulated values of rp19,20 (not reproduced here), Substitution of eq 3 or 4 into eq 1gives" Zrp2/(ZT,)2
= (nkT/2)[(U'l
-
= 0.206 for h = =
allo)/(~12
-
u12°)21
03
0.400 for h = 0
(6)
which may be used for estimation of the hydrodynamic interaction parameter h from the normal and shear stress data. In the same notations, the storage and loss moduli in shear, G' and G", are written as
+
G' = nkTZP[w2rp2/(1 w2rp2)]
+
G" = n k T Z p [ w r p / ( l w2rp2)]
+ wq,
(7)
where w is the angular frequency. Turning to the phenomenological theory of the normal stress effect, we may refer to the recent theory of Coleman and No11 for the second-order viscoelastic fluids.21 In this theory, components of stresses in steady shear flow are given as second-order terms in the rate of shear g12
= 7oK
Ull
- U22
=
-2yK2
c22
-
=
(p
(733
+ 2Y)K2
(8)
where 70, p, and y are the material constants. In terms of the same constants, G' and G" are given (again with second-order terms in w)22 (16) Y. Ikeda, Kobunshi, 5 , 635 (1957). (17) M.C.Williams, J . Chem. Phys., 42, 2988 (1965). (18) J. G.Kirkwood and J. Riseman, ibid., 16, 565 (1948). (19) N.W.Tschoegl, ibid., 39, 149 (1963). (20) In a good-solvent system, the excluded volume effect also affects the distribution of T=. See N. W. Tschoegl, J . Chem. Phys., 40,473 (1964). (21) B.D.Coleman and W. Noll, Arch. Rational Mech. A d . , 6 , 355 (1960); Ann. N . Y . Acad. Sei.,89,672 (1961).
Volume 70, Number 2 February 1966
M. TAMURA, 111. KURATA, K. OSAKI,AND K. TANAKA
518
lim (G’/02) = - 7 -0
lim ( G / ’ / w ) =
-0
(9)
qo
As will be shown later, eq 8 and 9 offer a possibility of separate determination of ul1 - 433 and 4 2 2 - u33 which are measured only in a combined form in typical normal stress measurements.
3. Materials The polystyrene used in the present study was prepared by thermaI polymerization at 95’. Fractionation was performed a t 30’ from a 0.5% benzene solution using methanol as precipitant. Three fractions, F1, F3, and F4, were used for measurement. Their viscosity-average molecular weights were 5.0 X lo6, 1.2 X lo6, and 6.6 X lo6, respectively, as evaluated by23 =
1.13 x
1 0 ~ ~ ~ 0 . 7 3 (benzene,
25’)
(10)
Commercial dioctyl phthalate (DOP ; Nakarai Chemicals, Ltd., CP grade) was used as a solvent without further purification. The viscosity was 1.10 poises at 20°. According to Ferry, et aL14the 8 temperature is about 12’. Polymer solutions were prepared by storing weighed amounts of polymer and solvent at 50’ with occasional stirring with a spatula. Several weeks were necessary for obtaining homogeneous solutions. Evaporation of the solvent was negligible.
4. Apparatus and Method Measurements of the normal stress effect were carried out witah a parallel-plate rheogoniometer. Details of this apparatus have been reported previously.14 The radial distribution of the pressure exerted normal to the stationary (upper) plate was measured. This is related to the normal stress components as
- (bP/b In r )
= 411
-
433
f
[d(a22
- 433)/b
K]
(11)
with K
= rO/l
(12)
Here r is the radial distance from the axis of rotation, D is the angular velocity of the rotating (lower) plate, and 1 is the gap between two plates. Measurements of the shear stress were carried out with a coaxial cylinder rheometer with rotating outer cylinder. The single-bob method of Krieger and Maron was used for determination of the steady shear viscosity.24 The dynamic shear moduli, GI and G”, were also The Journal of Physical Chemistry
measured with the same rheometer. Details of the method for measurement were given in the previous report by Kotaka and Osaki.26 For separate determination of two normal stress differences, ull - 433 and 4 2 2 - 43a,we used the following procedure. The substitution of eq 8 into eq 11 yields
-bP/d In r = (3a
+ 47)K2
(13)
Then, from eq 8,9, and 13, we obtain
dP
-2
h 3 ) =
3(P f 27’) = 3(422 - 433)/K2 (14) This equation allows us to evaluate u22 - 433 through the comparison of the normal pressure P with the storage modulus Application of the method is of course limited to the range of sufficiently small K where the second-order fluid approximation remains valid.
5. Results and Discussion Comparison of the stresses in steady shear with the dynamic shear moduli are illustrated in Figure 1 where the data obtained for a 2 wt % solution of polystyrene F1 in DOP at 20’ are shown by various types of circles and lines. The thin line represents the loglog plot of the incremental shear stress attributable to the polymer, 412 - KV@, against the rate of shear K . The plots of the corresponding dynamic data, i.e., log (G” wqB) us. log o,are shown by the small white circles. The large white circles represent the loglog plots of - (1/2)(bP/d In r ) against K , and the large black circles represent those of G‘ against o. Thus, the observed features of these properties are as follows. (i) The incremental shear stress g12 - K q s is proportional to K over the whole observed range of K. (ii) The normal pressure gradient - ( b P / b In r ) is proportional to K~ in the range of K smaller than about 20 sec-I. (iii) G” - alls is also proportional to o over a wide range of w . The proportional constant (G” q J / w is compatible with the prediction of the secondorder fluid theory that
-
-
(22) B.D.Coleman and H. Markovitz, J . A p p l . Phys., 35, 1 (1964). (23) W.R.Krigbaum and P. J. Flory, J. Polymer Sci., 11, 37 (1953). (24) I. M.Krieger and S.H. Maron, J. Appl. Phys., 23, 147 (1952). (25) T.Kotaka and K. Osaki, Bull. Inst. Chem. Res. Kyoto Univ., 39, 331 (1961). (26) A test of the method has recently been made by the present authors for concentrated polymer solutions: K. Osaki, M. Tamura, T. Kotaka, and M. Kurata, J. Phys. Chem., 69, 3642 (1965).
519
NORMAL STRESS EFFECT IN DILUTE POLYMER SOLUTIONS
lim (G" w+o
- wqs)/w =
(412
- K ~ ~ ) / =K q - qa
(15)
(iv) G' is proportional to u2in the range of w smaller than about 2 sec-l. I n the limit of small K and w, two quantities G ' / d and ( - 1 / 2 ~ 2 ) ( d PIn / b r ) come into coincidence within experimental error. The last finding shows that, for this system, 4 2 2 u~~is very much less than gll - 433in the range of K where -bP/b In r is proportional to K ~ . Thus, in this range of K , our measurements appear consistent with the molecular theory (eq l).27 Figure 2 shows the log-log plot of 412 - K q a and -bP/b In r against K for a series of polystyrene (Fl) solutions in DOP with various concentrations. 412 is proportional to K for all solutions over the whole range of variables. The value of K at which - b P / b In r begins to deviate from the K~ dependence increases with decreasing concentration. At concen102
101 K [sea-11.
Figure 2. Effect of concentration on normal and shear stresses in solutions of polystyrene F1 in dioctyl phthalate a t 20". Concentrations are 2, 1.5, 1, and 0.8% in weight from left to right, respectively. Small white circles represent u12 - K V ~ , and large white circles represent - d P / b In r.
108
n
8, k 0
P 102
E I
6 i
T
%* a I
s
10'
I
t t
100
i
100
10'
108
w , x [BBc-1].
Figure 1. Comparison between stresses in steady shear and dynamic moduli of a 2 w t % solution of polystyrene ( F l ) in dioctyl phthalate at 20": thin line, incremental steady shear UIZ - q splotted against rate of shear K ; large white circles, normal stresses measured as - (1/2)(dP/d In T) and plotted against K ; small white circles, loss moduli GI' wqs plotted against angular frequency w ; large black circles, storage moduli G' plotted against w ; broken line, limiting K Z dependence of the quantity - (1/2)( d P / b In r ) .
-
trations below 1%, the normal stress ull - 433is proportional to K~ over the whole range of K observed. Figure 3 illustrates the effect of molecular weight on the normal and shear stresses. A series of 2% solutions of polystyrene F1, F3, and F4 in DOP was tested. Now, the most important prediction by eq 1 is that both 412 - ~q~ and q 1 - g1lo are proportional to the number of polymer molecules n in unit volume and, hence, to the ratio of the weight concentration and the molecular weight, c/M. Here c is expressed in grams per cubic centimeter. If this same relationship holds for all M and all concentrations it would predict that all data obtained for the solutions with different combinations of c and M could be superposed on a composite curve in the plot of ( - b P / b In r ) M / c against K(V qa)M/c. This prediction is roughly supported by the present data as is shown in Figure 4. This is in sharp contrast to the case of concentrated polymer solutions, in which the superposition of the normal stress data was possible only within a very limited range of small K and the reduction factor was MO/c2, or simply c-2, instead of M/c.I5 (27) This result does not necessarily imply that uzz = cas. However, the difference between czz and c33, if any, is so small that the present measurements cannot detect it at low rate of shear.
Volume 70,Number 9 February 196%
M. TAMURA, M. KURATA, K. OSAKI, AND K. TANAKA
520
d
I
M1.S -=
OD
MIJp(M)dM
(18)
where p(M) is the distribution function of M in number. Then, assuming the Schulz exponential distribution for q(M) approximating M , by M,, we obtain from the observed value y = 1.10
M w / M , = 1.81 = 1.83
(freedraining case) (nonfree-draining case)
(19)
where M, is the number-average molecular weight. It is notable that an almost identical estimate for M,/M, was obtained irrespective of the assumptions on the strength of hydrodynamic interaction, h = 0 or QI, This implies that for polydisperse samples, the ratio y could be a convenient measure for the
I
1
I
I
101 K
[sec-l].
102
Figure 3. Effect of molecular weight on normal and shear stresses in 2 wt yo solutions of polystyrene F1,F3, and F4 (from left to right) in dioctyl phthalate a t 20'. Molecular weights are 5.0 X 108 for F1, 1.2 X 106 for F3, and 6.6 X 106 for F4,respectively. Small white circles represent ~ 1 2 KV:, and large white circles represent - d P / b In r.
From the data obtained, we can estimate the steady shear compliance or, more conveniently, the quantity 2 , ~ , ~ / ( 2 , by ~ , the ) ~ formula &rP2 y=-=--
(&,T,)~
1 cRT (-?)P/aln~) 2 kf (W - K d 2
(16)
The observed value of this ratio was about 1.1 or, more precisely, 1.10, 1.34, and 1.06 for F1, F3, and F4 polymers, respectively. These values are about 5 or 2.5 times as large as the theoretical values 0.206 (nonfree-draining case) or 0.400 (free-draining case; see eq 6). An analogous result was also obtained by Frederick, et al., in the study of the dynamic moduli for dilute polystyrene solutions in DOP.4 The ratio y is strongIy affected by the heterogeneity in molecular weight. The correction factor is given as28 ybetero/~homo =
M,M,+l/MwM,
(free-draining case)
= M,Mw3/(M'.6)2M,
(nonfree-draining case)
(17)
where M,, M,, M,, and M,+l are the viscosity-average, weight-average, z-average, and(z 1)-average molecular weight, respectively. M1a6is defined by
+
The Journal of Physical Chemistry
K(T
10" - q s ) M1/0'c0 [dyne cm-1 mole-' oma].
Figure 4. Reduced plot of the normal stress obtained for polystyrene in dioctyl phthalate with various combinations of M and c: small white circles, F1 with c = 1.96 X g/cc (or 2 wt %); small black circles, F1 with c = 1.47 X 10-3 (or 1.5 wt %); large white circles, F1 with c = 0.98 X lo-* (or 1 wt %); large black circles, F1 with c = 0.785 X 10-1 (or 0.8 wt yo);triangles, F3 with c = 1.96 X lo-*; squares, F4 with c = 1.96 X 10-8; solid line, composite curve corresponding to y = 1.10 (seee q 16). (28) S. E. Lovell and J. D. Ferry, J . Phys. Chem., 65, 2274 (1961).
ION-EXCHANGE PROCESSES IN AQUEOUS DIMETHYLFORMAMIDE MIXTURES
heterogeneity in molecular weight, but not for the strength of the hydrodynamic interaction h. I n conclusion, in the present system near the 0 temperature, the normal stress component ull - u33 was proportional to K~ and c / M over a wide range of variables, and the component u~~- u33was negligible.
521
The effect of solvent power on the normal stresses will be studied in the following paper. Acknowledgment. We are indebted to Professor Tadao Kotaka for his kind advice and stimulating discussions. Thanks are also tendered to the Ministry of Education of Japan for a grant-in-aid.
Ion-Exchange Processes in Aqueous Dimethylformamide Mixtures'
by A. Ghodstinat, J. L. Pauley, Teh-hsuen Chen,2 and M. Quirk3 Department of Chemistry, Kamas State College of Pittsburg, Pittsburg, Kansas
(Received August 90,1966)
Lithium-sodium, potassium-sodium, cesium-sodium, and sodium-cesium exchange on Dowex 50W X-1 in 0, 5, 10, 25, 50, 75, and 90% dimethylformamide-water mixtures were studied. In general, the logarithm of the exchange coefficients varied with the reciprocal of the dielectric constant as would be predicted for coulombic interactions. However, a minimum was observed for the potassium-sodium and the cesium-sodium exchanges with increasing dimethylformamide concentration. The sodium-cesium exchange showed a corresponding maximum. Radioactive tracer techniques were used to determine exchange coefficients. Solvent uptake and solvent distribution data were also obtained. In general, water is preferred in the resin phase. The preference decreases with higher dimethylformamide concentrations. Total solvent uptake also decreased with increasing dimethylformamide concentration, although much less markedly for the Li resin than for the others.
Introduction Nonaqueous solvents as ion-exchange media have been investigated in recent years by several I n general, mixed solvents uskg water 8s one C O D ponent have been used rather than strictly nonaqueous media for systems using organic exchangers. Equilibrium is reached very slowly in most pure organic the presentinvestigation, the dimethylsolvents. fomamide-water (DMF-H20) system was chosen since this system provided reasonably good salt sohbilities, a good range of dielectric constants, and waterlike properties' A low cross-1inked exchanger was to attainment Of in the DMFrich mixtures where resin swelling was limited. Experimental Section MateriaZs. Solutions were prepared using reagent
grade DMF and ion-free water. Potassium chloride, sodium chloride, lithium chloride, and cesium chloride were all reagent grade and were dried under V a m m n before use. The radioactive sodium-22 and cesium134 were carrier free in the chloride f0m-1. The ex(1) Presented before the Division of Physical Chemistry of the American Chemical society at the Southwest Regional Meeting, Shreveport, La.,Dec 1964. (2) Taken in part from the dissertation of Teh-hsuen Chen to the Graduate School of Kansas State College of Pittsburg in partial fda m e n t of the requirements for the Master of Science degree. (3) National Science Foundation undergraduate research participant. (4) R. G. Fessler and H. A. Strobel, J. Phys. Chem., 67,2562 (1963). ( 5 ) P. C. Huang, A. Mizany, and J. L. Pauley, ibid., 68,2575 (1964). (6) R. Gable and H. Strobel, ibid., 60,513 (1956). (7) D. D. Bonner and J. C. Moorefield, {bid., 58, 555 (1954). (8).A. Materova, Zh. L. Verts, and G. P. Grinberg, Zh. Obshch. Kham., 24,953 (1954).
Volume 70,Number I February 1966
