4183
COMPLEX MODULUS (OFCONCENTRATED P O L Y MSOLUTIONS ~R IN STEADY SHEAR
Complex Modulus of Concentrated Polymer Solutions in Steady Shear’
by Kunihiro Osaki, Mikio Tamura, Michio Kurata, and Tadao Kotaka Department of Industrial Chemistry and Institute for Chemical Research, Kyoto University, Kyoto, Japan (Received June 18, 1966)
Complex moduli of concentrated polymer solutions in steady shear were measured with a rheometer of the coaxial cylinder geometry. The outer cylinder was given a synthetic motion of pure rotation and sinusoidal oscillation, and the inner cylinder was suspended concentrically in the outer cylinder by a torsion wire, The oscillatory part of the inner cylinder motion was detected separately from the stationary part of the motion to determine the complex modulus G,*(w) as a function of angular frequency w for various fixed values of the rate of the superimposed shear K . Measurements of the complex modulus with superimposed steady shear, G,*(w), were performed a t 30” on four polymer-solvent systems, ie., 15% solutions of polystyrenes with molecular weight M = 1.95 X loe and 2.51 X lo6 in toluene; a 10% solution of poly(methy1 methacrylate) with M = 1.60 X loR in diethyl phthalate, and a 5% solution of poly(n-butyl methacrylate) with an extremely high molecular weight, 1.2 X lo’, in diethyl phthalate. From these measurements, the following results were found. (1) I n the low-frequency range, both the real and imaginary parts of the complex modulus in steady shear, G,’(w) and G,”(w), decreased with increasing rate of shear K. The effect was more remarkable in the real part than in the imaginary part, and a t very low frequencies, log [G,,o‘(w)/G,’(w)] was appreciably larger than log [G,=o”(w)/G,”(w)I2. (2) I n the range of high frequency which corresponds to the rubbery plateau region of the relaxation spectrum, G,‘(w) still decreased with increasing rate of shear, whereas G,”(w) increased. (3) The effect of superimposed shear on G,* was practically negligible for Newtonian fluids. The molecular weight dependence of the steady-shear viscosity was also studied on a series of polystyrene-toluene systems under various k e d values of K. Based on the results obtained, a discussion is given of the entanglement couplings in concentrated polymer systems.
Introduction Polymer molecules in bulk or in concentrated solution exhibit a marked non-Newtonian effect in steaclyshearing flow as well as the normal stress effect. Both effects are observed allso in dilute solution, but usually to a far lesser degree in magnitude. Another feature of bulk polymers or concentrated polymer solutions is the so-called box-typle distribution of relaxation times which characterizes the linear viscoelastic behavior in the range of a relatively long time scale. I n all of these features the presence of the entanglement couplings between polymer chains seems to be playing an important role. The characteristic modes of chain motion participating in the box-type distribution of relaxation times may be ascribed to cooperative motions of many molecules joined by the entangle-
mentsa2 The non-Newtonian effect in the steadyshear viscosity may also be explained as a result of a decrease in degree of entanglement with increasing rate of shear,* and the normal stress effect as a result of a large elastic deformation of the quasi-network structure formed by the entanglement^.^^^ The role of the entanglement coupling in viscoelasticity is qualitatively well recognized, but molecular theoretical understand-
(1) A brief preliminary note has been published in J . SOC.Mater. Sci. Japan, 12, 339 (1963). (2) See, for example, J. D. Ferry, “Viscoelastic Properties of Polymers,” John Wiley and Sons, Ino., New York, N. Y., 1961. (3) M.Yamamoto, J . Phys. Soc. Japan, 11, 413 (1956); 12, 1148 (1957); 13, 1200 (1958). (4) A. S. Lodge, Trans. Faraday SOC.,52, 120 (1956).
Volume 60, Number 1.9 December 1966
4184
ing of the coupling still remains at an early stage of development in spite of several pioneering works. a-6 I n terms of linear viscoelasticity, the simultaneous motion of polymer segments can be expressed as a sum of a series of characteristic modes of cooperative segmental motions in a phase space, each being specifled by its own relaxation time. The distribution of the (characteristic) relaxation times depends on the nature of the system, especially on the nature of coupling between segments, and is most conveniently determined by analyzing the complex modulus as a function of the angular frequency of excitation. Thus measurements of the complex modulus provide important information on the nature of segmental coupling. In this way, Ferry and co-workers' have recently been able to estimate the magnitude of the hydrodynamic coupling between polymer segments in dilute solutions with the aid of the molecular theories developed by Rouse8 and Zimm.9 I n concentrated solutions, on the other hand, quantitative interpretation of the boxtype distribution of relaxation times is seriously hampered by the lack of molecular theoretical knowledge of the entanglement coupling. Undoubtedly, other types of experimental information are needed for obtaining a more clear understanding of the entanglement. The normal stress effect and the non-Newtonian effect are related to an over-all contribution of all the modes of segmental motion, and no information on the separate contribution of each individual mode is obtained by this type of experiment. However, as was mentioned, the existing data on these effects indicate that an entanglement network is probably formed in concentrated polymer systems and is degraded to some extent as the rate of shear is increased.1° This suggests that the steady-shearing rate can be adopted as a state variable affecting the degree of the entanglement coupling.ll Consequently, measurement of the complex modulus with superimposed shear can be used as a tool for analyzing the distribution of characteristic modes associated with the entanglement coupling in polymer solutions subjected to steady shear. This paper describes the results of such measurements performed on some typical polymer-solvent systems, i.e. , polystyrene in toluene, poly(methy1 methacrylate) in diethyl phthalate, and poly (rz-butyl methacrylate) in diethyl phthalate.
Apparatus and Preliminary Measurements a. Apparat~s.1~113The apparatus used is of the coaxial cylinder type with rotating outer cylinder. Two sets of driving devices are used, one being for pure rotation and the other for rotational oscillation. Tho Journal of Physical Chemistry
K. OSAKI,M. TAMURA, R4. KURATA, AND T. KOTAKA
Each set consists of a 0.5-h,p. induction motor, a gear box of the ring-cone type with ~l gear ratio of 1:4, and an eight-step gear box of a planetary gear system with gear ratio of 1:3 per step. These two devices are connected through a superimposer with the drive shaft of the main assembly, on which the outer cylinder is mounted. The superimposer consists of a central cogwheel, two planetary gears, and a circular rack. The cogwheel is mounted on the shaft of the drive unit for pure rotation. Two planetary gears are fitted into an annular space between the cogwheel and the circular rack. The axes of the gears are attached to a disk, on the center of which the drive shaft of the main assembly is attached. On the rim of the circular rack a lever arm of the drive unit for oscillatory motion is attached. The whole cog engagement must be very precise so as not to leave any slight gap between them; otherwise smooth motions of the main drive shaft cannot be attained. The axes of the planetary gears and accordingly the disk on which the main shaft is attached can be either rotated unidirectionally by rotation of the central wheel or oscillated sinusoidally by oscillatory motion of the circular rack. The unidirectional rotation and the oscillation can be superposed to give an oscillatory rotation by simultaneous application of both motions to the superimposer.14 By this means, three types of motion can be given to the outer cylinder, depending upon the purpose of measurement. These are: (1) pure rotation within the range of speed between 1.24 X lovaand 10 revolutions per sec. (r.p.s.) for measurement of the steadyshear viscosity 7 as a function of the rate of shear K ; (2) sinusoidal oscillation within the range of frequency ( 5 ) (a) M. 8. Green and A. V . Tobolsky, J . Chem. Phys., 14, 80 (1946); (b) F. Bueche, ibid., 20, 1959 (1952). (6) 8.Hayashi, J . Phys. SOC.Japan, 18, 249 (1963). (7) N. W.Tschoegl and 5. D. Ferry, Kolloid-Z., 189, 37 (1964); N. W.Tschoegl and J. D. Ferry, J. Phys. Chem., 68, 867 (1964); J. E. Frederick, N. W. Tschoegl, and J. D, Ferry, {bid., 68, 1974 (1964). (8)P. E.Rouse, Jr., J. Chem. Phys., 21, 1272 (1953). (9) B. H.Zimm, ibid., 24, 269 (1956). (10) (a) T. Kotaka, M. Kurata, and M. Tamura, J . Appl. Phys., 30, 1705 (1959); (b) T. Kotaka, M. Kurata, and M. Tamura, Rheol. Acta, 2, 179 (1962). (11) The temperature as a state variable affects the hydrodynamic coupling between segments, but does not affect the entanglement coupling. See ref. 10. (12) The apparatus was designed and completed with the cooperation of Iwamoto Seisakusho Co. Ltd., Kyoto. A brief description of the apparatus was first published by M. Tamura, M. Kurata, and T. Kotaka, J . SOC.Mater. Sei. Japan, 8,335 (1959). (13) A description of the apparatus, particularly when it was used for dynamic moduli measurements, also appeared in T. Kotaka and K. Osaki, Bull. Inst. Chem. Res., Kyoto Univ., 39, 331 (1961). (14) M.Tamura, M.Kurata, T. Kotaka, and G . Iwamoto, unpublished.
COMPLEX MODULUS OF CQNCENTRATED POLYMER SOLUTIONS IN STEADYSHEAR
and 30 c.p.8. for measurement of between 4.5 X the complex modulus G* as a function of the angular frequency w ; and (3) superimposed motion of rotation and oscillation for measurements of the complex modulus with superimposed steady-shear GK*as a function of w and K. The outer cylinder is enclosed with a jacket in which oil is circulated from a thermostated bath to keep the temperature of test fluids constant in the range between room temperature and 100' to within an accuracy of f0.2'. An inner cylinder is concentrically suspended in the outer cylinder by a wire of known torsional stiffness. The dimensions of two cylinders are
R1 (radius of inner cylinder)
= 19.0 mm.
Rz (radius of outer cylinder) = 20.0 111111.
L (immersed length of inner cylinder) = 150 mm. (1) A differential transformer unit is attached to the lever which transmits oscillatory motion from one of the drive units to the superimposer. This allows us to detect only the oscillatory part of the outer cylinder motion regardless of whether or not a steady rotation is being superimposed. A pair of differential transformers is also used for detecting the inner cylinder motion. Here two cores are attached to the ends of a beam whose center is in turn attached to the top of the inner cylinder. TWGsolenoids are mounted on adjustable saddles so that each core is located a t the center of the hollow of each solenoid when the inner cylinder is a t rest. The output voltage of the transformers is directly proportional to the torque acting on the cylinder. 'In steaay-shear experiments the torque is measured as a function of the velocity of rotation. I n dynamic measurements, the alternating voltages generated in the transformer units a t the oscillator lever and a t the inner cylinder are simultaneously fed to the horizontal and the vertical plates of a cgthode ray tube, respectively. The .Lissajous pattern thus obtained on the screen is recorded. Then the real, GI, and imaginary, G", components of the complex modulus can be determined as shown later. I n dynamic experiments superimposed on steady shear, the torsion wire is twisted up to a certain angle in response to the superimposed steady shear and oscillates around this angle in response to the oscillatory part of the outer cylinder motion. First, on imposing only a steady rotation, the position of the solenoids on the adjustable saddle is adjusted to cancel the output voltage due to the steady shear. Then the oscillatory motion is superimposed, and oscillahory components of the output voltages are treated in the same way as above.
4185
b. Principle of Measurements and Determination of Apparatus Constants. (i) Steady-Shear Viscosity. Determination of the steady-shear viscosity by this type of apparatus is now a routine procedure, and the detailed description is not reproduced here. The socalled "singlebob" method of Krieger and Maron was used throughout this work.16 The range of shear rate covered in this study was from 1.7 X 10-1 to 3.3 X lo2sec.-l. (ii) Ordinary Complex The oscillatory motion of a viscoelastic fluid in an annular space between walls of two coaxial cylinders has been investigated in detail by Markovitz.16 Giving the amplitude and frequency of the outer cylinder, and assuming the end effect of both cylinders to be negligible, he has shown that
1 - p-qcos
$9
+ isin 9) +
~ ( - l ) n ( p ~ z / G * ) n[ ( Il C W - ~ )P-lAn
n-1
+BnI
=0
(2)
G* = G' Jr iG'' where p is the complex ratio of amplitudes of two cylinders, 9 is the phase difference between two cylinders, i is the imaginary unit p is the density of the test fluid, w is the angular frequency, 1 is the moment of inertia of the inner cylinder assembly, and IC is the torsional constant of the suspending wire. G* is the complex modulus, and G' and G" are its real and imaginary components, respectively. A, and B, are geometric constants of the apparatus. Their first two terms are
m,
(Rz2 - R I ~ ) / ~ T L R ~ ~ R ~ ~ ;
Ai B1
= (RZ2
Az
=
- R12)2/8R2z;
[4In (Ri/RJ
+ (Rzz/Riz)
B2 = (1/192R22)[(R2z-
(RI~/R~~)]/~~TL; Rl2)(Rz4- 5R22R12 -
-
+
2 ~ ~ 4 )1 2 ~ 2 ~n ~ (R~/RJI ~ 4 (3) where R1, Rz, and L represent the dimensions of two cylinders as defined in eq. 1. In our case,ls using the values given in eq. 1, we found
-41= 1.432 X B~ = 4.75
x
A2 = 2.39 X lo-'
10-4; B%= 2.80
x
10-6
(4)
The terms with indexes higher than n = 3 were all negligible. The moment of inertia I of the inner cylinder assembly including the core of the differential trans(15) I. M. Krieger and S.H. Maron, J. A p p l . Phys., 25, 72 (1954). (16) H. Markovitz, ibid., 23, 1070 (1952).
Volume 69,Number 12 December 1966
K, OSAKI, N. TAMURA, Pi. KURATA, AND T,KOTAKA
4186
formers was determined from the frequency v of free oscillation by using the well-known relation that v (i/2a)(IC/I)”*where IC is the torsion constant of the wire employed, For the wire with IC = 1.615 X 106 dynecm./rad, the observed value of v was 1.111sec.-l, which led to I = 3.314 X IOa g. cm.%. The above test was performed without applying an exciting current to the solenoidal coils of dBerential transformers. I n the presence of the exciting current, it was recognized that the frequency of free oscillation was increased up to v’ = 1.124 sec.-l for the same torsion wire. This increase in v may be attributed to the influence of restoring force acting on the cores due to their interaction with the magnetic field of the solenoids. In order to take this effect into account, we used an effective wire constant k’ defined as k’ = k Ak instead of k where Ah (Zn)21(v’2- v 2 ) = 3.6 X IOa dyne-om./rad. It was found that the correction Ak did not depend on the wire constant k, nor on the frequency so long as the displacement of the cores was not very large, The real wire constants IC of the torsion wires used were 1.578 X IO8 for the most rigid one and 1.015 X 106dyne-cm./rad for the least rigid one. The amplitude ratio p and the phase difference cp of two cylinders can be determined at each angular frequency of oscillation w from the Lissajous ellipse and its circumscribed rectangle which is determined by lines parallel t o the horizontal axis and t o the vertical axis of the oscilloscope: that is, p can be given as the ratio of two sides of the rectangle, and p can be estimated by the use of the relation17 Isin pI = (4/a) { ( A ) / [ A ] ] Here , ( A ) and [ A ] are the area of the ellipse and of the rectangle, respectively. In this way, by the analysie of the Lissajous pattern at various frequencies, two components, real and imaginary, of the complex modulus can be determined as function of angular frequency (3. (iii) Complex Modulus with Superimposed Steady Shear. The principle for measurement of the dynamic complex modulus with superimposed steady shear is essentially the same as above. When the outer cylinder is given a constant angular velocity, the inner cylinder is rotated up to an equilibrium angle where the torque exerted by the torsion wire is counterbalanced by the viscous force. A sinusoidal oscillation superimposed on the rotation of the outer cylinder then gives rise to an oscillation of the inner cylinder around this equilibrium angle. The oscillation of the inner cylinder is also sinusoidal if the amplitude of oscillation of the outer cylinder is kept sufficiently small. A complex quantity corresponding to the ordinary complex modulus can then be determined by analyzing the oscillatory parts of motions of two cylinders, again
-
The Journal of Physical Chemistry
+
with the use of eq, 2. This quantity is denoted by G2*, and is temporarily termed the complex modulus with superimposed steady shear. The coinplex modulus with superimposed steady shear varies not only with the angular frequency W , but also with the rate of shear K , the latter being determined by the steady motion of the outer cylinder. C. Preliminary Test of Dynamic Measurement with Superimposed Steady Shear. Castor oil, as an example of purely viscous fluids, was used for a preliminary test of the dynamic measurements with superimposed steady shear. The steady-shear viscosity of the castor oil was 5.8 poises. A series of tests was performed a t 25’ by using a series of torsion wires with different torsion constants. A typical result is given in Figure 1. In thia particular example, a wire with k: = 9,253X lo6 dynes cm, rad-1 was employed. I n Figure 1, the open circles represent the experimental points obtained for pure oscillation and three types of shaded circles represent those obtained for oscillation super-
0.4 Q
q
0.3
B
-0 I
0.2
0.1
n
0
-0.1
15
10 *-
a
s
$
0
.9
-5
P
-10
- 15 0
100
50
150
w2, 800. -2.
Figure 1. w sin rp/p and 1 - (cos q / p ) plotted against a2 for castor oil at 25” (7 = 5.7 poises). Superimposed rate of shear, K, in sec.-l: 0, 0; @, 0.162; 0, 0.482; e,1.46. Solid lines: calculated values by eq. 5 and 6.
(17) L.A. Wood, Rev. Sci. Instr., 2, 644 (1931).
4187
COMPLEX MODULUS OF CONCENTRATED POLYMER SOLUTIONS IN STEADY SHEAR
imposed on rotation. Different marks correspond to different values of rate of shear. I n the case of a purely viscous fluid with a oonstant viscosity v j the real part of the complex modulus, G*’,should be identicajlly zero irrespective of the rate of shear K , and the imaginary part GKIt should be given by qw. Equation 2 then becomes
+ B l p ) ~-.2 kAI]/q [(A21 3. B2P)PW2
- J&P1/q2
= w
sin q/p
(5)
= 1 - (cos V / P > (6)
Accordingly, the plots of w sin q / p and 1 - (cos q / p ) against ~2 both should give straight lines whose slopes and intercepts are determined by eq. 5 and 6. These lines are shown in Figure 1 by the solid lines. Agreement between the calculated and experimental values was satisfactory in the plot of w sin cp/p us. w 2 over a wide range of frequency. On the other hand, in the plot of I - (cos p/p) ug. wZ, the agreement was limited to within a narrow range of frequency, for u 2 about 30 to 100. The frequency range of application of eq. 6, of course, depended on the torsion constant of the wire employed. A series (oftorsion wires with a wide variety of torsional constant was needed for covering a wide range of frequency. Generally, the thinner wire was suitable for lower frequencies. When steady shear was superimposed, however, a, thin wire was often twisted too much by the torque due to the steady motion of the fluid. This introduced a difficulty in the measurements for the combination of a low frequency and a high rate of shear. However, for a suitable combination of w and K, we were able to determine G,* as a function of w and K . The results will be shown below.
Experimental Section
Table I : Molecular Weights of Polystyrene Samples con-
Polymerization temp.,
version,
%
OC.
115 95 80 60 40
5.1 10.7
8.5 8.6 7.3
Designated as PS(A).
Mv
0.56 X 0.72 x 1.04 X 1.95 X 2.51 X
106 108 106 10Ba lO@
* Designated as PS(B).
temperature, 230’; these samples were kindly given us by Dr. H. Utiyama, Kyoto University). Their viscosity-average molecular weights were 1.18 X lo5, 8.62 X 104, 2.43 X lo4, and 1.05 X lo4, respectively, as evaluated by eq. 7 . Poly(methy1 methacrylate) was prepared by thermal polymerization at 47’ without solvent or catalyst. The conversion was 8.5%. The viscosity-average molecular weight was 1.60 X lo6 as was evaluated bylg
[VI = 5.7 X
10-6M,0.76
(benzene, 25’)
(8)
Poly(%-butyl methacrylate) was prepared by emulsion polymerization with the use of potassium persulfate as initiator and sodium lauryl sulfate as emulsifier. This sample has an extremely high intrinsic viscosity, 8.17 dl./g., in methyl ethyl ketone at 2 3 O , which corresponds to a viscosity-average molecular weight of 1.18 X 10’ as evaluated by the equation of Chinai, et al. 2o [q] =
1.56 X 10-5Mv0.81
(9)
a. Materials. Polymers used in this investigation were polystyrene, poly(methy1 methacrylate) , and poly(n-butyl methacrylate). Nine samples of polystyrene were used, of which five samples with relatively high molecular weights were whole polymers specially prepared for this study by thermal bulk polymerization a t various temperatures. Conversions were kept below 10% in order to avoid the possibility of chain branching. The viscosity-average molecular weight M , was evaluated from the intrinsic viscosity in cyclohexane a t 35’ with the use of1*
Toluene and diethyl phthalate, G.R. grades, were used as solvents without further purification. Each solution was prepared in a sealed ampoule at an elevated temperature (about 50’). Several weeks was necessary before obtaining homogeneous solutions. b. Superimposed Oscillation Measurement in Polymer Xolutions. Measurements of the dynamic modulus with superimposed steady shear were performed at 30’ for four types of polymer solutions, a 15% solution in toluene of polystyrene A (PS(A)-Tol), a 15% solution in toluene of polystyrene B (PS(B)-Tol), a 5% solution in diethyl phthalate of poly(%-butyl methacrylate)
= 8.2 x 10-4~,0.50 (7) Polymerization temperatures, conversion, and M , of these samples are given in Table I. Four other samples with relatively low molecular weights were the fractions of a low conversion product of thermally polyrnerized styrene (polymerization
(18) W. R. Krigbaum and P. J. Flory, J.Polymer Sci., 11,37 (1953). (19) P. J. Flory, “Principles of Polymer Chemistry,” Cornell University Press, Ithaoa, N. Y . , 1953, p. 312. (20) S. N. Chinai and R. N. Guzzi, J. Polymer Sci., 21, 417 (1956). The use of eq. 9 in such a high molecular weight range as this is some-
what questionable. This value of M v is therefore only a tentative estimate.
Volume 69,Number 19 December 1966
4188
K. OSAKI, M. TAMURA, M. KURATA, AND T,KOTAKA
10 -1
100
10'
10'
x , 880. -1.
Figure 2. Steady-shear viscosity, n, plotted against rate of shear K a t 30" for four polymer-solvent systems: 0, PS(A)-toluene; (PI PS(B)-toluene; c), PMMA-diethyl phthalate; e, PBMA-diethyl phthalate.
I
I
100
101 W,
(PBMA-DEP), and a 10% solution in diethyl phthalate of poly(methy1 methacrylate) (PMMA-DEP). The viscosity data in steady shear of these solutions are shown in Figure 2, where the viscosity qn is plotted against the rate of shear K in double logarithmic scale. As is seen from the figure, these four solutions are examples of polymer solutions which show the nonNewtonian effect to a different degree; $,e. the effect increases in the order PS(A)-To1 < PS(B)-To1 < PMMA-DEP < PRMA-DEP. Figure 3 shows the result of superimposed oscillation measurements obtained for the 15% toluene solution of polystyrene A, where the open circles represent the real and the imaginary parts of the complex modulus, G' and G", as function of w, and the four types of shaded circles represent those of the complex modulus with superimposed steady shear, G,' and G,", obtained at various rates of shear K . Apparently, the superimposed steady shear produces a rather minor effect on the complex modulus so far a8 the rate of shear is sufficiently small to keep the nonNewtonian effect in the viscosity less significant. The effect of the rate of shear becomes significant in the real part of the complex modulus only at the highest shear rate investigated. The imaginary part seems insensitive to the steady shear. Figure 4 shows the result of the superimposed oscillation measurements for the 15% solution of high molecular weight polystyrene B in toluene. This polymer exhibits a stronger non-Newtonian effect than does polystyrene A. A remarkable effect of the rate of shear on the complex modulus is observed in this figure. As the rate of the applied shear is increased, the real part of the complex modulus is significantly The Journal of Physical ChemktTy
I
seo.-l.
Figure 3. Real and imaginary parts of complex modulus in shear, GK'and QK", plotted against angular frequency, O, for 15% solution of polystyrene A in toluene at 30". Superimposed rate of shear in aec.-l: 0, 0; c), 0.162; 0, 0.482; 0 , 1.46; 0 , 4.60.
104
108
I
d6
109
$
.-
i
10-1
1
I
100
101
I
w , sec.-I.
Figure 4. Real and imaginary parts of complex modulus in shear, GK'and Gd', plotted against angular frequency, o, for 15% solution of polystyrene B in toluene at 30". Superimposed rate of shear in sec.-l: 0, 0; (3, 0.162; c), 0.505; 0, 1.62; 8 , 5.13.
COMPLEX MODULUS OF CONCENTRATED POLYMER SOLUTIONS IN STEADY SHEAR
Figure 5. Real and imaginary parts of complex modulus in shear, GK' and GK", plot,ted against angular frequency, w, for 10% solution of polg(methy1 methacrylate) in diethyl phthalate a t 30". Superimposed rate of shear in sec.-l: 0, 0; a, 0.188; 0, 0.560; 0, 1.71.
16
I
I
I
c-
Figure 6. Real and imaginary parts of complex modulus in shear, GK'and GK", plotted against angular frequency, W , for 5% solution of poly(n-butyl methacrylate) in diethyl phthalate a t 30". Superimposed rate of shear in sec.-1: 0, 0; @, 0.190; (3, 0.580; 0, 1.76.
diminished over almost the whole range of freqyency observed. However, the imaginary part is again less sensitive to the rate of shear, particularly at high frequencies. The lower the frequency becomes, the larger is the relative decrease in G,' or G;" observed. I n addition, the curves for the complex modulus with superimposed steady shear are different in shape from those for the complex modulus without superimposed steady shear, especially at a high rate of shear. Simple sliding of the curves for different shear rates
4189
along the coordinate axis does not provide a composite curve as the WLF shift does for data of different temperatures.21 This fact implies that the effect of the steady shear on the complex modulus is more complicated than is the effect of temperature. Figures 5 and 6 show the superimposed oscillation data obtained for the 10% solution of poly(methy1 methacrylate) in diethyl phthalate and for the 5% solution of poly(%-butyl methacrylate) in diethyl phthalate, respectively. These solutions show stronger non-Newtonian behavior than do the polystyrene solutions. As is seen in Figures 5 and 6, the effect of the steady shear on G,* is very significant in these solutions. The qualitative feature of the effect is the same as in the polystyrene solutions described above. A closer inspection indicates, however, that the inzaginary part of the complex modulus increases with increasing shear in the high-frequency range. Now returning to Figure 4, we may observe the same effect at the highest frequency, though to a slight degree. Thus we may conclude that this effect is not a characteristic of a special type of polymers, e.g., poly(methy1 methacrylate) and poly(%-butyl methacrylate), but is common to all concentrated polymer solutions. c. Steady-Shear Viscositg. The effect of molecular weight on the steady-shear viscosity was also investigated at various rates of shear for 15% solutions of polystyrene in toluene at 30'. The results are shown in Figure 7. The molecular weight dependence of the zeroshear viscosity shows the well-known feature2; that is, the log-log plot of 70 vs. M , displays a sharp turn at about M , -= 3 X lo6, and beyond it, increases with 3.4 power of M,. This turning point is often referred to as the critical molecular weight M , for the onset of the entanglement network. At a given rate of shear, say K = 0.482 sec.-1, the polymers with a relatively low molecular weight, say less than about lo6, are practically Newtonian. The non-Newtonian deviation becomes significant at and above a certain molecular weight. It is of course unlikely that, in the log-log plot of vK vs. M , with a given rate of shear K, a sharp turning point is defined between the Newtonian and non-Newtonian regions. The transition from one region t o the other is gradual as r ] , decreases gradually with increasing K for a fixed M , (see Figure 2). It is nevertheless worthwhile to note that the log qx vs. log M T plot most closely resembles type A in Figure 8, but neither type B nor C, and that the boundary between Newtonian and nonNewtonian regions shifts to lower M., as K increases. (21) M. L. Williams, R. F. Landel, and J. D. Ferry, J. Am. Chem. Soc., 7 7 , 3701 (1955).
Volume 69,Number 18 December 1965
4190
K. OSAKI,M. TAMURA, M. KURATA, AND T. KOTAKA
of the two following features obtained in the three regions of different time scales. (i) In the flow region, the decrease in the real part of the complex modulus due to the superimposed shear is larger than the square of the decrease in the imaginary part. In fact, the value of n, defined by
was about 3-4 in Figure 4. (ii) In the plateau region, the imaginary part of the complex modulus increases with increasing rate of shear (see Figures 5 and 6). Rough estimates of the relaxation spectrum can be based on either component of the complex modulus, G,’(w) and G,”(w). The results are such that the relaxation spectrum H(ln 7 ) as a function of logarithmic relaxation time is markedly cut off at the long time end and is heightened in the plateau region of shorter time scales. Figure 9 gives a schematic representation of the result.
Log M. A
Log M. B
Log M . C
Figure 8. Schematic representation of three possible types of log 7 us. log M relationship. Solid line: zero shear viscosity; dashed line: viscosity under a constant value of the rate of shear.
Discussion The present measurements were performed only within a rather restricted range of frequency for each given system. It is, however, clear from the shape of the double logarithmic plots for G’ and G” us. w that the frequency range covered in Figure 4 roughly corresponds to the so-called flow region of the relax& tion spectrum, while the frequency range covered in Figure 6 corresponds to the rubbery plateau region. Figure 5 may then be regarded as an example for showing the shear rate effect in the intermediate region between the above two. Thus, we are able to deduce a general picture of the shear rate effect on the relaxation spectrum from comparison or synthesis The Journal of Physical Chemistry
7 Log A.
Figure 9. Schematic representation of shear effect on the relaxation spectrum H. Solid line corresponds t o zero rate of shear, and dashed line to a finite rate of shear.
The type of behavior observed has a unique feature different from that observed for the effect of temperature. I n the latter case, a single shift factor aT can be assigned for all relaxation times (the WLF shift2I), and the relative effect of the variable on G’ and G” can be given as
Here To is a reference temperature. The observed value of n, wm found to be much larger than 2 as mentioned above. I n a sense, the effect of the rate of shear resembles the effect of molecular weight depression &g exemplified by the sharp cut of the long time end of the relaxation spectrum. Quantitative analysis of the
MEMBRANE POTENTIAL IN NONISOTHERMAL SYSTEMS
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experimental results in terms of the existing theories such aa Marvin’s22 will be published elsewhere.
valuable discussions. Thanks are also tendered to Prof. S. Onogi of this university for stimulating discussions and to the Ministry of Education of Japan for a grantin-aid.
Acknowledgment. Talking with Ilr. R. s. Marvin, who WM a visiting professor of this university in the at an period 1961-1962’ was stage of this study. We are indebted to him for many
(22) R. 6. Marvin and H. Oser, (196~);6 7 ~ 87 , (1963).
J. Res. Natl. Bur. Std., 66B, 171
Membrane Potential in Nonisothermal Systems
by Masayasu Tasaka, Shoji Morita, and Mitsuru Nagasawa Departments of Applied and Synthetic Chemistry, Faculty of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan (Received June 21, 1066)
Applying nonequilibrium thermodynamics, a theory of membrane potential in nonisothermal systems was obtained. Measurements of membrane potential across an ionexchange membrane separating two solutions which are at different temperatures were carried out to be compared with the theory. Agreement between the theory and the observed results is satisfactory. The present experimental results are in agreement with the results reported by Tyrrell, et al., who used strips of ion-exchange resin, but are different from those reported by Ikeda, et al., using collodion membranes.
Introduction It is well known that an electrostatic potential difference appears across a membrane separating two electrolyte solutions of different concentrations. However, since not only B concentration gradient but also a temperature gradient across the membrane is a driving force for the permeation of ions, a potential difference must also appear when there is a temperature difference on both sides of the membrane. The study of such “thermal membrane potentials” is important because they are often observed in physiology, in eleetrodialysis, or in desalination operations, though little attention has thus far been devoted to this phenomenon. Some experimental results were reported by Ikeda, et aL,l who found that the thermal membrane potential across oxidized collodion membranes separating two solutions of 0.1 M X:C1 at different temperatures is quite low, lower than 0.05 mv./deg. On the other
hand, Tyrrell, et al., measured thermal diffusion potentials in strips of ion-exchange resin and reported a value about 10 times higher than that reported by Ikeda, et al., at 0.1 M KCI. Since the fundamental principles in these two experiments are the same except for the shapes of the ion exchangers used, the difference observed between both results is quite a dilemma. Hills, et aL13derived an equation of nonisothermal membrane potential by using nonequilibrium thermodynamics, but the effect of temperature difference on membrane potential could not be clearly predicted from their theory, since their theory includes the heat ~~
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(1) T. Ikeda, J . Chem. Phys., 28,166 (1958);T.Ikeda, M.Tsuchiya and M. Nakano, Bull. Chem. Soc. Japan, 37, 1482 (1964). (2) H. J. Tyrrell, D. A. Taylor, and C. M. Williams, Nature, 177, 668 (1956). (3) G. J. Hills, P. W. M. Jacobs, and N. Lakshminarayanaiah, Proc. Roy. SOC.(London), A262, 246 (1961).
Volume 60, Number 12 December 1066
