536
~ M A L L I EBISNsgIW, , FREDERICK, TSCITOEIUL, AND FERRY
vertical displacement from the curve on which “normal” acids fall fairly precisely is about the same as for maleic acid.14 These two acids have much larger isotope effects than “normal” acids because of the additional hydrogen frequency lowering in the respective anions, I and 11, due to the
formation of a strong ivitramolecular hydrogen bond. “Normal” acid anions hydrogen bond only weakly with a water molecule (hence, a highei stretching frequency prevails in the hydrogen bond formed) leading to a smaller total OH frequency lowering in going from the acid to the anion, and therefore leading to a smaller isotope effect, I n himaleate ion the OH stretching frequency falls near 1650 om.-’, a region characteristicla of hy(14) G. Dahlgren, Jr., and E. A., Long,
J . Am. Chem. floc., 82, 1303
Vola 66
drogen bonds of the single, symmetric minimum variety. In agreement with the observed isotope effect, the OH stretching vibIation in I1 also is found16 in the neighborhood of 1660 cm.-1, Another criterion fiupporting the conclusion that a strong hydrogen bond exists in I1 is that Kz, although it could not be directly measured, can be estimated (Fig. 1) to be a factor of 106 smaller than K1. Hydroquinone is a good model for comparison since electrostatic effects in the dianions should be similar in the two compounds, the distance between oxygens being nearly the same in hydroquinone as in the trans conformation of the dihydroxybiphenyl; hydroquinone can form no intramolecular hydrogen bond in the monoanion, however, and the observed1’ value of K 1 / K S is 45. This large stabilimtion of the monoanion I1 may be compared with the situation in maleic (II,/Kz = 26,000)us, fumaric (K1/Ka 32) acid.l4
-
(15) R. Blino, D, Had& and 4. Novak, 2. Etektrochem., 64, 567 (1Q60), (16) D, Had?& private commun1oatlon. (17) H. Staude and M. TeupP1, Z Blekfvochem., 61. 181 (1957).
limo).
VISCOELASTIC PROPERTIES
OF DILUTE POLYSTYRENE SOLUTTONS AND VERIFICATION OF THE ZIMM THEORY’ BY RICHARDB. DE MALLIE,JR., MEYERH.BIRNBOIM, J. E. FREDERICK, N. W. TSCHOEGL, AND JONN D. FERRY Department of Chentistry, University of
Wisconsin, dfadison, Wisconsin
Received November 9, 1961
Storage (0’) and loss (0”) shear moduli have been measured over a wide frequency range with the a paratus of Birnboirn and Ferry for dilute solutions of a pol styrene with sharp molecular weight distribution,. M , = 26S,f100, in a chlorinated
diphenyl. The high viscosit of the soyvent (2.2 poises a t 25*) ensured that the viscoelastic dispersion fell nrithin the experimental frequency region. d e concentration range was 0.6 to 4% and the temperature range from 0 to 40’. The results are in close accord with the theory of Zirnm, as follova: (a) the ratio (G” w l . ) / G l , where w is circular frequency and n. solvent viscosity, agrees with the theoretical value of 1.73 at higher frequencies; ( b ) GI‘ and G‘ are proportional to w*/a in this region, though some deviation appears a t higher concentrations; (c) the evpenmentally determined terminal relaxation times agree with those calculated from the aolution viscosity within experimental error; (d) the molecular weights calculated from the Zimm theory are correct at low concentrations, though somewhat too high et the higher concentrations; ( e )the local effective vizsoeities calculated from the terminal relaxation times, a t low concentratione, are close to the solvent viscosity. The sharp molecular weight dintribution and the high solvent viscosity, which should minimize effectsof internal viscosity of the polymer chhin, probably m e important in achieving the good agreement with the theory.
-
Introduction The Rouse theory2 for viscoelastic properties has been applied rather successfully t o a variety of experimental data on concentrated polymeric systems. However, it would not be expected to give quantitative agreement with data on dilute solutions, since it neglects hydrodynamic interaction between different segments of the same molecule and it reduces to an unsatisfactory expression for the intrinsic viscosity a t zero frequency. As Cerf3 has pointed out, it is surprising that some measurements on dilute solutions appear to fit the Rouse theory quite well. Nevertheless, the available data are sparse and rather widely spaced on the frequency scale, especially a t high (1) Part XXXVIII of a aeries on Mechanical Properties of Substances of High Molecular Weight. Presented at the Society of Rheology, Oct. 30, 1961. (2) p. E. Rouse, J . Chem. P h ~ s . 21, , 1272 (1963). (3) R. Cerf, Aduances zn Polymer So;., 1 , 382 (1959).
-
frequencies.4 More measurements are needed to make critical comparison of the Rouse theory with that of Zimm,5 which takes hydrodynamic interaction into account, (Neither theory accounts for internal viscosity of the polymer chain, which may be important in solvents of low viscosity.s) An essential difference between the predictions of the two theories is illustrated in Fig. 1, where the dimensionless reduced shear modulus components are plotted against a dimensionless frequency WR = an. Here G‘R = G‘M/cRT and G”R = (GI’ - wrle)M/cRT; G‘ and G” are the storage and loss shear moduli, iLf the molecular weight (assumed homogeneous), c the polymer concentration in g./ml., w the circular frequency, and rlB the solvent viscosity; r1 is the terminal or longest relaxation time in each theory.6 At high fre(4) W. P. Mason, “Handb. d. Physik,” Edited by S. Fliigge, Vol. X I h , p. 361, Springer-Verlag, Berlin, 1961. (5) B. H. Zimm, J . Cham. Phus., 24, 269 (1956).
March, 1962
VISCOELASTIC PROPERTIES OF DILUTEPOLYSTYRENE SOLUTIOXS
quencies according to the Rouse theory, GIB = C f f Rand both are proportionaJ to d'a; according to the Zimrn theory, the ratio GYtt~/Gt~ is not 1 but 1.73, andl both quantities are proportional to For an adequate experimental test, it is desirable to measure the shear moduli GI and 0'' at many rather closely spaced frequencies through a range of about two decades on each side of w = 1/71, and to eimploy a polymer with the sharp molecular weight distribution which is assumed in the theories. Dilute solutions can be measured over a continuous frequency range from 0.01 to 400 C.P.S.in the apand the desired paratus of Birnboim and Feri-~r,~ region near W T ~= 1 can be encompassed by using a highly viscous solvent in which the terminal relaxation time is lo2to lo3longer than in ordinary organic liquids. The high viscosity probably has another advantage in that the frictional resistance to configurational changes provided by the environment of the polymer molecules is much higher than any intramolecular hindrance (internal viscosity) which might be present but is ignored in the theories. The present paper describes some measurements on a sample of polystyrene with a sharp molecular weight distribution, dissolved in a chlorinated diphenyl, in a concentration range from 0.5 to 4% polyiner by weight. Materials.-The polystyrene, E$-108, was generously provided by Dr. H. W. McCormick of Dow Chemical Company. It had been prepared by anionic polymerization followed by terminating the chains with mater.g Its weight-average molecular weight wz.s 267,000 and the ratio of weight to number average was 1.08. The chlorinated diphenyl, Aroclolr 1248, was donated by the &Ionsento Chemical Company through the kindness of Mr. C. M. Williams. Its viscosity at 25' was 2.2 poises, determined by the falling sphere method with spheres of synthetic ruby, and its density was 1.442 g./ml. Viscosities ( q s ) determined a t other temperatures between 1 and 25" followed the empirical equation log qs = 2.63 - 5.893/ (39.2 t ) , where t is Centigrade temperature. This equation was used to obtain values of qa for subsequent calculations. The intrinsic viscosity of the polymer in this solvent also was determined by the falling ephere method t o be 0.85 dl./g. at 25'. This value is quite clone to the predicted intrinsic viscosity of a polystyrene of this molecular weight in o-dichlorobenzene, a more conventional solvent of similar chemical nature-namely, 0.86 dl./,g. The latter value is estimated from the data of Streeter and Boyer'o and established viscosity-molecular weight relationships11 in benzene. Method.-A few measurements were made with the apparatus of Birnboim and Ferry as originally described,8 but unless otherwise identified the data reported here were obtained after two important changes in procedure. A flexible yoke was attached to the lower pole piece in such a way that by raising it the driving coil mounting could be forced up against the central pole piece and immobilized. A long threaded rod, projecting outside the brass case and the thermostat bath, is used to raise and lower the yoke while the apparatus is in operation. Thus, for operation in the impedance mode, the components of
+
-
(6) The numerical evaluations were made by Dr. S. E. Lovell'of the
Theoretical Chemistry Laboratory. Tables of values may be obtained upon request (7) 9. E. Love11 and J. D. Ferry, J . Phys. Chem., 66,2274 (1961). (8) M. H. Birnboim and J. D. Ferry, J . d p p l . Phys., 92, 2305 (1961). (9) H.W. McCormick, F. M. Brower, and L. Kim, J . Polymer Sci., 89,87 (1959) (10) D. J. Streetw and R. F. Boyer, I n d . Eng. Chem., 43, 1790
(1951). (11) P. J. EIory, "Piinciples of Polymer Chemistry," Cornell University Press, Ithaca, N. Y.,1953.
537
t.22
h
F
3
1
I 4
0
s3 -1 I
2 -2
is -3
-
M
0
-4
1 2 -2-10 1 2 3 log WT'. Fig. 1.-Logarithmic plots for the contributions of a polymer solute to the components of the complex shear modulus, w predicted by the theories of Rouse and Zimm. -3-2-10
0 -
%4 += 0
a
E- 3
h
f I
@ M
,o
6
Ffl
3
0
I
0
2 3 4 5 log WaT. Fig. 2.-Logarithmic plots of G' and G" - uqSfor 1% polystyrene 5-108 in Aroclor 1248, referred to 25": pip up, measured a t 9.4"; pip right, measured a t 24.8'. Curves represent the Zimm theory with the coordinates adjusted so the cross corresponds to the origin of the reduced plot in Fig. 1.
1
the clamped impedance, Ro and X O ,can be determined a t every frequency immediately before or after the impedance in motion, R and X . The estimation of Ro ftnd X Oat various frequencies and temperatures from empirical power series based on earlier calibration measurements thereby is eliminated. Smaller values of R ROand X X o (corresponding to samples with higher viscosity and/or rigidity) now can be determined with adequate precision. The method for determining the coefficient C in the phase meter mode also has been modified so that frequent calibrations can be made while a sample is in the cell. When the sample is it viscoelastic liquid, the forces arising from displacement of the moving system to a new fixed position all will relax eventually, except for that due to the spring stiffness EM. Under such conditions equation 11 of reference 8 reduces to C = SMRrdt/dz (1) where dl and ds are maximum recorder trace heights for the displacement and force signals, respectively, and Rd is the variable resistance in series with the driving coil. To determine C, the coil is energized with a square wave of frequency 0.01 c.p.5. from the 202A oscillator, and the appropriate measurements are made. For most samples, relaxation is complete within the half period. For longer relaxation times, a constant voltage of 20 v. from dry cells can be substituted for the square wave input. (This method is not applicable to gelatinous samples, however, for which there i a non-relaxing force contribution f r o q the sample.)
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DEMALLIE,BIRSBOLM, FREDERICM, TECROEGL, AND FERRY
538
I
I
I
Vol, 86
I
log W U T . Fig. 3.-Logarithmic plots of G' and G" - wvs for 2% po1ystgr;ne S-108 in Aroclor 1248, referred to 25". Black p1p up, 10.2 ; pip right, 18.1"; pip down, 25.1'. Open circles, circles, measurements before apparatus improvement!: measurements after improvements: pip up, 10.0 ; pip right, 20.0"; pip down, 30.0". UT = (7
- %)TOP0/(17
- ?Ii)OTP
(2)
where the subscript 0 again refers to the reference temperature. The temperature dependence of r] - qs was nearly the same as that of q 8 . Logarithmic plots of G' and G" wvS reduced in this manner for solutions of 1.0, 2.0 and 4.0% polymer by weight are shown in Fig. 2-4. At each concentration, the measurements a t different temperatures superpose extremely well to provide composite reduced curves. Moreover, the data of 2% taken before and after the improvements in apparatus and procedure agree very well, though the latter are expected to be more reliable. Similar plots at concentrations of 0.5% (measurements at 0.1 and 25.0') and 3% (measurements a t 25.0 and 40.2'), not shown, gave comparable superposition, though the data at 0.5% were more 01 1 I I I scattered and it evidently was not practical to -1 0 1 2 3 4 attempt measurements a t still lower concentralog W U T . Fig. 4.-Logarithmic plots of G' and G" - :os for 47, tions. polystyrene S-108 in Aroclor 1248, referred t o 25 : pip up, The separation between G' and GI' - wvplat higher measured a t 24.8; pip right, measured a t 40.0". frequencies in Fig. 2-4 shows a t once that the results conform to the Zimm theory rather than that Results of Rouse. 'In fact, the data fit the Zimm theory, Each solution x7as studied at two or more tem- which is represented by the solid lines in these peratures, and the results were reduced to 25.0' by figures, with remarkable precision. The theoretical the method of reduced variables, plotting G'Topo/Tp curves are drawn by matching the right side of and (G" - q s ) T o p o / T p against WUT. Here Fig. 1 to the experimental points, with suitable po and p are the solution densities at the reference horizontal and vertical shifts. The cross in each temperature T o and the temperature of measure- of Fig. 2-4 corresponds to the origin in the diment T . The shift factor aT was determined mensionless plot of Fig. 1; its position on the from the contribution of the polymer to the steady- abscissa scale is -log T~ and its position on the ordiflow viscosity, ~ - r ] , , which is the limiting value of (12) J. D. Ferry, "Viscoelastic Properties of Polymers," John G"/w laat low frequenciesI2 Wiley and sons, Xiew York, N. Y,, 1861,pp. 169, 204.
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-
March, 1962
VISCOELASTIC PROPERTIES
nate scale is log cRT/M. The values of the log M and log T~ thus determined are listed in Table I, together vith some other derived quantities. The values of these parameters also serve to verify the Zimm theory, as described below. TABLE I PARAMETERS OBTAINEDFROM ZIMM THEORYAND DERIVED CALCULATIONS Reference temp., 25’ %
polyc, mer p./rnl. 0.5 0.01372 1.0“ .0144 1.0 0144 2 0 .0286 3 0 0430 4 0 ,0588
log r)
-
qs
log 1M
0bsd.b
log 71
0bsd.c
log
log 71
log
ca1cd.d
7~
L
z
2
6
[?]e
OF
DILUTEPOLYSTYRENE
SOLUTIONS
539
of (GI’ - wrs)/oat low frequencies. The agreement between observed and calculated values of r1 is quite precise, reflecting the close fit of the experimental data to the Zimm functions in each case. (Of course, if the true value of i7f is used in this calculation, the observed and calculated values of log 71 will differ by the error in the observed log M incolumn 4.) Local Effective Viscosity.-The terminal relaxation time of Zimm also can be expressed5 as 71
=
0.200 ?.a3Za/zlkT
(4)
Here q, is the effective local viscosity opposing motion of a chain segment, taken in the Zimm theory to be identical with qs but distinguished here 60 its numerical value can be calculated sepa-2.01 rately; u2 is the mean square molecular length per Measurements made before apparatus improvements. monomer unit, and 2 the degree of polymerizaFrom position tion, The characteristic length a for polystyrene From position of cross on ordinate scale. of cross on abscissa scale. d F r o m equation 3, using M in the Aroclor is cplculated from the intrinsic from column 4. viscosity14to be 9.3 A. (about 26% higher than in Discussion a @solvent). Values of log vs obtained from Frequency Dependence of G;‘ and G”.-While equation 4 are included in Table I. Since log qs = the experimentally determined frequency de- 0.34, the local effective viscosity a t lower polymer pendence follows the theoretical curves very concentrations is indeed very close to the, solvent closely over most of the range, a systematic devia- viscosity as inherent in the Zimm theory. At tion does appear a t higher frequencies in the 4% higher concentrations, it is somewhat higher solution, where both G‘ and G” increase somewhat (2.5-fold a t 4%)) but it remains far smaller than less rapidly with frequency than predicted. Since the solution viscosity. the theory treats the polymer molecules as indeCalculation of qa provides somewhat the same pendent of each other, it is natural that there sort of informatSon as comparing the contribution should be some deviations at concentrations where of polymer to the steady-flow viscosity, 7 - qg, overlapping of the polymer coil domains becomes with that expected on the basis of direct proporsubstantia,l. The transition from dilute to con- tionality to the intrinsic viscpity, [q]vsc. Petercentrated solutions will require further study, lin15 has defined an effective local viscosity in this however. sense as (9 - v s ) / [ q ] c . Logarithms of the latter Molecular Weight from Viscoelastic Measure- quantity are given in the last coluinn of Table I. ments.-Since log M , froin ultracentrifuge The effective viscosity of Peterlin is somewhat measurements a t Dow Chemical Company is larger than q B ; it increases in a similar manner, 5.43, it is clear that the values derived from appli- though somewhat faster, with increasing concencation of the Zimm theory a t concentrations of 1% tration. and below are in excellent agreement. At higher Applications of the Zimm theory to viscoelastic concentrations, the molecular weights obtained in properties of polymers of different molecular this manner become progressively too large. This weights and molecular weight distribution, as deviation again no doubt reflects the overlapping as the effects of solvents of different viscosiof the polymer coils, though it is not clear why well ties, will be reported subsequently. this should affect the apparent molecular weight Acknowledgments.-This work was supported in the manner observed. Terminal Relaxation Times.--The terminal re- in part by the Office of Naval Research under Conlaxation time of the Zimm theory can be expressed tract N7onr-28509, and in part by the Research in terms of the polymer contribution to viscosity Committee of the Graduate School of the University of Wisconsin from funds supplied by the Wisas follows,13noting that the coeffkient if1 is 4.04 consin Alumni Research Foundation. We are TI 0.422(7 - q,)AW/~RT (3) indebted to Pro€essor Bruno H. Zimm for his comValues calculated in this manner, using the apparent ments. values of M , also are given in ‘Table I. For this (14) Refeience 11, p. 618. calculation, v - v8 is obtained as the limiting value 0.31 0.64 0 58 1.10 1.42 1.69
5.44 5.46 5.44 5.60 5.61 5.81
-2.89 -2.83 -2.90 -2.60 -2.38
( l a ) Reference 12, p. 1.59, equation 18.
-2 87 0.37 -2.82 .36 .41 -2.90 .53 -2.52 .64 -2.37 .71 -2.02
0 52 ..55 .49 -71 .86 1.01
(16) A . Petellin. “Proc. 2nd Inst. Congr. Rheology,” Ruttermorths, London, 1954, p. 343.
