J. E. FREDERICK, S.W. TSCHOEGL, A N D J. D. FERRY
1974
Dynamic Mechanical Properties of Dilute Polystyrene Solutions; Dependence on Molecular Weight, Concentration, and Solvent’
by J. E. Frederick, N. W. Tschoegl, and John D. Ferry Department of Chemistry, Unicersity of Wisconsin, Madison, Wisconsin
(Receiwd February 24,1964)
Storage (G’) and loss (G”) shear moduli have been measured for dilute solutions of several polystyrenes with sharp molecular weight distributions in three chlorinated diphenyls (Aroclors) and in di-2-ethylhexyl phthalate at the &temperature. The molecular weights ranged from 82,000 to 1,700,000 and the concentrations from 0.5 t o 475 polymer. Measurements were made between 0.016 and 400 C.P.S. and usually a t two or three temperatures between 5 and 40”. The effect of temperature was in every case described by the method of reduced variables. The viscoelastic properties were expressed either as the polymer contributions to the complex modulus, G’ and G” - u l ~ v a ,or as the relative contributions to the complex viscosity, (?’ - u l v a ) / ( ? - ulva) and ~ ’ ’ / ( v- ul?a), where w is the circular frequency, u1 the volume fraction of solvent, the solvent viscosity, and 7 the steady-flow solution viscosity; all results except those in the &solvent mere reduced to 25”. The observed frequency dependence ranged from the predictions of the Zimm theory t o the predictions of the Rouse theory with varying degrees of intermediate behavior. For every solution it could be described by the Tschoegl extension of the Zimm theory using a parameter E calculated from intrinsic viscosities and a parameter h, representing the strength of the hydrodynamic interaction, chosen empirically. The value of h changes progressively from (Zimm behavior) to 0 (Rouse behavior) with increasing concentration, increasing molecular weight, and increasing solvent power. The influence of concentration is probably the key to all three effects The molecular weights derived from the viscoelastic measurements by the theory are all a little high, by a larger factor than could be attributed to a small degree of molecular weight heterogeneity. Measurements on one solution of a polystyrene with a most probable molecular weight distribution gave a frequency dependence of the real component of the complex viscosity whose shape agreed closely with B theoretical evaluation for this case on the basis of the Zimm model by Peticolas.
Introduction Measurements of the dynamic mechanical properties of dilute solutions of polystyrene (a sample with a sharp molecular weight distribution, M w = 267,000) in a solvent of rather high viscosity have been reported earlier. The frequency dependence of the storage and loss shear moduli conformed closely to the theory of Zimm.3 By the “Zimm theory” we mean, as customarily, the case of dominant hydrodynamic interaction treated by Zimin, corresponding to the KirkwoodRiseman derivation of intrinsic viscosity a t high molecular ~ e i g h t s . Later ~ studies of dilute polyisobutylene solutions4J showed that polymers with molecular weight The Journal of Physical Chemistry
of the order of lo6again followed the Zimm theory, but that for very high molecular weight the frequency dependence approached rather the predictions of the theory of Rouse.6 Extensions of the Zimm theory were (1) Part XLVI of a series on mechanical properties of substances o f high molecular weight Presented in part at the Fourth International Congress on Rheology, Providence, R. I., August 27, 1983. (2) R. €3. DeMallie. Jr., SI. H. Birnhoim. J. E. Frederick, N. W . Tschoegl, and J. D. Ferry, J . P h y s . Chem., 6 6 , 536 (1962). (3) B. H. Zimin, J . Chem. P h y s . , 24,289 (1956). (4) N.W. Tschoegl and J. D. Ferry, Rolloid-Z., 189, 37 (1963). (5) N. 15’. Tschoegl and J. D. Ferry, J . Phys. Chem., 6 8 , 887 (1984). (8) P. E. Rouse, Jr., J . Chem. Phys., 21, 1272 (1953).
DYNAMIC AfECHAKICAL
PROPERTIES O F
DILUTEPOLYSTY R E K E
then developed7)8with the conclusion that either the Rouse or the Zimm preldictioits, or a continuous range of behavioral types intermediate between these two, may be fulfilled depending on the degree of hydrodynamic interaction am0 ng different segments of the polymer molecule. The interaction is expected to be influenced by the molecular weight and also by the character of the thermodynamic interaction between polymer and solvent. I n the present study, several polystyrene samples with sharp molecular weight distributions and weightaverage molecular weights ranging from 82,000 to 1,700,000 have been investigated in several different solvents to elucidate these effects and 1,o make comparisons with the new theory. The effect of polymer concentration has been studied in the range from 0.5 to 4%. Rleasurements have been made also on one polystyrene with a molecular weight distribution closely approaching the "most probable" form. The solvents were chlorinated diphenyls and di-2-ethylhexyl phthalate, all of which had sufficiently high viscosities to cause the dispersion of the modulus to fall in an experimentally accessible frequency range.
Experimental Materials. Four polystyrene samples with sharp distributions were generously provided by Dr. H. W. SlcCormick of Dow Chemical Co.; thley had been prepared by anionic polymerization. Their weight-average molecular weights (determined a t Dow ultracentrifugally) and Ii4,.,/lWn ratios are given in Table I. The sample with a most probable weight distribution (19F) had been kindly furnished by hlr. R. F. Boyer of Dow some years before, and has been the subject of several other investigations in this Y a b o r a t ~ r y as ~ , ~well ~ as a t Dow. " Recent ultracentrifuge stud'ies by Blair and Williams12have shown that the distribution is close to the most probable form. In addition, two highly fractionated samples were given us by Professor Bruno H. Zimm (MDP-2, MDP-1). Table I :
Characteristics of Polystyrenes --[q1,
Sample
53-102 s-111 S-108 S-1163 MIIP-2 MDP-1 19F a
At 10.0".
M
W
82,000 259,000 267,000 400,000 1,000,000 1,700,000 360,000
iW,,,/M,,
A-1232
25.0" in-A-I248 DOP
dl./g., a t
1.05 1.08 1.08 1.05
... .,.
0.385
...
...
...
.
...
2.05 2.70
0.85 1.00 1.94 2.05 1.11
,
I
,
1.90
, ,
... ...
... 0.52" ... ...
...
1975
SOLUTIOKS
Aroclors 1232, 1248, and 1254-partially chlorinated diphenyls with average degrees of chlorination increasing in the order listed-were kindly furnished by Mr. 7111. F. Waychoff of Monsanto Chemical Co. Since the viscosities vary somewhat from one lot to another, a single lot of each was used for the experiments reported here. The viscosities were determined a t several temperatures by capillary viscosinieters, by the falling sphere method, or directly in the Birnboim apparatus (see below); the values agreed within a few per cent when more than one method was used. The viscosities a t 25" were 0.143, 2.67, and 90.4 poise, respectively, for the three Aroclors; they all increased rapidly with decreasing temperature. Densities were measured pycnometrically a t several temperatures. Coniplete data for viscosity and density are given elsewhere.I3 Di-2-ethylhexyl phthalate (DOP, from Union Carbide Cheinicals Corp.) was chosen as a rather viscous solvent with the Flory teinperature in a convenient range. The &point was estimated by Mr. James Sanders as follows. The intrinsic viscosity [ q ] of Sample S-1163 in DOP was measured a t several different temperatures, using a n Ubbelohde viscosimeter with a suitable flow time. The expansion factor a was calculated from the equation a 3 = [q]/KM"', where M is the molecular weight, taking K as a function of temperature from calculations of F l ~ r y . ' ~ ,Then, '~ following F l ~ r y . ' K(a5 ~ , ~ ~- a3) was plotted against l/T; an approximately linear relation was obtained, crossing the &point a t zero on the ordinate axis. Alternatively, following Kurata, Stockmayer, and Roig,I6 K ( a 3 a)(l 1/3a2)a'2was plotted against 1/T. Values of 10-13" were obtained for the %-point, and 12" was selected for the subsequent viscoelastic measurements. The solvent viscosity a t this temperature was 1.29 poise. Because of the low viscosity, measurements were restricted to solutions of the sample of highest molecular weight (MDP-l), since a minimum solution viscosity of the order of 5 poise is needed for satisfactory results (and, unlike the case of the Aroclors.
+
(7) N. W. Tschoegl, J . Chem. Phys., 39, 149 (1963). (8) N. W. Tschoegl, ibid., 40. 473 (1964). (9) L. D. Grandine and J. D. Ferry, J . A p p l . Phys., 24, 679 (1953). (10) J. D. Ferry, L. D. Grandine, Jr., and D. C. Udy, J . Colloid Sci., 8, 529 (1953). (11) D. J. Streeter and R. F. Boyer, Ind. Eng. Chem., 43, 1790 (1951). (12) J. E. Blair and J. W. Williams, J . Phus. Chem., 6 8 , 161 (1964). (13) J. E. Frederick, Ph.D. Thesis, University of Wisconsin, 1964. (14) P. J. Flory, "Principles of Polymer Chemistry," Cornell University Press, Ithaca, N. T.,1953 (15) P. J. Flory, J . Am. ChPm. Soc., 73, 1915 (1951). (16) XI. Kurata, W. H. Stockmayer, and A. Roig, J . Chem. Phys., 33, 151 (1960).
Volume 68, Number 7
J u l y . 1.964
1976
specification of the &point prevented increasing the: viscosity by working a t lower temperatures). Intrinsic viscosities of several of the polymers were also measured in Aroclor 1248, and two in Aroclor 1232. These are also included in Table I. It is evident that the Aroclors are moderately good solvents. Method. The storage and loss moduli, G' and G", of each solution were measured by the apparatus of Birnboim and Ferry, l7 with modifications which have been described in subsequent publication^.^,^ The frequency range was 0.016-400 c.P.s., and measurements were usually made at 25.0" and one or two other temperatures. For calculating the polymer contribution to loss modulus, G" - u1wqS, where 211 is volume fraction of solvent, w the circular frequency, and q 9 the solvent viscosity, the latter was always taken as that measured directly in the Birnboim apparatus for the sake of consistency.
J. E. FREDERICK, N. W. TSCHOEGL, AND J. D. FERRY
given in the figure legends and the footnotes to Table 11. Comparison of Molecular Weights in a Good Solvent. I n Fig. 1, the two components of the complex modulus are plotted for 2% solutions of four samples of different molecular weights in Aroclor 1248. I n agreement with the earlier, less extensive results on p o l y i ~ o b u t y l e n e , ~ ~ ~ the frequency dependence changes gradually from Zimmlike to Rouse-like behavior n ith increasing molecular weight.
Results The results are presented partly as the polynier contributions to the modulus components, G' and G t ' vlwqs, and partly as the contributions to the reduced complex v i s ~ o s i t yV'R , ~ = (7' - u l q s ) / ( q - vlqs) and q " ~ = q " / ( q - vlqs), where 7 is the steady-flow solution viscosity obtained from the dynamic data as the value of G"/w at low frequencies where this ratio becomes constant. The two representations are of course equivalent, though V'R and q " ~provide a more sensitive comparison with theory. The modulus components are plotted logarithmically against w ; the reduced complex viscosity components against w ( q - U~T&CRT, where c is the polymer concentration in g./ml., for ease of comparing with t h e ~ r y . ~ For such logarithmic plots, the Rouse theory predicts that at high frequencies the storage and loss components will coincide, with a slope of l / %for the modulus and - ' / z for the viscosity ; the Zimm theory predicts that the loss component is higher by a constant factor and the slope is z/3 for the modulus and - l / 3 for the viscosity. The characteristic frequency dependences are shown in Fig. 1 of ref. 2 and Fig. 3 of ref. 5 . For almost every solution, measurements at two or more different temperatures have been reduced to a reference temperature of 25" in the usual manner2 from the temperature dependence of (G" - vlwqs)/w a t low frequencies where that ratio becomes constant and equal to the polymer contribution to the steady-flow viscosity. I n a few cases where the ratio had not reached constancy a t the lowest frequencies of measurement, the temperature shift factors were determined empirically. The exact temperatures of measurement are The Journal of Physical Chemistry
0
1
2
3
4
5
-1
lcg maT
0
I
e
3
4
Figure 1. Logarithmic plots of G' (points top black) and G" - vlwvs (points bottom black) for four samples with molecular weights as indicated, in A-1248. All concentrations 27,; all data reduced to 25.0" from measurements at the following temperatures: 82,000, pip down, 8.4"; left, 10.0"; up, 25.0'. 239,000, pip down, 5.1"; left, 25.0". 400,000, 25.0"; 1,700,000, pip down, 25.0'; left, 40.0". Theoretical curves are drawn with the following values of e, in order: 0.065, 0.100, 0.100, 0.135; and of h, in order: m j 100, 8, 0.
For each solution a pair of theoretical curves is shown, calculated for a particular choice of certain parameters h and E which enter the theory of TschoegP as described in the Discussion. The theoretical calculations provide dimensionless reduced moduli G'R = G'M/cRT and G''R = (GI' - vlwqs)M/cRT as functions of a reduced frequency W T ~where T~ is the terminal relaxation time. Logarithmic plots of these are matched to the experimental data by suitable horizontal and vertical shifts of the scales, and the cross shown on each graph corresponds to the origin of the dimensionless theoretical plot. From its position, the molecular weight and terminal relaxation time can be obtained as described earlier. (17) M . H. Birnboim and J. D. Ferry, J . A p p l . Phys., 32, 2305 (1961).
1977
DYNAMIC MECHANICAL PROPERTIES OF DILCTE POLYSTYRES'E SOLUTIONS
Table I1 : Parameters from' Extended Zimm Theory and Derived Calculations (All Reduced to 25", except Solutions in D O P a t 12") M Solvent
1248
x
10-3
82
0 ,090b
267d
0 ,090b
1000 1700
1254
I>OP 1248
0.065
239
40 0
1232
Conon., wt. %
€
0 . 10Fib
0.131" 0 . 147E
e
1000 1700
e
267 1700
b 0
190h
. .
2 4" 2 4" 1 2 3 4 1 2 4 1 0.5j 1 2 1 1 2 28 2 3 2
c
x
102,
g. /ml
2 86 5 68 2 86 5 68 1 44 2 86 4 30 5 68 1 44 2 86 5 68 1 44 0 72 1 44 2 86 1 27 1 27 2 53 3 05 1 98 2 97 2 86
s
h m m
100 15 m Q)
1 1 63
8 2 15 8 2.5 0 15 2.5 0 m
m
2 ...
2 279 2 279 2 219 2 160 2 235 2 235 1 886 1 886 2 211 2 118 1 974 2 133 2 097 1 993 1 845 2 133 1 993 1 645 2 235 2 368 1 985
log
log
'M,/M
log w"e
0.01 0.04 0.20 0.23 -0.01 0.18 0.05 0.11 0.21 0.29 0.32 0.30 0.15 0.08 0.13 0.33 0.21 0.17 0.19 0.67 0.83 0.57
4 92 5 03 5 58 5 61 5 42 5 61 5 48 5 54 5 81 5 89 5 92 6 30 6 38 6 31 6 36 6 33 6 54 6 39 5 62 6 90 7 06 5 85d
71,
sec.
-3.55 - 3 23 -2.62 -2.24 -2.86 -2.49 -2.41 -2.17 -2.24 -1.96 -1.52 -1.30 -1.29 -1.06 -0.39 -2.55 -2.00 -1.72 -0.85 -1.16 -0.43
...
Theoretical curves calculated from e = 0,100. Not shown graphia S o t shown graphically; measurements a t 10.0" and 25.0'. Data from ref. 3, q.v. for individual temperatures of measurement. e Theoretical cally; measurements a t LO", 11.7", and 25.0". curves calculated from E = 0.135. Wot shown graphically; measurements a t 10.0" and 25.0". S o t shown graphically; measureSample 19F with most probable distribution; ilil values for ments a t 25.0°, 34.8", and 49.5'.
an.
Comparison of Solvents for High Molecular Weight. I n Fig. 2, the components of the complex modulus are plotted for 2y0 solutions of Sample MDP-1, with the highest molecular weight, in three different solvents. Whereas in the two Aroclors, both moderately good solvents, the frequency dependence is indistinguishable from that predicted by the Rouse theory, in the DOP &solvent it conforms closely to the Zimm theory despite the very high molecular weight. This agrees with the expectation of Peticolasls that the Zimm theory should apply in &solvents regardless of the polymer molecular weight. Comparison of Solvents and MolecuEar Weights af Low Concentration. In Fig. 3, complex moduli of the samples with molecular weights 1,000,000 and 1,700,000 are compared a t 1% concentration in A,roclors 1232 and 1248. Just as a t 2% concentration (Fig. 2), the shapes of the frequency dependence in these two Aroclors cannot be clearly distinguished, although they are now both intermediate between the Rouse and Zimm predictions. The shift toward Rouse-like behavior with increasing molecular weight is evident a t this lower concentration in both solvents. (The data a t high fre-
quencies in A-1248 have been disregarded in .matching the curves, because the divergence between G' and G" - z ' l ~ q 8is not explained by any current theory and its origin is uncertain.) Comparison of Diferent Concentrations. I n Fig. 4, the components of the reduced complex viscosity are plotted for Sample MDP-1 in DOP a t the Flory temperature a t concentrations of 2% (same data as a t the top of Fig. 2) and 3Y0. This plot has an expanded vertical scale and is a more sensitive test of agreement with theory. At 2%, the Zinim theory, represented by the solid curves, is rather closely followed; but a t 3% there is a definite shift toward Rouse-like behavior even a t the &point. The same effect is seen in a good solvent in Fig. 5, where the components of the reduced complex viscosity a t molecular weight 400,000 in Aroclor 1248 shift progressively from Zimm-like toward Rouse-like behavior with increasing concentration. In these plots, the match with theoretical curves in(18) W. L. Peticolas, ,J. Chem. Phys., 35, 2128 (1961), and private communication.
Volume 68, Number 7
Jzil?~,1.964
1978
J. E. FREDERICK, N. W. TSCHOEGL, AND J. D. FERRY
0
1
2
3
A
O
1
2
3
4
Figure 3. Logarithmic plots of G' and G" = V , W ~ , for Sarnples MDP-1 (molecular weight 1,700,000) and MDP-2 (molecular weight 1,000,000) in A-1232 and A-1248. 411 concentrations 1%; all data reduced to 25.0" from measurements a t the following temperatures: both samples in A-1232, pip down, 0.0'; left, 10.0"; up, 25.0'. Both samples in A-1248, pip down, 10.0"; left, 25.0"; up, 40.0". Theoretical curves are drawn with e = 0.135; values of h are 15 for left pair and 2.5 for right pair. 1
I
I
1
I
A-1232 -1
0
1
2
3
109 Figure 2 . Logarithmic plots of G' (points top black) and G" - ulwq, (points bottom black) for Sarnple MDP-1, molecular weight 1,700,000, concentration 2%, in three solvents as indicated. Data on DOP measured a t &temperature of 12.0' and unreduced; all other data reduced to 25.0". In A-1248, pip down is 25.0°, pip left 40.0'; in A-1232, pip down is 10.0", pip left 25.0'. Theoretical curves are drawn with B = 0 and h = m in DOP, and e = 0.135 and h = 0 in the Aroclors.
volves a horizontal shift only, since the vertical position is fixed by the reduction process which specifies that the reduced real component is unity a t low frequencies. The cross represents the origin of a dimenThe Journal of Physical Chemistry
sionless theoretical plot, and from its position the molecular weight can be calculated. Effect o j Molecular Weight Distribution. The components of the reduced complex viscosity are plotted in E'ig. 6 for Sample 19F in Aroclor 1248 a t 2% concentration. For this moderately low molecular weight, the hydrodynamic interaction can be taken as dominant to avoid extreme complications, and the results are compared with the Zimm theory for a sample homogeneous with respect to molecular weight, represented by the dashed lines matched to the experimental points a t high frequencies. The values of V'R deviate only to a moderate degree at intermediate frequencies, where a more gradual curvatuve is produced by the molecular weight distribution; those for 7 " ~deviate somewhat more, showing a lower maximum and an extension to lower frequencies as compared with the monodisperse behavior. The data for agree very well with the frequency dependence evaluated for a most probable distribution of molecular weights on the basis of the Zimm theory modified to account for coil expansion in a good solvent, as recently calculated by P e t i c ~ l a s ' ~ ; his calculated curve is shown as a solid line in Fig. 6. Several additional solutions of the monodisperse polymers were studied, including Samples 5-102 and S-111 at 4% concentration in A-1248, and 5-108 a t 2% in A-1254.20 These data are not shown graphically, but (19) W. L. Peticolas, private communication
1979
DYNAMIC r\/IECHANICdL PROPERTIES O F DILCTEPOLYSTYRERTE SOLUTIOXS
0
0
-0.5
-0.5
$e
4x
-1.0
Y o
0
2%
eon
.a
!h
cr,
3% .E
-0.:
I
0
m
-0
-0.5
-a
-1.c
cn 2
-1.c -1.5
0
- 0.: I i I I I-
8
*-
7
-5
-6
1
-4
Jog ~ ( ~ - v , y . J / c RJ 7
Figure 4. Logarithmic plots of 7 ' and ~ q ' ' ~for Sample MIIP-1 in dioctyl phthalate at the &temperature of 12.0" a t two concentrations as indicated. Theoretical curves are drawn with B = 0 ; values of h are, in order: m, 2.
they were fitted in each case to theoretical curves with selected values of the parameters h and E, which are given together with other derived quantities in Table 11. The earlier data2 on a sample with molecular weight 267,000 have also been re-evaluated in this manner and are included in Table 11.
Discussion Determination of Parameters h and
The theory of TschoegP extends the Zimm theory to take into account both varying hydrodynamic interaction and coil expansion in good s'olvents, and it predicts the frequency dependence of G'R,' G"R, V I R , and 7 ' ' ~for cases intermediate between Rouse-like and Zimm-like behavior, in terms of the quantities e and h. The parameter E is a measure of the expansion of the polymer coil in good solvents, as first proposed by PeterlinZ1and used in the theory of Ptitsyn and E i z n e P ; the mean-square distance between any two points on. the chain separated by n bonds is proportional to n1+ e.
1.c
- 1.:
I
t
-a
-7
o
I
-6
I
-5
I
-4
109c+-v,?,)/cRT Figure 5 . Logarithmic plots of ?'E and 7 ' ' ~for Sample S-1163 in A-1248 a t 25.0" a t three concentrations as indicated. Theoretical curves are drawn with s = 0,100; values of h are, in order: 63, 8, 2.
instead of to n as it would be in a &solvent. The limiting values of E are 0 and 1/3. Values for the different molecular weights in Aroclor 1248 were calculated from the intrinsic viscosity data of Table I as described in an earlier paper.5 Following Stockmayer and Fixman,2a [~]/fVI1" was plotted against and the intercept a (20) We are much indebted to Dr. Robert S. Moore for these measurements. (21) A . Peterlin, J . Chem. Phys., 2 3 , 2464 (1955). (22) 0. B. Ptitsyn and Yu. E. Eimer, Z h . Fiz. Khim., 32, 2464 (1958).
(23) W. H. Stockmayer and M. Fixman, J . Polymer Sci., C1, 137 (1963).
Volume 6 8 , Number 7
J u l y , 196.4
1980
J. E. FREDERICK, S.W. TSCHOEGL, AND J. D. FERRY
I
I
I
I
I
0 3%
I 0-
h
0 -0.5 n
F -0m -1.C I
-8
0,lI
f
/
-7
I
I
-6
-5
P
-4
'09 4'-"1J/CRT Figure 6 . Logarithmic plots of 7 ' and ~ V ' / R for Sample 19F in A-1248, concentration 27& reduced to 25". Pip down, 6.5"; left, 15.0"; up, 25.0". Solid curve drawn for theory of Peticolas with theoretical origin a t cross. Dashed curves are Zimm theory for sample homogeneous with respect to molecular weight ( E = 0, h = m), matched at high frequencies.
viscoelastic properties is governed by h alone. Equally good fits to the data were obtained with identical or slightly higher values of h. However, the molecular weights derived from fitting the data (see below) with 6 = 0 were somewhat more discrepant from the correct values than those derived with the finite values of E in Table 11, so the use of E appears to be a significant improvement in the theory. Dependence of h on Molecular Weight and Concentration. It is evident from Fig. 1, 4, and 5 that the frequency dependence shifts with increasing molecular weight and/or concentration from Zimm-like toward Rouse-like behavior, corresponding to decreasing values of h. The latter are portrayed in Fig. 7 for all the solutions of monodisperse polymers in Aroclor 1248. There is an essentially monotonic change in h with molecular weight and concentration. In a 6-solvent, on the other hand, h would be independent of molecular weight and decrease with increasing concentration only. =
0.065
0.090 0.105
0.131 0.147
and slope b of the best straight line mere determined. Then for each sample E was obtained from the equation618 E
=
+
(M"z~bja)/[(3M"'b/a 0.497) 2.308(3N "'b/a
+ + 0.497)1'3]
0.
0
8
0
0
0
(1)
These values are listed in Table 11. Because the theoretical frequency dependence curves do not depend critically on the value of E , compromjse values were used for economy in computing as specified in the footnotes. Also, it was assumed that E would be the same in Aroclors 1232 and 1254 even though the solvent power is somewhat different. The parameter h is a measure of the strength of the hydrodynamic interaction, and varies between 0 (corresponding to Rouse-like behavior) and m (corresponding to Zimm-like behavior if E = 0). The magnitude of h was determined for each solution by first calculating theoretical curves for T'R and 7 " ~(in a few cases G'R and G"R) with several h values and the appropriate value of E . and then selecting the pair of curves which most closely matched the experimental data. The results are also listed in Table 11. The theory is quite successful in describing the frequency dependence in all the solutions, as evidenced by the good fit of the theoretical curves in Fig. 1-5. The shape of the frequency dependence is actually primarily determined by h. To gage the role of e in the theory, the calculations and matches mere all repeated for-€ = 0, corresponding to the earlier theory of Tschoeg17 in which the frequency dependence of the T h e Joiirnal of Physical Chemistry
100-
Do
8
0
I
A-1248 10
3 x lo5
lo6
M (loqarithmic) Figure 7 . Map of values of h as a function of concentration and molecular weight plotted logarithmically, for polystyrenes in Aroclor 1248.
While the experimental results are very successfully described by different values of h in this manner, it remains in a sense an adjustable parameter, since there is no clear theoretical reason why it should change with concentration and molecular weight in the manner observed. The theory applies to infinite dilution only and predicts no concentration effects; while from current treatments of intrinsic viscosity24 it would be expected that h should be for all except very low molecular weights. (24) 11. Kurata and W. H. Stockmayer, Adrian. Polymer Sci., 3 , 196 (1963).
DYNAMIC RIECHANICAL
PROPERTIES O F
DILUTEPOLYSTYRESE
Actually, it appears from Table I1 and Fig. 7 that h may approach m corresponding to Zimm-like behavior for all molecular weights even in good solvents if the concentration is sufficiently low. Thm, all deviations from the Zimm theory, except for the relatively minor effects of E which can clause a slight shift in the direction of Rouse,6 may reflect a concentration effect. This effect appears even in a &solvent if the concentration is sufficiently high; it appears at lower concentrations in good solvents because the coils pervade larger volumes, and for the same reason it alppears a t a lower Concentration the higher the molecular weight as portrayed in Fig. 7. If this interpretation held quantitatively, h would be expected to be approximately a unique function of e [ ? ] ,taking [ y ] as a measure of the coil volume, following the corresponding-state treatments of Simha.26126 This is not quite the case; at eqnal values of c [ v ] , a solution with higher rnolecular weight still has a lower value of h, as can be concluded by examination of Tables I and 11. Nevertheless, we are inclined to believe that the effects of finite concentration are the key to the deviations from the Zimm theory, and that these must be elucidated before the effects of molecular weight and solvent power on the parameter h can be understood. Calculalion of ,from Viscoelastic illeasurements. The abscissa position of the theoretical cross of origin5 in a matched logarithmic plot of 7 ’ and ~ 7 ” ~(Fig. 4 and 5 ) is log L%~V, where = z T k / T 1 , T k being the kth relaxation time specified by the theorg; S is a function of h and E , and values of it are included in Table 11. I n this way, M is readily calculated. Similarly, when a logarithmic plot of G’ and G” (Fig. 1-3) is matched to theory, the ordinate position2 of the theoretical cross of origin is log cRT,’iV!. The logarithms of molecular weights obtained thus from Fig. 1-5 and similar graphs for the other solutions are given in the table as log ,$Ive. As in previous studies,*J the molecular weights from the viscoelastic measurements are all a little too high; log AT!ve/Lll is generally of the order of 0.2. The data suggest that for very low molecular weights and low concentrations the discrepancy should disappear. Otherwise, however, the variations reveal no clear dependence on concentration or molecular weight. I n the &solvent, the deviation is surprisingly great. The interpretation of these effects must await further work. The case of the sample with molecular weight distribution will be discussed below. Terminal Relaxation Times. The abscissa position of the cross of origin on a match of G’ and G” is -log 71. From the data of a match of V’R and v “ ~ ,7 1 is obtained as (7 - vlvs)Mve/cRTS.Values of log r1 for all
1981
SOLUTIOrVS
the solutions are also given in Table 11. At high dilution, r1 in different solvents would be expected to be proportional to v s when compared at the same molecular weight and concentration. This expectation is fairly well confirmed when S-108 a t 2% is compared in A-1248 and A-1254, and when MDP-1 at 2% is compared in A-1232 and A-1248. For the fern other comparisons available, the agreement is less satisfactory. It may be noted that rl is proportional to (7 - u 1 ~ J M / c(except for the small changes in S associated with different values of h and E ) ; and 7 - vlvs is essentially vspvs, where vsp is the specific viscosity. The dependence of r l / v s on solvent, concentration, and molecular weight is therefore equivalent to the dependence of specific viscosity on these variables a t moderate concentrations, which has been treated extensively by Simha and coll a b o r a t o r ~ ~ 4and - ~ ~is beyond the scope of this work. Effect of Molecular Weight Distribution. The theory of P e t i c o l a ~ ’extending ~ the Zimm theory to a most probable distribution of molecular weights assumes that h = ~0 and takes into account the different coil expansions of the different molecular weight species by using the Flory formulation of the expansion factor a as a function of M rather than the e of Ptitsyn and Eizner. It is assumed that the relaxation times of the ith species are proportional to illls/2a13 as consistent with the original Zimm theory. The necessary parameters were chosen to correspond to the specific case of polystyrene 19F in Aroclor 1248, from the intrinsic viscosities of Samples S-102 and S-108 in Table I. The theory provides T’R as a function of a dimensionless reduced frequency w r 0 , where ro is defined as (7 - s 8 ) ~ , / 0 . 5 8 6 . C R T ( C Y , ~ X ,and ” ~ )x, ~ . ,= M , / B n . The factor 0.586 is S/XI’, where 11’ ( = 4.04) is a coefficient arising in the Zimm theory. For polystyrene 19F, ja,3x1’’2)w= 2.9. The component 7 ‘ ‘ ~was not evaluated since it involves a much more difficult calculatioii. The agreement in shape of the calculated curve with the experimental data is gratifying. The match determines ro, and hence in turn W,. This value (Table 11) is too high by a somewhat larger factor than usual; log = 0.57. The source of this discrepancy is not understood at present. It is evident that data for a sample with molecular weight distribution could be matched approximately to a theoretical curve for homogeneous mclecular weight, shown by the dashed curve in Fig. 6. The question then arises as to what kind of molecular weight average is obtained when the molecular weight is calculated in (25) R. Simha and J. L. Zakin, J . Colloid Sci., 17, 270 (1962). (26) L. Utracki and R. Simha, J . Polymer Sci., A I , 1089 (1963). (27) L. Utracki and R. Simha, J . Phys. Chem., 6 7 , 1052 (1963).
Volume 68, Ahmher 7
J u l y , 1964
1982
the usual manner. The type of average actually depends on whether the match is made at high or low frequencies. Averages for several different cases have been evaluated and will be published elsewhere. For the Zimm theory in a &solvent ( h = a ,E = 0), a lowfrequency match from the lower limbs of G' and G" or the left limb of q"R gives the average MzMw~Wn2,' where MS/t is the number-average of M"/" (equal to $?n(fW'~z)xv), For a most probable distribution, this is 3.38$?,. When the match is made a t high frequencies, however, from the right limb of q " ~or the upper limb of G' together with G", the average obtained is ( M 5 / 2 ) z / $ ? n 2 J which for a most probable distribution is 1.78&fn. Thus, a small residual heterogeneity in the samples which have been assumed to be monodisperse could make the apparent molecular weights calculated from viscoelastic measurements come out too large, but the distribution is probably sufficiently sharp in our samples to make this effect negligible. Evidently the observed discrepancies must be attributed to some other source. Relation to Measurements at High Frequencies. Dynamic viscoelastic measurements have been recently made by Lambz8 on similar polystyrenes with sharp molecular weight distribution in solvents such as toluene whose viscosities are of the order of lop3 smaller than those of our hroclors. Since his frequency
(m~)~,
The Journal of 1' h y ~ i c a lClicnaistry
J. E. FREDERICK, K. W. TSCHOEGL, A K D J. D. FERRY
range is about lo3 greater than ours (lo3to 106 c.P.s.), his measurements also encompass the region where oq 1 and reveal dispersion of G' and G" - u 1 q 9 very similar to that observed in Fig. 1-3. I n particular, they also show a shift from Zimm-like toward Rouselike behavior with increasing molecular weight, though complete conformity with the Rouse theory is not attained. One possible difference between the two types of measurement is that effects of 'intramolecular chain stiff ness should be more prominent in Lamb's highfrequency, low-viscosity studies than in our low-frequency, high-viscosity work, where the resistance to chain motions is stroiigly dominated by the high viscous resistance of the environment. A more detailed.comparison of the respective results remains to be made.
Acknowledgments. This work was supported in part by the Office of .Kava1 Research under Contract Konr 1202(19), by the U. S. Public Health Service under Grant GM-10135-02, by the Kational Science Foundation, and by the Research Committee of the Graduate School of the University of Wisconsin from funds supplied by the Wisconsin Alumni Research Foundation. We are indebted to Mrs. Roger Binkley for help with some of the experiments. (27) J. Lamb, presented at the Fourth lnternational Congress of Rheology, Providence, R.I., August 30, 1963.
