2524
Macromolecules 1986,19, 2524-2532
assumes that the tube-forming 1-chains undergo diffusion and relax only by reptation. Thus the theory predicts an MWl3dependence of J z , b B ? / z , b ~(eq lb"'), in agreement with our present results. The theory also predicts an Mw1-2 dependence of the diffusion coefficient D G 1 of monodisperse entangled polymers (the 1-chains), again in agreement with reported experimental results.l1J2 However, their theory also predicts an MWl3 dependence of the viscosity qlpl instead of the well-known Mw3.5 dependence for the monodisperse systems (Figure 4). The question is why the reptation picture accounts the Mwl dependence of J2,bBq2,bB and DG1 satisfactorily but not that of the linear viscoelastic quantities of the 1-chains such as 711,m.13 One possible (but yet uncertain) interpretation is as follows. In the framework of the generalized tube model, the reptation is only one possible mechanism accounting for the polymer diffusion and relaxation, and some other mechanisms such as tube leakage must be operating simultaneously in an entangled system, in which the actual relaxation and diffusion processes take place as a result of competition of all such possible mechanisms. To explain the above-mentioned difference in the Mwldependence of qlm, DG1,and J z , ~ B in?the /~ framework , ~ B of the generalized tube theory, we may assume that the mechanical properties are strongly affected, but diffusion is not much affected, by coupling of reptation and other mechanisms. The 2chain in the blend relaxes when 1-2 entanglement becomes ineffective as the result of diffusion but not of stress relaxation of the 1-chain. In other words, the surrounding 1-chains confine the 2-chain even when the 1-chains have mechanically relaxed. This probably leads to the relation J2,bS%,bB OC M w I 3 8s long as DG1 OC Mw1-2. Another possibility for answering this question is to give up the reptation picture completely and employ some other picture to explain the Mwldependence of ql,m, D G ~and , J2,bBV2,bB.l5 Such attempts have been made, but as far as we know, they have not yet been successful. Even in such a model, the J2,bB?)2,bB should probably be explained by a mechanism such as directly related to DG1but not nec-
essarily to vl,m. Further theoretical study of this problem is necessary.
Acknowledgment. We acknowledge with thanks the financial support of the Ministry of Education, Science, and Culture (Mombusho), Japan, under Grants 60470107 and 60303018. We also thank Dr. Mitsutoshi Fukuda, Toyo Soda Mfg. Co. Ltd., for supplying us with most of the monodisperse PS samples used in this study. Registry No. PS, 9003-53-6; DBP, 84-74-2.
References and Notes (1) Watanabe, H.; Kotaka, T. Macromolecules 1984, 17, 2316. ( 2 ) Watanabe, H.; Sakamoto, T.; Kotaka, T. Macromolecules 1985,18, 1008. (3) Watanabe, H.; Sakamoto, T.; Kotaka, T. Macromolecules 1985, 18, 1436. (4) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (5) Riddick, J. A.; Bunger, W. B. In Techniques of Chemistry, 3rd ed.; Weissberger, A., Ed.; Wiley: New York, 1970; Vol. 11. (6) Rudd, J. F. In Polymer Handbook, 2nd ed.; Brandrup, J., Immergut, E. H., Eds.; Wiley: New York, 1975; Chapter V. (7) Daoud, M.; de Gennes, P.-G. J. Polym. Sci., Polym. Phys. Ed. 1979, 17, 1971. (8) . . Doi. M.: Edwards. S. F. J. Chem. SOC..Faraday Trans. 2 1978. 74,1789, 1802, 1818. (9) Klein, J. Macromolecules 1978, 11, 852. (10) Graessley, W. W. Adu. Polym. Sci,' 1982, 47. (11) Green, P. F.; Mills, P. J.; Palmstrrzm, C. J.; Mayer, J. W.; Kramer, E. J. Phys. Reu. Lett. 1984, 53, 2145. (12) Tirrell, M. Rubber Chem. Technol. 1984, 57, 523. (13) Einaga and Fujita14,15collected the currently available DG data on monodisperse polymers reported by many workers and pointed out that the relation DG a M,-2 is not well established yet. According to their interpretation, the M , dependence of D G becomes stronger with increasing M,, and finally at suffiwhich is conciently large M,, it might be given by Mw-2.5, sistent with the viscosity data. However, in the M,, (>>M*) range examined in this study, the D G 1 data seem to be better represented by M,1-2, while ql,m is proportional to M,?6. Hence, an inconsistency between DG1 and ql,m exists, at least in the M,, range examined. (14) Einaga, Y.; Fujita, H. Nihon Reoroji Gakkaishi 1984,12, 136. (15) Fujita, H.; Einaga, Y. Nihon Reoroji Gakkaishi 1984, 22, 142, 147; Polyn. J. 1985, 17, 1131, 1189.
Rheological Properties of Randomly Branched Polystyrenes with Different Molecular Weights between Branch Points Toshiro Masuda,* Yasuhiko Ohta, and Shigeharu Onogi Department of Polymer Chemistry, Kyoto University, Kyoto 606, Japan. Received March 18, 1986 ABSTRACT Randomly branched polystyrenes were prepared by copolymerization of styrene and divinylbenzene. Fractionated samples were characterized in terms of total molecular weight, molecular weight distribution, and average molecular weight between branch points (Mb). Frequency dependences of the storage shear modulus G'and loss modulus G"for 50 w t % solutions of two homologous series of randomly branched polystyrenes were measured. Each homologous series is built upon Mb = 4.55 X lo4 or 1.74 X lo5. Data for the series M b = 9.28 X lo4 in our previous paper are included here for discussion. Zero-shear viscosity qo and steady-state compliance J," for the 50 wt % solutions of these series were compared with those for linear and star-shaped polystyrenes. qo of randomly branched samples fit on the line for linear polymers a t M , < 2Mb but not a t M , > 2Mb. The molecular weight dependence of T~ for randomly branched polymers is much lower than that of linear ones. With decreasing Mb, qo of randomly branched samples a t the same molecular weight decreases. Data on randomly branched polymers having M,/Mb of about 3.5 fall well on the line for 3-4-armed-star polystyrene solutions. The steady-state compliance J," is proportional to M , a t M , < Mb, is independent of M , in the range M b < M , < 2Mb, is again proportional to M , in the range 2Mb < M , < (5-6)Mb, and increases rapidly with molecular weight a t M , > (5-6)Mb. Moreover, the effect of Mb on the relaxation spectra for randomly branched samples with different M b are compared.
Introduction The effect of branching on viscoelastic properties is one of the most important problems to be solved in polymer 0024-9297/86/2219-2524$01.50/0
industry as well as in polymer science. Recently, preparation of model branched polymers with a narrow distribution of molecular weights has become possible, and 0 1986 American Chemical Society
Rheological Properties of Branched Polystyrenes 2525
Macromolecules, Vol. 19, No. 10, 1986 Table I Copolymerization Conditions of Randomly Branched SamD1es
molar ratio polymn series styrene DVBa AIBNb time, h 1 0.49 1.5 RB46 142 1 0.98 4 RB93 277 RB174 211 1 0.50 9.5 a
Divinylbenzene.
conver- temp, sion "C 0.30 75 0.29 65 0.13 50
cup'-Azobis(isobutyronitri1e).
comb-shaped, star-shaped, and H-shaped polymers have been prepared. Many of the rheological properties of these branched polymers have been performed, and it has been found that there are some rather striking differences from linear polymer, especially a t high concentrations where entanglement interactions are important. As is well-known, there are other types of branched polymers besides comb-, star-, and H-shaped polymers; randomly branched polymers occur in most commercial polymerizations, and there has been a considerable amount of worksm motivated by strong industrial interests. Only a few on rheological properties, however, employed randomly branched polymers that are well characterized with respect to number and length of the branches. In this study, viscoelastic properties have been measured for concentrated solutions of a series of narrow fractions of randomly branched polystyrenes prepared by copolymerization of styrene and divinylbenzene (DVB). Experimental results on the rheological properties for two series of randomly branched polystyrenes, joined by another series reported in a previous paper,23are compared with linear and star-shaped counterparts.
Experimental Section Materials. Two homologous series of randomly branched polystyrenes, the RB46 series and the RB174 series, were prepared by copolymerization of styrene monomers with a small amount of DVB, using a,a'-azobis(isobutyronitri1e) (AIBN) as initiator.
Table I1 and Steady-State Compliance (J,")of Randomly Branched Polystyrenes J," X lo6, M, X [a]! dL/g Mb X M,/Mb qo X lo4, P cm2/dyn 790 0.381 24.6 237 121f 649 0.345 16.7 116 35.5' 385 0.311 10.7 66 18.6 189 0.266 45.5 5.36 17.1 7.4 11.0 4.42 131 0.245 3.67 74.2 0.207 2.20 5.6 3.10 64.0 0.191 1.54 4.13 3.31
Molecular Characteristics, Zero-Shear Viscosity sample RB464 RB465 RB466 RB467 RB468 RB469 RB4610
M,
X
1120 762 489 244 167 100 70.2
RB934 RB935 RB936 RB937 RB938 RB939 RB9310 RB1744 RB1745 RB1746 RB1747 RB1748 RB1749 RB17410
MJls)
X
758 166 71.0 495 307 204 124 76.9 61.0 37.0
566 436 287 191 162 122
74.5
410 286 155 125 75.4
The polymerization conditions and conversion are given in Table I, together with those of the RB93 series, which was used in the previous paperz3and whose rheological results are also discussed in this study. The fractionation of whole polymer thus obtained was repeated 3-4 times in a benzene-methanol system, and seven fractions having the desired molecular weights were selected as the samples. Weight- and number-average molecular weights, intrinsic viscosities [ q ] in cyclohexane a t 34.5 "C, the average molecular weights between branch points Mb, and M w / M bfor samples employed in this study are listed in Table 11. This table also contains data of the RB93 seriesa for the reader's convenience. The sample codes (RB4-RB10) in the previous papera were renamed RB93X ( X = 4-10) here. M , of all samples was obtained by the osmotic pressure technique using a high-speed membrane osmometer (Mecrolab Model 502), and M , was determined from a universal calibration curve of the gel permeation chromatogram obtained in tetrahydrofuran a t 45 "C on a Waters Model 200 chromatograph. M,(ls) was also measured by a low-angle light scattering photometer (Chromatix KMX-6) in toluene a t room temperature. Judging from the tetrafunctionality of DVB and M w / M bvalues of the styrene-DVB copolymers in Table 11, we expect samples with M,/Mb < 2 (RB4610, RB937-RB9310, and RB1746RB17410) to be linear polymers and those with M , / M b > 7 (RB464-RB467 and RB934) to be randomly branched ones. Samples having intermediate M,/Mb values can be imagined t o be star-shaped ( M w / M b= 2-5) and H-shaped ( M w / M b= 5-7). This subject will be discussed later in terms of rheological properties of the samples. The dynamic viscoelasticity of 50 wt % solutions of all samples was measured. Kaneclor 500 (KC5; partially chlorinated diphenyl) was used as the solvent. The required amount of polymer and solvent (KC5) was completely dissolved in dichloromethane, and then the dichloromethane was completely removed by evaporation from the mixture in a vacuum oven. Measurements. The rheological behavior of the samples was investigated by use of a concentric cylinder-type rheometer (CCR) and an R-18 Weissenberg rheogoniometer (WR). All WR testing was performed with'a plate-and-cone combination having a 2.5-cm diameter and a 4' cone angle. Oscillatory measurements for CCR and WR testing were confined to the angular frequency (w) range 2.3 X lo-' t o 3.0 s-l and 1.2 X lo-* to 6.0 s-l, respectively. The measuring temperature ranged from 40 to 100 "C. The viscoelastic functions, storage shear modulus G'and loss modulus G", were superposed into respective master curves a t 50 "C by use of the
(qo),
330 231 136 85.8 53.3 42.0 22.6
0.335 0.290 0.258 0.210 0.185 0.156 0.115
416 327 216 192 138 95.0 66.8
0.465 0.427 0.373 0.321 0.300 0.260 0.209
92.8
174
5.33 3.30 2.19 1.33
76 34.0 18.7 7.3 6.6 3.12 1.35
20 7.1 5.9 5.5 3.4 2.55 2.30
3.25 2.51 1.65 1.10
463 388 141 41.9 34.4 18.2 4.82
22.8 17.1 10.0
10.9 9.8 4.9 3.40
Weight-average molecular weight obtained from the GPC calibration curve (see text). Weight-average molecular weight determined by the light scattering method. Number-average molecular weight determined by the osmotic pressure method. Measured in cyclohexane at 34.5 "C. eAverage molecular weight between branch points. (Estimated by the extrapolation of log J' vs. log w plots in Figure 9.
2526 Masuda et al.
Macromolecules, Vol. 19, No. 10, 1986 ?-
I
I
PS OL
0
0
e
LB
e LS
6-
e LM o RB
1 5-r P
-
\
4-
E l u t i o n Vol.,ml
Figure 1. Gel permeation chromatograms for copolymers of styrene and divinylbenzene at various polymerization times (solid lines) and for typical fractions (broken lines). time-temperature superposition principle. The shift factor aT obtained can be expressed by the following WLF-type equation? log U T = -9.3(T - Tr)/[88.6 + (T- T,)];T,= 50 "C (1) There was no difference between uT for branched polymers and linear ones.
Results and Discussion Copolymerization. Figure 1 shows gel permeation chromatograms (GPC)at various polymerization times for copolymers of styrene and DVB under the conditions in Table I (solid lines). Although linear and branched polymers of the same molecular weight should give different elution volumes because of their different chain dimensions, the change in the molecular weight distribution with polymerization time can be qualitatively seen from this figure. The arrows in Figure 1 indicate the elution volumes corresponding to the average molecular weights between branch points, Mb,estimated from the kinetics of copolymerization, as discussed in the next section. For the RB174 series, the GPC curve at shorter time (6 h) shows a single peak, where the elution volume corresponds to the molecular weight of a linear polymer of 2.8 X lo5. This value of Mb agrees well with that estimated from the kinetics of copolymerization (arrow on 6-h curve). At longer times (8 and 9.5 h), however, the peak disappears with increasing polymerization time and the distribution curve begins to show a broad plateau, shifting to the higher molecular weight side. For the RB46 series, on the other hand, two peaks appear on the curve with increasing polymerization time. The peak a t lower molecular weight (higher elution volume) remains at constant molecular weight. Molecular weight at the peak of the 1-h curve is lower than Mb estimated from the kinetics of copolymerization (arrow for the RB46 series at 1 h). This difference may imply that the branched polymers already exist a t 1 h. Gel Permeation Chromatography. The whole polymers prepared under the polymerization conditions (see Table I) were fractionated by the precipitation method described before to obtain seven fractions for each series (Table 11). The GPC curves for typical fractions on arbitrary scales of intensity are also shown by broken lines in Figure 1. In Figure 2, the relation between log [TIM, and elution volume for linear and branched polystyrenes is shown, where [ T ] in cyclohexane at 34.5 "C was used. Open circles in this figure indicate linear polymers, left-black circles, mixtures of three- and four-branched star polymers,12
-3
75
100 125 Elution Vol ,ml
0
Figure 2. 'Relation between [q]MWand elution volume for linear and branched polystyrenes. L, LS, LM, and RB indicate data for linear, six-branched, multibranched, and randomly branched polymers, respectively. LB indicates results for star polymers with average number of branches of near 3.5. top-black circles, six-branched polymers," bottom-black circles, multibranched polymers18having more than seven branches, and right-black circles, randomly branched polymers.23As seen from this figure, [TIM, for branched polymers fit well on the solid line for linear polymers, regardless of the type, number, and length of branches. Similar results were reported by According to these experimental results and to theoretical conside r a t i o n ~ , the ~ * determination of M , from the GPC calibration curve and intrinsic viscosity [ T ] would be valid. In fact, M , values determined from the GPC calibration curve agree well with M,(ls) values determined by light scattering, shown in the third column of Table 11. Molecular Weight between Branch Points. In the copolymerization of styrene and DVB, the conversion p can be written as
P = (4+ n2)/(N1 + N2) = n1/N,
(2)
where n and N denote the number of monomers converted into polymer and in initial mixtures, respectively. Subscripts 1and 2 are used for styrene and DVB, respectively. On the other hand, the density of branch points in the polymer chain, Pb is given as35 Pb = v/nl = -C[1
+ (1/p) In (1 - p)]
c = (h2'/71h2)(~2/~1)
(3) (4)
where u is the total number of branch points in the system, k12 is the reactivity constant of the first DVB bond, k12/ is that of the second, and y1 is the monomer reactivity ratio. When we assume k12/ = k12 and y1 = 0.57, which is the average value36 of the three isomers of DVB, and substitute the values of N 1 / N 2given in Table I into eq 3 and 4, we find Pb = 0.00229 for the RB46 series, Pb = 0.00112 for the RB93 series, and Pb = 0.000597 for the RB174 series. Then the branching parameter and the molecular weight between branch points Mb,defined by the equation = 1/Mb = Pb/hfO
(5)
can be calculated as X = 2.20 X and hfb = 4.55 X lo4 for the RB46 series, X = 1.078 X 10" and it& = 9.28 X lo4 for the RB93 series, and X = 5.75 X lo4 and h f b = 1.74
Rheological Properties of Branched Polystyrenes 2527
Macromolecules, Vol. 19, No. IO, 1986
P S-Cyclohexane
~
I
01-
1
2’
-5
-1
-3
-1
-1 logwaT .sec-l
0
1
2
1
Figure 5. Master curves of G’ for 50 wt % solutions of the RB174 series in KC5. The reference temperature is 50 O C .
I
I
I
I
0.81
-
05
t
03
1 OL
06
08
10
9,
Figure 4. g? = ( [ ? ] b / [ ~ ] # / ~ plotted againstg, = ((s$)b/(so2)1)”2 for randomly branched polystyrenes. The symbols are the same as those in Figure 3. 3’
the RB174 series, where Mo is the molecular weight of the monomers. In spite of the assumption of equal reactivity for the two double bonds in DVB (kli= k I 2 ) the , M b values are consistent with the GPC curves of the whole polymer (Figure l ) , the intrinsic viscosity-molecular weight relations (Figure 3) and the zero-shear viscosity-molecular weight relations (Figure 12). Thus, the number code of the RB46 series, the RB93 series, and the RB174 series denotes polymers having M b of 4.55 X lo4, 9.28 X lo4, and 1.74 X lo5, respectively. Intrinsic Viscosity and Chain Dimensions. In Figure 3 the intrinsic viscosity [a] is logarithmically plotted against M,, where triangles indicate linear polymers, filled circles, the RB174 series, squares, the RB93 series,23and unfilled circles, the RB46 series. As seen from this figure, [?II for linear polystyrenes is proportional to Mw0.5to give the following equation:23 X lo5 for
= 7.1 x 1 0 - 4 ~ ~ 0 . 5
(6)
[qlb values for the RB series fit well on a straight line for linear polymers at low molecular weights and then deviate from the line to be proportional to Mw0.243.25 at high molecular weights, suggesting that RB samples having low M , are linear and high molecular weight polymers are randomly b r a n ~ h e d . ~It~should - ~ ~ be pointed out here that the molecular weight a t which [ a J b for the RB series deviates from the linear line agrees well with the Mbevaluated from the kinetic theory of copolymerization. Figure 4 shows the relation between two branching paSO^)^)'/^, for rameters, gg= ( [ a h , / [ a I l Y 3 and g, = ((S:)b/ randomly branched polystyrenes, where ( s o 2 ) is the mean-square radius of gyration. In this figure, filled circles, squares, and unfilled circles represent the RB174 series, the RB93 series, and the RB46 series, respectively. The values of g, were calculated by using the value of k f b es-
-5
-1
-$
-1
-1
1
1
0
1
2
log wa+cc-’
Figure 6. Master curves of G”for 50 wt % solutions of the RB174 series in KC5. The reference temperature is 50 “C. timated from the kinetics of copolymerization and the Zimm-Stockmayer equation40 for randomly branched polymers: gS2= [(l+ AM/6)lI2
+ (4/3~)AMl-~/~
(7)
The solid line in this figure has a slope of 0.42, to give gg a g,0.42.It is interesting that this relation agrees well with gv a gSo4,which is expected from eq 7 (at high M) and the experimental result [ a ] b a. iW2.A similar result was reported by Kurata and co-workers for randomly branched polystyrene^.^' The above relation, g, = g?2.4,is similar to that for star-shaped polystyrenes with fewer than six branches.21 Frequency Dependence of Viscoelastic Functions. In Figures 5 and 6, master curves of G’ and G ” for the RB174 series are shown, respectively. The master curves of G’ and G ” for RB1744 are considerably different from those for the others and have a break in the rubbery-toflow transition region. A similar tendency can be seen in results reported for four- and six-branched star polystyrenesghaving high molecular weights and for H-shaped polystyrenes.22 The master curves of G’ and G ” for RB1745-RB17410 show a typical viscoelastic behavior of highly entangled systems of linear polymers, because the Mw/Mevalues of these samples are from 3 for RB17410 to 16 for RB1745, where Me, the average molecular weight between entanglements, is 28 000, estimated from ( M e ) b d k = 18 100 for bulk p o l y ~ t y r e n eand ~ ~ ,the ~ ~concentration c = 0.641 g/cm3 for 50 wt % solution in KC5, using the relation Me = (Me)bulk/C. The molecular chain structure of RB1745 having M W / M b = 2.51 (see Table 11) would be
2528 Masuda et al.
Macromolecules, Vol. 19, No. 10, 1986 I
I
,
-9
0
2
1
log waT.sec-l
Figure 7. Master curves of G ’for 50 wt % solutions of the RB46 series in KC5. The reference temperature is 50 OC.
I
-6
I
-5
I -4
1
-3
I
-2
1
-1
I
0
1
1
’
-6
I
1
I
I
-5
-4
I
1
1
-3 -2 I ogw9,sec-1
I
-1
0
1-7
Figure 9. Master curves of the storage compliance J’for 50 wt % solutions of PS712 (linear polymer), RB1744 (branched polymer), and RB464 (branched polymer).
I
2
log ma., , sec-1
Figure 8. Master curves of G ” for 50 wt % solutions of the RB46 series in KC5. The reference temperature is 50 “C.
just on the transitional stage from linear to star shape. The star-like character of the sample, however, cannot be distinguished by the G ’and G“ curves. Master curves of G’ and G ” for the RB46 series are shown in Figures 7 and 8, respectively. Samples included in this series, except for RB4610, are branched polymers, as indicated in Table I1 and Figure 3. The terminal zone of these master curves shifts to the lower frequency side with increasing molecular weight, but all curves, including that for RB464 with the highest molecular weight (1.12 X lo6),show no clear rubbery plateau. The rubbery zone for RB464-RB466, whose M,/Mb = 10-25, can approximately be represented by straight lines with a slope of about 1/2(G’ 0: c J / ~A ) .similar result was reported by Roovers and Graessle9 for comb-shaped polystyrenes with 30 branches. This behavior is much different from that of the RB174 series, which demonstrates a clear plateau as seen in Figure 5. This remarkable difference is caused by the short distance between branch points with small Mb (Mb/Me = 1.63). The presence of many branch points prevents4 the backbone chain from relaxing by either r e p t a t i ~ nor~path ~ fluctuations4 and instead the backbone relaxes by a constraint release mechanism, giving rise to a Rouse-like beh a ~ i o in r ~the ~ rubbery zone. For low molecular weight polymers (RB467-RB4610), the rubbery-to-flow relaxation in G ‘and G“ occurs more sharply in comparison with that for materials having M , higher than 10Mb. Taking a look carefully a t the curve for samples of high molecular weight, for example, RB464, we realize that the G’curve shows a shoulder in the longer time (lower fre) . quency) region than the Rouse-like region (G’ a w ~ / ~This
means that there is a new relaxation mechanism a t long times besides the relaxation of backbone chains due to the constraint release mechanism, as will be shown in the relaxation spectrum. In Figure 9, master curves of the storage compliance J’ for 50 wt % solutions of linear and randomly branched polystyrenes are shown. J’ was calculated from G ’and G ” using the relation42
J ’ = G‘/(G‘2
+ G”2)
(8)
As seen from this figure, the master curve of J’ for linear polymerB PS712 (M, = 20.5 X lo4,M,/M, = 1.13) shows a two-step change, giving the steady-state compliance Jeo and the entanglement compliance JeNo,respectively. J’ for randomly branched polystyrene RB1744, having M , lower than &&,, shows a behavior similar to that for PS712, and JeNoof this sample is almost the same as that of PS712. However, the master curve for RB464 ( M , N 25Mb) does not show any clear plateau a t intermediate frequency. Relaxation Spectrum. Relaxation spectra of the RB174 series determined from the master curve of G ” using Tschoegl’s equationG are shown in Figure 10. These spectra agree well with those determined from G’. As is evident from Figure 10, spectra of the RB174 series except for RB1744 show no peak but rather a plateau in the rubbery region and extend gradually to a very long time with increasing molecular weight. The spectrum for RB1744 (highest molecular weight and M,/Mb = 3.25), on the other hand, shows a shoulder at log T = 3.5 and is lower than that of RB1745 with lower molecular weight in the
Rheological Properties of Branched Polystyrenes 2529
Macromolecules, Vol. 19, No. 10, 1986 I
I
log
I
1
I
I
. scc
Figure 11. Relaxation spectra for 50 wt % solutions of the RB46 series in KC5 at 50 “C.
rubbery-to-flow transition region, which appears a t the almost same relaxation time for RB1745 and RB1744. Although the molecular weight of RB1744 is about 1.3 times higher than that of RB1745, the longest relaxation time for RB1744 increases only a little, compared with that for RB1745. Graessley and Rooversgfound that two sets of relaxation times appear for four- and six-branched star polystyrenes with long branches and that G’and G”curves for star polymers fall off less rapidly in the mbbery-to-flow transition region, compared with that for linear polymers. On the other hand, we found18that relaxation spectra for star polymers having constant molecular weight but different number of branches show a broad peak a t the relaxation time where the linear polymer of 2 X (molecular weight of branches) has a peak, indicating that the span molecular weight determines the characteristic time for rubbery-to-flow relaxation. These two experimental facts imply that the structure of the RB174 series changes from linear-liketo star-like with increasing molecular weight and that RB1744 is a star-like polymer. Relaxation spectra for the RB46 series are shown in Figure 11. The shape of the relaxation spectra for three samples with high molecular weights, RB466-RB464, is clearly distinguishable from the others. The low molecular weight samples, RB4610-RB467, which are considered to be linear (RB4610), star-shaped (RB469 and RB468), and H-shaped (RB467) polymer from the average number of branches per molecule (Mw/Mb)in Table 11, seem to have a single relaxation mechanism. The average molecular weight between branch points for these samples (Mb = 4.55 X lo4)and the value of M b / M e (=1.63 by Me = 2.8 X lo4 for 50 w t % solution) are not enough to exhibit a long-time relaxation mechanism as found in high molecular weight four- and six-arm star polymersgand H-shaped polymers.22 H ( T )for each of the three samples RB466RB464, which are randomly branched polystyrenes (Mw/Mb= 10-25; see Table 11),decreases gradually with increasing relaxation time (7) and clearly demonstrates an extra set of relaxation times in the long-time region. The intensity of H ( T )decreases and the average relaxation time ( T at the shoulder or break) increases with increasing degree of branching (Mw/Mi,). The slope of H ( T )for these samples is approximately -1/2, i.e., H ( T )a: T - ~ / * ,in the region log T = 2-3, suggesting a “Rouse-like”mechanism is taking part in the relaxation. Graessle? proposed a constraint release concept for linear polymer systems and star and linear polymer mixtures. Using the random flip model presented by Orwoll and Stockmayer,46he obtained a “Rouse-type” distribution of relaxation times for the constraint release contribution to
the long-time relaxation behavior. The situation is not exactly the same as but looks similar to the case of the randomly branched polymers, in which entanglement coupling takes place mainly on the free chain and little on the backbone chain (not linear in this case). The relaxation of stress (conformational orientation) on the treelike backbone chain will be preceded by relaxation of free chains outside the molecule and constraint release, giving rise to Rouse-like b e h a ~ i o r . ~ According to our rheological s t u d i e ~ l ~on! ~multi~ branched star polystyrenes having 15-40 branches, the relaxation spectrum of such a polymer has a shoulder or break on the long-time side in the same manner as that for higher molecular weight samples of the RB46 series (Figure 13 in ref 18). It was concluded47that the multibranched polymer has a core (the less flexible part a t the center of the molecule) owing to the increased segment density near its center. It was also suggested4’ that the shoulder of the relaxation spectra of multibranched polymers is caused by a translational movement of each molecule back to the equilibrium position. On the basis of reptation ideas, it is expected that many branching points on the backbone chain (it is indeed treelike) of highly branched samples such as RB466RB464 (Mw/Mb= 10-25) would be very effective a t suppressing rept?tion, even more so than for multibranched star- and comb-shaped polymers. In other words, the relaxation of the backbone seems to be impossible by a reptation mechanism. Therefore, the above-mentioned shoulder appearing a t the long-time end of the relaxation spectra of RB466, RB465, and RB464 may be considered to be due to a movement of each highly branched macromolecule as a whole. This long-time relaxation due to a translational motion of the whole molecule (particle-like behavior) was indeed observed for highly branched comb-shaped polystyrene melts by Roovers and Graessley (Figure 5 in ref 4), but was not mentioned by the authors. The long-time relaxation that appears in well-dispersed filled polymers, for example, rubber-modified polymeric materials,48 is also from the same origin. To summarize, our speculation on relaxation processes for randomly branched polymers with high Mw/Mbis as follows: (1) The free chain outside the molecule relaxes first. (2) The relaxation of the backbone chain near the free chain takes place by a release of the constraints due to the free chains. (3) The whole molecule undergoes translational movement back to the equilibrium position. The relaxation times for processes 1-3 correspond to (1) log T = 1-2, (2) log T = 2-3, where H ( T ) a: T - ~ / ~and , (3) the bump in the longest time region (Figure 11). Combshaped polymers with 30 branches4 show processes 1 and 2, plus partly process 3, star-shaped p o l y m e r ~ lwith ~ ? ~10 ~ or more branches show processes 1 and 3, and stars with 6 or fewer branches and H-shaped polymers show process 1. So-called viscosity enhancement can be considered as a retarded process of (1). Rheological Behavior in t h e Terminal Zone. The rheological behavior of polymer liquids in the terminal zone can be characterized by zero-shear viscosity qo and steady-state compliance J,”.These parameters are defined as42 qo = lim [ G ” / w ] (9) L-0
J,” = lim J’= lim [ G ’ / ( W ~ ~ ) ~ ] (10) w-0
w-0
The numerical values of qo and J,” at 50 “C obtained in this work are listed in Table I1 for 50 wt % solutions of randomly branched polystyrenes in KC5. The J,” values
2530 Masuda et al.
I
Macromolecules, Vol. 19, No. 10, 1986
PS-KCS
I
5
51
6 log M w
Figure 12. Molecular weight dependence of zero-shear viscosity, qo, for 50 wt % solutions of the linear polymers (triangles),the RB46 series (unfilled circles),the RB93 series (squares),and the RB174 series (filled circles) in KC5 at 50 "C. The broken line indicates data for 50 wt % solutions of star polymers with an average number of branches of near 3.5. for the RB93 series in this table were reevaluated from J' master curves made by removing the data obtainedB above 100 "C, because it was found the material *formsgels at high temperatures. The effect of the gels is not serious in qo evaluation. In Figure 12, qo is logarithmically plotted against the weight-average molecular weight for randomly branched and linear polystyrenes. Triangles indicate linear polym e r ~filled , ~ ~circles, the RB174 series, squares, the RB93 series,23and unfilled circles, the RB46 series. For linear polymers (PS series) the molecular weight dependence of qo can be represented by a straight line having a slope of 3.5; qo is proportional to Mw3.5.qo curves for the RB series having molecular weights lower than 2Mb coincide with that for the PS series, indicating that polymers of the RB series having low molecular weights are linear polymers. This result is consistent with all results obtained from GPC measurements (Figure l),the [q]-M, relation (Figure 3), and the kinetics of copolymerization discussed before. At higher molecular weights, qo of randomly branched polymers is clearly lower than that of linear ones of the same molecular weight. The slope of qo curves for branched polymers is also lower than that for linear ones, though the former have a tendency to increase as the molecular weight increases. Similar results have been reported for randomly branched poly(viny1 acetate)49and randomly branched poly(dimethylsi1oxanes) Similar behavior has also been observed for star-shapedls and comb-shaped polymers having branches of constant 1ength.l This smaller dependence of qo on molecular weight for randomly branched polymers is in marked contrast with results obtained for star- and comb-shaped polymers having a constant number of branches per m o l e ~ u l e . ' ~ ~ * ~ - ~ J ~ In Figure 12, log qo of 50 w t % solutions of star-shaped polystyrenes having an average number of branches of about 3.5 is also shown (broken line).15 qo points for randomly branched polymers having M, = 3.5Mb (marked with pips) fit well on this viscosity curve for star polymers, suggesting that these randomly branched polymers are star-like polymers. On the basis of Bueche's idea,50the zero-shear viscosity of monodisperse branched polymers is given by qb
=
Vl&'s2)'
= Kk?s2M)"
(11)
where a = 1 for gS2M< M,, a = 3.5 for gS2M> M,, K is
I
I
5
6
I
l o g $Mlw
Figure 13. Zero-shear viscosity qo logarithmically plotted against the corrected molecular weight g:Mw, where:g was estimated from eq 7. The symbols are the same as those in Figure 12. 11,
I
I
1
PS-KCS
10~1%24b
Figure 14. Zero-shear viscosity q logarithmicall plotted against the corrected molecular weight gj.4Mw,where gJ4 was estimated from eq 7 . The symbols are the same as those in Figure 12. a constant, M is the molecular weight, and M, is the critical molecular weight for linear polymer. Subscipts b and 1 denote branched and linear polymers with the same molecular weight, respectively. Therfore, qo of branched polymers should be plotted against gS2M, instead of M, in order to agree with that of linear polymers. As seen from Figure 13, in which log qo is plotted against log g,2MW,plots of qo for branched polymers seem to fit the curve for linear polymers (solid line and triangles), where polymers of the RB series with lower molecular weight are assumed to be linear polymers. It was reported that plots of log qo against log g,2.4M, for star polymers fit better to the plot for linear polymers :, do.8,9*21*47 The plots than plots of log qo against log gM of qo against g,2.4Mwfor randomly branched polymers, however, deviate a little from the line for the linear polymers, as can be seen from Figure 14, where gs2.4was estimated from eq 7. As mentioned before, the relation g, = g,O can be applied to randomly branched p o l y ~ t y r e n e s As . ~ ~long ~ ~ as ~ this relation is valid, qo for linear and branched polymers must be expressed in the following general form: 70 = f k s 2 M w )= F([V"'~M,''~)
(12)
When gs2,4is better than gS2as a correction factor for Mw,
Macromolecules, Vol. 19, No. 10, 1986
Rheological Properties of Branched Polystyrenes 2531
log M,
log [l15'3M4'6, (dllg)"' Fi re 15. Zero-shear viscosity qo logarithmically plotted against [q]eMw1/6. The symbols are the same as those in Figure 12. The intrinsic viscosity was measured in cyclohexane at 34.5 O C .
Figure 17. Molecular weight dependence of steady-state compliance Jeofor 50 wt % solutions of randomly branched and linear polystyrenes at 50 "C. The symbols are the same as those in Figure 12. The straight lines indicated by Y and X are drawn ' solutions of three- and four-armed star polystyrenes. for 50 wt %
comb-shaped polymers: and Mb,/M, = 2-3 for H-shaped polymers;22here Mbr is branch molecular weight and MEE 50%,50'C 10 is the average end-bend molecular weight of comb-shaped mole~ules.~ As seen from Figures 15 and 16, all samples examined in this study show no viscosity enhancement on the basis of the qO-[t]correlation. The experimental fact for the RB46 series that the viscosity can be reduced onto the to-Mw line for linear polystyrenes indicates the samples are not highly entangled; Mb/M, is only 1.63 for the RB46 series. The branches are short enough to be relatively unentangled themselves and the backbones are also weakly entangled. Relaxation processes 1-3 in this system seem not to play any role in viscosity enhancement. If samples RB1745 and RB1746 are assumed to be three-armed stars, they have Mb/M, = 6.2 and should logll12,(dllg)2 begin to show viscosity enhancement according to the F re 16. Zero-shear viscosity qo logarithmically plotted against criterion mentioned above. This is indeed seen in Figures [v] . The symbolsare the same as those in Figure 12. The intrinsic 13 and 14, but we choose to believe the results shown in viscosity was measured in cyclohexane at 34.5 O C . Figures 15 and 16. The reason these samples do not show any enhancement in viscosity is still not clear. I t might as in the case of star polymers, t oshould be expressed be attributed to an imperfect star due to the polymerizaas899J6functions of g,2.4M, and [tI2 tion method. In Figure 17 the molecular weight dependences of the (13) t o = f(g,2.4Mw)= F ( [ t l 2 ) steady-state compliance J," for randomly branched and using the definition of 8.13 = [ t ] b / [ t ] l . Figures 15 and 16 linear polystyrenes are shown. J," was evaluated as a show plots of eq 12 and 13 for branched polymers. The limiting value of the storage compliance J' at low fresolid lines in these f i i e s indicate data for linear polymers, quencies. As is evident from this figures, J," of linear and polymers of the RB series with lower molecular polymers (triangles) having M , > 2 X lo6 is independent weights are assumed to be linear polymers, as discussed of the molecular weight. Low molecular weight polymers before. As seen from these figures, the correlation by the having M,/Mb < 2 (RB4610, RB937-RB9310, and use of [ t ] s / 3 M w 1and / 6 [7712, particularly the former, gives RB1747-RB17410; see Table 11)are considered to be linear an excellent agreement between toof branched and linear and the J," values fall on the line for linear polymers. J," polymers. I t seems very difficult at the present stage, to of these polymers increases with increasing molecular decide hastily which factor is better for randomly branched weight, but it is about 2 times higher than that expected polymers. I t should be noted here that a big advantage for the 50 wt % solutions of monodisperse polystyrenes. of the plots shown in Figures 15 and 16 is that they use This is due to a somewhat broad distribution of molecular only experimentally determined variables ( M , and [SI), weight for these polymers. A t M,/Mb ranging from 1.10 instead of g,2 from a theoretical equation (eq 7) for tetto 1.65, J," of the RB174 series is independent of molecular rafunctional random branches. This gives rise to the reweight and the value is almost equal to that of linear markable difference in scattering of the data in Figures polymers (A). The molecular weight dependence of J," for the three 13 and 14 and those in Figures 15 and 16; especially it homologous series of randomly branched polystyrene would be significant for samples having small values of systems shown in Figure 17 is very interesting when we Mw/Mb. We decided to use the results obtained in Figures 15 and 16 for discussions of viscosity correlation and ensee it in terms of M,/Mb and Mb shown in Table 11. For hancement for this reason. the RB46 (unfilled circles) and RB93 (squares) series, the constant J," region is observed for M,/Mb = 1.5-2.2. In A number of works on well-characterized branched the range M,/Mb = 2.2-6 for the RB46 series and 2.2-4 polymers show viscosity enhancement occurs when Mb/ Me = 3-5 for star-shaped p ~ l y m e r s , ' MEE/Me ~ ~ ~ ~ ~zz ~4 Jfor ~ ~ ~ ~for the RB93 series, J," again becomes proportional to M,.
P
2532 Masuda et al.
Macromolecules, Vol. 19, No. 10, 1986
low,
Figure 18. Relation between odeo and molecular weight (M,) for 50 w t % solutions of linear and randomly branched polystyrenes at 50 "C. The symbols are the same as those in Figure 12.
The molecular weight dependence of J," for randomly branched polystyrenes in this regime is similar to that for star-shaped polymers with number of branches below six,911J6 as in the case of va. J," of polymers in the RB46 series having M, higher than about 6Mb (and probably the RB93 series in the range M , > 5Mb) increases abruptly, departing from a straight line, as shown by unfilled circles. This departure from the line may be attributed to a relaxation mechanism peculiar to randomly branched polymers having many branch points, such as RB464. The data for branched samples of the RB46 and RB93 series seem to be expressed by a single curve in the figure. As for the RB174 series, it is very difficult to conclude that the uppermost two black circles for RB1744 and RB1745 fall on the universal line, because the samples are lightly branched as M,+./Mb = 3.25 and 2.51, respectively. The characteristic relaxation time T~ of a polymer liquid is defined here by 170Jeo,and the molecular weight dependence of voJeo for 50 wt % solutions of linear and randomly branched polymers is shown in Figure 18. The symbols are the same as those in Figure 12. The molecular weight dependence of 170Jeofor linear and branched polymers with M , lower than 2Mb can be represented by a straight line with a slope of 3.5. On the other hand, the relaxation time for randomly branched polymers with M , ranging from 2Mb to (5-6)Mb is proportional to Mw2,and that for randomly branched polymer with higher molecular weights can be represented by a straight line having the same slope as that for linear polymers; ?deo is proportional to M,3.5. However, 7oJe0for branched polymers is about one decade lower than that for linear polymers at the same molecular weight. The transitions from the behavior of linear polymer to that of randomly branched polymer are also observed a t the high molecular weight end of the RB174 series and a t the low molecular weight end of the RB46 series. Registry No. (Styrene).(DVB) (copolymer), 9003-70-7.
References and Notes (1) Fujimoto, T.; Narukawa, H.; Nagasawa, M. Macromolecules 1970, 3, 57.
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