J. Chern. Soc., Furaduy Trans. I, 1989, 85(5), 1065-1074 Dielectric Relaxation in Concentrated Solutions of cis-Pol yiso prene Part 1 .-Effect of Entanglement on the Normal-mode Process Keiichiro Adachi," Y asuo Imanishi and Tadao Kotaka Department of Macromolecular Science, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan Dielectric measurements were carried out on concentrated toluene solutions of narrow molecular-weight distribution cis-polyisoprenes (cis-PI) with molecular weight of 1.6 x lo3 to 5 x 10'. Two loss maxima were observed in the temperature range 200-400 K. The low-frequency process was assigned to the normal mode process due to the fluctuation of the end-to-end distance of the cis-PI molecules; and the high-frequency process to the segmental mode process related to the glass transition of the polymer.The molecular- weight dependence of the relaxation time T, for the normal mode process at a fixed concentration was found to be similar to the viscoelastic relaxations : below the characteristic molecular weight M(,, 5, cc M2 but in the range above M,, t, cc M43. The M , varied in proportion to the inverse of concentration. The high-frequency segmental mode process exhibited the relaxation time t, almost independent of the molecular weight. The time-temperature superposition principle was applied to construct master curves of the dielectric loss factor E" over a wide frequency range. The E" curves broadened with increasing molecular weight and concentration in the high frequency side of the curves.However, the half width of the loss curves was almost independent of the molecular weight and concentration. Each monomeric unit of cis-polyisoprene (cis-PI) has two different types of components of the dipole moment: one is the parallel component aligned parallel to the chain and the other perpendicular to the chain contour.'*2 The former causes the dielectric normal mode process due to the fluctuation of the end-to-end vector, and the latter the segmental mode process known as the primary or cc relaxation process near the glass transition temperature. We have already reported the dielectric relaxation behaviour of cis-PI in dilute and semi-dilute ~ o l u t i o n s ~ - ~ as well as in bulk.'V2 For bulk cis-PI, we found that the relaxation time z, for the dielectric normal mode process shows the molecular weight M dependence similar to the viscoelastic longest relaxation time.'T2 In the range of A4 lower than the characteristic molecular weight M,, z, was proportional to M2 in accordance with the Rouse theory.6 In the range above M,, however, zn was proportional to kP7 due to entanglement of the molecules.This behaviour is similar to the 3.4 power law'.8 for the viscoelastic longest relaxation time and was explained semi-quantitatively in terms of the tube theory proposed by de Gennes' and Doi and Edwards." In dilute solutions, z, was proportional to M'.' and M'.74 in a theta solvent, dioxane, and a good solvent, benzene, re~pectively.~ This behaviour was essentially in agreement with the Zimm theory. l 1 On the other hand, the relaxation time z, for the segmental mode process is almost independent of M .We pointed out that for cis-PI with a certain M the ratio ZJZ, is independent of temperature T, although z, and z, themselves depend strongly on T. This behaviour suggests that the local segmental motions relate intimately with the 10651066 Entanglement in cis- Polyisoprene Solutions friction coefficient [ for the large scale motions. However, as far as we know, no experimental work on the relationship between t, and t, have been reported. As is well known, [ and the molecular weight Me ( I I MJ2) between entanglements are the key parameters to describe the dynamic properties of polymers. They depend strongly on the content of a diluent. To clarify these problems, we studied in detail the dielectric normal mode and the segmental mode processes in concentrated toluene solutions of cis-PI.The results are reported in this article and the succeeding two papers.12 This first paper is concerned with the effect of diluent on the normal mode process of cis-PI. The first purpose of this study is to clarify the M dependence of z, in concentrated toluene solutions of cis-PI by comparing the results with the data for the bulk cis-PI reported previously. The second purpose is to clarify the concentration C and M dependences of the distribution of relaxation times. In our previous study on bulk cis-PI,2 we observed that the width of the E” curves increased with M. We attempted to explain this behaviour based on the tube model.’O On the other hand, the width of the E” curves of cis-PI in dilute solutions was almost independent of M.3 In these experiments, we used cis-PI with the polydispersity factor, M,/M,, of 1.2 to 1.4.Thus, the broadening of the E” curves may be attributed in part to the distribution of molecular weight (MWD). In this study, we examined the effect of MWD on the width of the E” curves, using narrower distribution samples. Theory The theories of the dielectric normal mode process were described by Zimm,” Stockmayer and Baur,13* l4 and by o~rselves.~ Stockmayer and his coworkers carried out pioneering work on poly(propy1ene oxide) and called this process the low frequency process of type A polymer^.^^"^ Here, we briefly describe the basic equations of the process which we called the normal mode process.When a polymer has a dipole moment p proportional to the end-to-end vector r, the contribution of the parallel dipoles to the complex dielectric constant E*(w) at angular frequency o is given by4 where Cis the concentration in wt/vol., N A is the Avogadro constant, E , is the unrelaxed dielectric constant, p is the dipole moment per unit contour length (p = p-) and ( r 2 ) is the mean-square end-to-end distance. The relaxation strength, A E ~ [ = ~ ( 0 ) - E,], for the normal mode process is given by16 (2) Ae,/C = 4xN,p2 (r2)F/(3k, T M ) where ~(0) is the E’ value at zero frequency and F is the ratio of internal to external electric field. F i s close to unity.16 Both the Rouse theory for a non-entangled polymer and the tube theory for an entangled system predict an autocorrelation function # = (r(0) - r ( t ) ) / ( r 2 ) of the same form : # = C ( 1 / p 2 ) exp ( -p21/tl) p = 1 , 3 , 5 , .. . (3) where p is the number of the normal mode and t,, the relaxation time for the first mode. Since the first mode has the dominant contribution, the relaxation time z, for the normal mode process is approximately described by 7,. The difference between the theories is only in the M dependence in the expression of z,. The longest relaxation time 2, of the free-draining model proposed by Rouse is written as‘K. Adachi, Y. Imanishi and T. Kotaka 1067 where [ is the monomeric friction coefficient, and x, the number of the repeat unit. Since in the unperturbed state ( r 2 ) is proportional to the molecular weight M , zR is proportional to W even in bulk and in concentrated solutions as long as the polymer chains are not entangling.It is noted that zR for the dielectric normal mode is twice the viscoelastic longest relaxation time. Zimm’l described z, for the non-draining model and predicted that z, cc W 5 . However, we will not use the Zimm theory since the hydrodynamic interactions are screened in condensed systems. According to the tube model, z1 is equal to the relaxation time z, for the tube disengagement process :’ where L is the contour length of the tube, and D, the diffusion coefficient of the chain along the tube. According to the Doi-Edwards theory,1° eqn ( 5 ) is rewritten as z, = L2/D ( 5 ) where Me is the molecular weight between entanglements. Experimental Samples of cis-PI were prepared by anionic polymerization with sec-butyllithium and n-butyllithium as the initiator, and characterized by a gel permeation chromatography (GPC) as described previously.’+ The weight average molecular weight M , was determined with a low-angle light scattering monitor installed in GPC.The number- average molecular weight M , was determined for some low molecular weight samples by freezing point depression with benzene and cyclohexane as the solvent. The polydispersity index, M,/M,,, was determined from GPC chromatograms. The characteristics of the samples are listed in table 1 where the samples already reported are also listed.2 The number of the sample code indicates the weight average molecular weight in units of kg mol-’.The solvent toluene was dried with calcium hydride and then distilled by vacuum distillation. Sample solutions were prepared and stored under dry argon atmosphere to prevent contamination by moisture. Measurements of the dielectric constant E’ and loss factor E” were made in the frequency range from 20 Hz to 100 kHz with transformer bridges (General Radio 1615- A and Showa Denki). A capacitance cell was the same type as described previously.233 Results and Discussion Frequency Dependence of the Complex Dielectric Constants The representative frequency-dependence curves of E” are shown in fig. 1 and 2 for 50 O/O solutions of PI-05 and PI-53, respectively. In fig. 1 and 2, two loss maxima are seen: for example see the E” curve at I89 K in fig. 1. Comparing the loss curves for PI-53 and PI- 05 at 179 K, we find that the loss maximum frequency for the high-frequency process is almost independent of M,.On the other hand, the loss curves at 244 2 K indicate that the low-frequency process depends quite strongly on M,. Thus, the high- and low- frequency processes were assigned to the segmental mode process and the normal mode process, respectively. Separation of the two loss maxima for the 52% solution of PI-53 is more than 5 decades. Since the available frequency range was limited to only 4 decades, we could not observe the whole E’ and E” curves including both relaxation regions by one measurement at a fixed temperature. To avoid this difficulty, we constructed master curves of E” assuming the time-temperature superposition principle. We reported for bulk cis-P12 that the relaxation time z, for the normal mode process and that z, for the segmental mode process shift by the same amount parallel with temperature.In concentrated1068 code Entanglement in cis-Polyisoprene Solutions Table 1. Characteristics of cis-polyisoprene PI-02 PI-03 PI-05 PI-14 PI-32 PI-53 PI- I74 PI-09 PI-1 I PI- 16 PI-20 PI-24 PI-42 PI-59 PI-74 PI- 102 PI- 128 PI- 160 MW - - - 13.5 31.6 52.9 8.6 11.0 16.2 20.4 23.5 41.9 58.9 74.0 174 102 128 164 M , Mw/M, initiator" 1.1 1 1.09 1.07 1.08 1.05 1.08 1.13 1.29 1.17 1.23 1.20 1.20 1.34 1.33 1.20 1.18 1.37 1.17 S S S S S S S n n n n n n n n n n n " s: sec-butyllithium, n: n-butyllithium. l # ~ ~ l ~ ~ ~ l l l I l l 1 , n I 4r 179K 189 K log Cf/Hz) Fig. 1. Frequency, f, dependence of the dielectric loss factor E" at various temperatures indicated in the figure for 53 wt % of PI-05 in toluene.solutions, the same behaviour may be expected. This is supported by the Arrhenius plot for 7" and 7, reported in part 2 of this series of studies.12 We took 273 K as the reference temperature. The E" curve at a temperature T was multiplied by 273/ Tand then shifted in the horizontal direction to get best superposition. When the temperatures for the two loss curves were separated widely, vertical shift was also needed t o get best superposition. This is probably due to the change in density,K. Adachi, Y . Imanishi and T. Kotaka 1069 log CflHZ) Fig. 2. Frequency dependence of the dielectric loss factor E" for 52 wt YO of PI-53 in toluene.L 6 A 2 2 4 6 a 0 v l " ~ ' ' ' ' ~ ' ' ' ~ ' ' ' ~ log ( f l W Fig. 3. (a) Master curve for the 53 YO solution of PI-05 in toluene and (h) that for bulk PI-05, both at 273 K . Thus, the log E" us. log f plot was shifted both in the horizontal and vertical directions. Fig. 3 shows two examples of the master curves thus chnstructed for PI-05. We see that superposition is quite good in the region where the normal mode process was observed, while in the region of the segmental mode process, some scattering of data points is seen. Smoothed curves were drawn as shown in fig. 3 for the master curves. Fig. 4 and 5 show examples of the smoothed master curves all reduced to 273 K for solutions1070 Entanglement in cis- Polyisoprene Solutions log CflW Fig. 4.Smoothed master curve of the loss factor E" for solutions of PI-32 in toluene at 273 K. Dashed curve corresponds to eqn (9). 0, 41.4; 0, 52.2; A, 61.9; ., 85.5; 0, 100 wt YO. log Cf/Hz) Fig, 5. Smoothed master curve of the loss factor E" for toluene solutions of &PI with various molecular weights at 273 K. 0, PI-05, 52.2%; 0, PI-14, 49.2%; A, PI-32, 52.2%; 0 , PI-53, 52.3%. of PI-32 from the bulk to 41.4% concentration and for approximately 50 YO solutions of cis-PI with varying M,. Molecular Weight Dependence of the Relaxation Time The relaxation times z, and z, at 273 K were determined from the loss maximum frequency f, by z = l/(2;nfm). Fig. 6 shows the double logarithmic plots of z, and z, against M,. The data for the bulk samples reported previously are also plotted.2, l7 The slope of the double logarithmic plot of z, vs.M , changes at the characteristic molecular weight Me. On the other hand, z, is almost independent of M,. A slight M dependence of z, seen for the bulk samples with M , less than 5000 is ascribed to the increase in the free volume content as is generally known for oligomers.K . Adachi, Y. Irnanishi and T. Kotaka 1071 -2- -4- n \ W M - s -6- r -A- --A-- 50% -30% . - 10 Fig. 6. Weight average molecular weight M , dependence of the relaxation time z for the normal mode process (open symbols) and those for the segmental mode process (filled symbols) for the solutions and bulk. Dashed curve corresponds to the theoretical z given by eqn (6) and (7), and dot-dash curve to the one given by eqn (4), (6) and (8).According to the Rouse‘ and Zimmll theories, t, is expected to be proportional to ikP and W 5 , respectively. Although the experimental error is relatively high, the slope of the plot in the range of M, < M , is determined to be 2.0f0.2. Thus, the hydrodynamic effect does not contribute in the concentrated solutions with C > 0.2. In the range of M , > M,, the slopes of the tn vs. M , plots for 50, 30 and 20% solutions are 4.3 0.2. Generally, the viscoelastic relaxation time z, in the entangled state is proportional to the 3.5 f 0.2 power of Mw.73 * The present values of the power in concentrated solutions are significantly higher than the well- known empirical value of 3.5. Doi18 explained the deviation of the observed exponent of 3.5 from the theoretical value of 3.0 predicted by the tube theory considering the fluctuation of the contour length of the primitive chain confined in a tube.The theory predicts (7) where t, is given by eqn (6). Assuming Me = 5000 and using the observed zn at M , = 5000 for t,(M,), we calculated the theoretical t as given by the dashed line in fig. 6. It t = [ 1 - 1 .47(Me/M)fl2 t,1072 Entanglement in cis-Polyisoprene Solutions I ‘ I 1 1 1 I I I I I- -I 0 015 I log (C/g ~ r n - ~ ) Fig. 7. Concentration C (in g ~ r n - ~ ) dependence of the characteristic molecular weight M,. is seen that agreement is fairly good. Since M, increases with decreasing C, the data covered in the present study are limited in the range close to M,. Therefore, the high value of the exponent in the M dependence of z may be partly attributed to the fluctuation of the contour length.As the origin of the discrepancy between the theoretical and observed powers, we should consider the so-called tube renewal effect due to disintegration of tube-like constraint with time arising from the motions of the chains surrounding the test chain. Kleinlg proposed the relaxation time z for the tube disengagement process by taking the tube renewal effect into account. He assumed that the tube itself moves similarly to the Rouse chain and that an entangled chain relaxes through the two mechanisms, i.e. reptation and tube renewal : where 7, and z, are given by eqn (4) and (6), respectively. This equation is plotted by a dash-dot line in fig. 6. As seen in the figure, eqn (8) predicts a power close to 3.0 in the range M > M, and is not satisfactory to explain the present results.Concentration Dependence of the Characteristic Molecular Weight Although the experimental uncertainty is relatively large, we determined the M,, for the solutions at various C and plotted them in .Fig. 7. It is seen that M, is nearly proportional to the inverse of C . This behaviour is the same as the M , for viscoelastic relaxation in concentrated polymer solutions.* Distribution of Relaxation Time for the Normal-mode Process In fig. 3-5, we see that E” curves of the segmental and normal modes overlap partly. In order to see the concentration and molecular weight dependence of the shape of the E”K. Adachi, Y. Imanishi and T. Kotaka Table 2.The parameters of Negami-Havriliak equation for the segmental mode process and the parameter K for the normal mode process conc. code (wt %) a @ ~ , / l O - ~ s 102A& K PI-32 41.4 0.32 0.25 2.1 4.95 -0.40 PI-32 52.2 0.32 0.20 3.4 6.36 -0.37 PI-32 61.9 0.25 0.24 2.8 7.35 -0.34 PI-32 85.5 0.22 0.33 19 7.52 -0.32 PI-32 100 0.18 0.38 93 7.68 -0.31 - - - - -0.31 - - - - -0.41 PI-53 52.3 PI-14 49.2 I, I I . 1 I I I I . I . 1 ) . I - 2 0 2 4 1% (flfm) \ \ I I I . I , 1 , I \ , I -2 0 2 4 1073 Fig. 8. Double logarithmic plot of the normalized loss curve: (a) 50 YO solutions of cis-PI in toluene with various molecular weights and (b) solutions of PI-32 with various concentrations. The arrow indicates the half width. The dashed line indicates the results of eqn (3).curves for the normal mode process, we have to subtract the contribution of the segmental mode process from the observed E" curves. We estimated the contribution of the segmental mode process using the Hav- E*(w) - E , = A&/[ 1 - (iws,)'-']B (9) riliak-Negami equation :*O where 7, is the nominal relaxation time, and a and p are the constants. The dashed line in fig. 4 corresponds to a = 0.32, p = 0.20, to = 3.4 x lo-', and A& = 0.065. The same parameters were used for the 50% solutions shown in fig. 5. For the solutions of PI-32, we used the parameters listed in table 2. The E" curves thus estimated for the normal mode process are compared in fig. 8 where E" and frequencyfare normalized by the maximum value 8: and the loss maximum frequency f,, respectively. We recognize that the loss curve broadens with increasing M and C.Obviously, this may be attributed to the effect of entanglement. However, the broadening occurred only in the high frequency side of the loss peak. In the low frequency side, we see that the observed 8' agrees rather well with the theoretical curve calculated by eqn (3). This result does not agree with the theoretical prediction by either the bead-spring1074 En tanglement in cis- Polvisoprene Solutions model or the tube model. Both theories predict the same distribution of relaxation times given by eqn (3) and hence the same shape of the E” curve as given by the dashed line in fig. 8. In our previous studies,21 we discussed the distribution of the relaxation times for the normal mode process in bulk samples2 and dilute solutions3 of cis-PI, using the half width of the E” curves and the Davidson-Cole parameter.22 It was demonstrated that the MWD affects strongly the half width.In the present study, we used cis-PI samples with M,,/M,, less than 1.08 to discuss the shape of the E” curves. As is seen in fig. 8, the half width indicated by arrows depends little on M and C . Thus the half-width is an insensitive measure of the distribution of the relaxation times. On the other hand, the slope K of the double logarithmic plot of E” vs.fin the high frequency side is considered to be a better parameter than the half width to describe the broadening of the E” curve. The values of K for the present samples are listed in table 2. We see that the absolute value of K increases with decreasing M and C.It is noted that both the Rouse model for nonentangled systems and the tube model for entangled systems predict the slope of -0.50. At present there exists no molecular theory to explain the relaxation spectrum for the normal mode process. Summary 1. The relaxation time z,, for the normal mode process in concentrated solutions of cis-polyisoprene is proportional to Ww in the range below the characteristic molecular weight M , and to Mk3 above M,. 2. The characteristic molecular weight M , is inversely proportional to the concentration. 3. The distribution of the relaxation times for the normal mode process increases with increasing molecular weight and concentration. This work was supported in part by the Grant-in-Aid (6055062) for Scientific Research by the Ministry of Education, Science and Culture.Support from the Institute of Polymer Research, Osaka University is also gratefully acknowledged. References 1 K. Adachi and T. Kotaka, Macromolecules, 1984, 17, 120. 2 K. Adachi and T. Kotaka, Macromolecules, 1985, 18, 466. 3 K. Adachi and T. Kotaka, Macromolecules, 1987, 20, 2018. 4 K. Adachi and T. Kotaka, Macromolecules, 1988, 21, 157. 5 K. Adachi, H. Okazaki and T. Kotaka, Macromolecules, 1985, 18, 1687. 6 P. E. Rouse, J . Chem. Phys., 1953, 21, 1272. 7 J. D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York, 3rd edn, 1980), chap. 10, p. 224. 8 W. W. Graessley, Adv. Polym. Sci., 1982, 47, 67. 9 P. G . de Gennes, J . 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