Dielectric Relaxation of Monodisperse Linear Polyisoprene

Article pubs.acs.org/Macromolecules

Dielectric Relaxation of Monodisperse Linear Polyisoprene: Contribution of Constraint Release Yumi Matsumiya,† Kazuki Kumazawa,† Masahiro Nagao,† Osamu Urakawa,‡ and Hiroshi Watanabe*,† †

Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan Department of Macromolecular Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

‡

S Supporting Information *

ABSTRACT: cis-Polyisoprene (PI) has the type A dipole parallel along the chain backbone so that the end-to-end fluctuation of PI chains results in slow dielectric relaxation. Comparison of dielectric and viscoelastic data of PI has revealed several interesting features related to the entanglement dynamics, for example, success and failure of the full dynamic tube dilation (DTD) picture for monodisperse linear and star PI, respectively [see a review: Watanabe, H. Polym. J. 2009, 41, 929, for example]. For monodisperse linear PI, recent modeling [Glomann et al. Macromolecules 2011, 44, 7430] and single-chain slip-link simulation [Pilyugina et al. Macromolecules 2012, 45, 5728] suggest that the constraint release (CR) mechanism has negligible influence on the dielectric relaxation time τε in the entangled regime, which appears to disagree with the previous data. Thus, we revisited the classical problem: CR contribution to the dielectric relaxation of PI. Specifically, we made dielectric and viscoelastic measurements for PI/PI blends in a wide range of the molecular weights of long and short components, M2 = 1.1M and M1 = 21K−179K, and with a small volume fraction of the short component, υ1 = 0.1 and/or 0.2, to examine the CR contribution in the experimentally clearest way. It turned out that τε of the short component was longer in the blends than in respective monodisperse bulk even for M1 = 179K. Furthermore, the viscoelastic and dielectric data of the short components (M1 ≤ 43K) in the blend exhibited identical mode distribution and relaxation time, which confirmed that the CR mechanism was fully suppressed for these components in the blends. These results demonstrate that the CR mechanism does contribute/accelerate the dielectric relaxation in monodisperse bulk PI systems even in the highly entangled regime (M1/Me = 36 for M1 = 179K). This CR-induced acceleration was found to be consistent with the empirical equations for the terminal relaxation time and CR time of monodisperse PI available in the literature, as noted from a simple DTD analysis of the terminal relaxation process (reptation along a partially dilated tube that wriggles in a fully dilated tube).

1. INTRODUCTION In concentrated systems of long flexible polymer chains, deeply overlapping/penetrating chains mutually constrain their largescale (global) motion because they are uncrossable with each other. This entanglement effect on the chain motion has been an important subject in polymer physics.1−5 In the tube model formulated on the basis of a dynamic mean-field concept, the topological constraint for a focused chain due to surrounding chains is represented as a tube-like constraint for the chain motion.2−4 The motion of the chain along the tube is combined with the motion of the tube (the multichain effect incorporated in the mean-field picture) to analyze the global chain dynamics and corresponding relaxation of various physical properties. This approach of the tube model has been widely utilized in rheological analysis, although an importance of more basic description of the chain motion has been emphasized5 and some questions have been posed on the tube picture in nonlinear regime.6 The motional mechanisms of the chain in the tube model, e.g., reptation, contour length fluctuation (CLF), and constraint release (CR; equivalent to the tube motion) for linear chains at equilibrium,3,4 have not been derived from a more fundamental © 2013 American Chemical Society

expression of the intermolecular forces (or potential) but are introduced in a rather ad hoc sense. Thus, it is important to experimentally examine the validity of these mechanisms. For this purpose, observation of the motion with spectroscopic methods (such as neutron spin echo7) is useful. At the same time, it is very informative to compare different physical properties of a given system because the stochastic chain motion is differently averaged in different properties. We have been focusing on this difference between dielectric and viscoelastic properties to experimentally extract characteristic features of entangled polymers, as summarized below. Polyisoprene (PI) chains with high cis content have the socalled type A dipole parallel along their backbone, so that their global motion results in slow dielectric relaxation.8−11 For ordinary linear PI without dipole inversion, the local dipole is proportional to the bond vector (end-to-end vector) u of the submolecule, and the total dipole of the chain is proportional to its end-to-end vector R (= sum of u along the chain backbone). Received: March 24, 2013 Revised: June 26, 2013 Published: July 25, 2013 6067

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2.1−17.9 and their blends with longer PI (M2 = 1.1 × 106). The volume fraction υ1 of the short PI chains in the blend was kept small (mostly υ1 = 0.1) so that the CR effect on the viscoelastic/dielectric relaxation of those samples in monodisperse bulk can be almost fully erased in the blends. This strategy is similar to that in previous studies examining the CR effect on diffusion22,23 and viscoelastic relaxation.24,25 (In fact, comparison of the viscoelastic and dielectric relaxation of short PI in our blends confirmed that the CR mechanism was fully suppressed for the case of M1 ≤ 4.3 × 104 ≪ M2.) It turned out that the relaxation of the short PI component is slower in the blends than in monodisperse bulk for both viscoelastic and dielectric data. This fact strongly suggests that the CR mechanism does contribute to/accelerate both viscoelastic and dielectric relaxation of monodisperse linear PI in bulk, at least for 10−4M1 ≤ 17.9 (M1/Me ≤ 36 with Me being the entanglement molecular weight), no matter what the results of modeling/simulation19,21 are. In addition, a simple DTD analysis utilizing empirical equations of the terminal relaxation time of monodisperse bulk PI10,11,13 and the CR time26 suggested that the PI chain in monodisperse bulk exhibits the reptation/CLF motion along a partially dilated tube that wriggles in the fully dilated tube (thereby satisfying the fullDTD relationship between the viscoelastic μ(t) and dielectrically evaluated φ′(t)) and that the suppression of CR in the blends does not allow the tube to dilate even partially, thereby retarding the viscoelastic/dielectric relaxation of PI. Details of these results are presented in the paper.

For such PI chains, the dielectric relaxation reflects an orientational memory of the end-to-end vector R at times t and 0, ⟨R(t)·R(0)⟩eq, that directly involves the cross-correlation between different submolecules, as fully explained in previous reviews.10,11 In contrast, the viscoelastic relaxation reflects the orientational anisotropy of the submolecules not directly detecting the cross-correlation.10,11 This difference results in the difference of the normalized dielectric and viscoelastic relaxation functions in the linear response regime, Φ(t) and μ(t), that has been experimentally observed for monodisperse PI having various topological structures.10−16 Specifically, for entangled monodisperse linear PI,11,12 μ(t) was found to be close to {Φ(t)}α with α ≅ 2 (see also Supporting Information). Within the context of the tube model, this result suggests that the dynamics of the entangled linear chains is strongly affected by reptation/CLF along the tube as well as by CR, the latter being activated by the motion of surrounding chains. Furthermore, the dielectric Φ(t) is close (though not identical) to the survival fraction of the tube, φ′(t).10,11 A simple analysis of the Φ(t) and μ(t) data (utilizing the Gaussian feature of the chain) indicated that a relationship μ(t) = {φ′(t)}1+d (d ≅ 1.3) is valid for entangled monodisperse linear PI. This empirical relationship agrees with the relationship deduced from the molecular picture of full dynamic tube dilation (full-DTD),17,18 a coarse-grained version of the CR molecular picture: The CR mechanism activates large-scale motion of the entanglement segments in a direction perpendicular to the chain backbone. This motion results in mutual equilibrium (exploration of all possible conformations) within a given sequence of those segments to enhance the viscoelastic relaxation. The same motion also enhances the dielectrically detected end-to-end fluctuation, though less prominently. (In contrast, reptation/CLF equally enhances the viscoelastic and dielectric relaxation.) Those equilibrated segments as a whole can be regarded (coarse-grained) as a dilated segment that serves as a stress-sustaining unit.3,4,11 The DTD picture focuses on this dilated segment (that enlarges with time) to describe the chain dynamics/relaxation in long time scales. The validity of the full-DTD relationship, μ(t) = {φ′(t)}1+d (d ≅ 1.3), suggests that the CR/DTD mechanism significantly contributes to the dynamics of the entangled monodisperse linear PI.10,11,13 However, recently, Glomann et al.19 extended the tube model of Likhtman and McLeish20 (originally formulated for viscoelastic relaxation) to dielectric relaxation by eliminating the CR ef fect on the end-to-end fluctuation to claim that the model describes consistently the viscoelastic and dielectric data for monodisperse PI. In addition, a recent singlechain slip-link simulation (making a mean-field treatment of the entanglement) reported by Pilyugina et al.21 suggests no significant CR contribution to the end-to-end fluctuation of monodisperse linear PI, as explained in more detail in the Supporting Information. The results of these modeling and simulation appear to disagree with the previous experimental data.10,11,13 It is highly desired to resolve this disagreement because CR is the important mechanism of chain dynamics. Thus, we have revisited the classical problem, the CR contribution to the dielectric and viscoelastic relaxation of monodisperse linear PI, on the basis of a purely experimental approach, not relying on models/simulations that unavoidably include some molecular assumptions. Specifically, we measured the viscoelastic and dielectric data for monodisperse linear PI chains with 10−4M1 =

2. EXPERIMENTAL SECTION 2.1. Materials. Table 1 summarizes the characteristics of narrow molecular weight distribution PI samples utilized in this study. For

Table 1. Molecular Characteristics of Linear PI Samples code PI PI PI PI PI

a

21K 43Kb 99Kc 179Kb 1.1M

10−3Mw

Mw/Mn

21.4 43.2 98.5 179 1120

1.04 1.03 1.04 1.02 1.13

a Synthesized/characterized in ref 13. bSynthesized/characterized in ref 26. cSynthesized/characterized in ref 27.

simplicity, those samples are hereafter referred to as monodisperse. The sample code indicates the molecular weight. All samples have the molecular weight larger than the entanglement molecular weight Me (= 5K for PI) and are entangled in respective monodisperse bulk systems as well as in their blends. The highest-M sample, PI 1.1M, was synthesized in this study via living anionic polymerization in vacuum utilizing sec-butyllithium and benzene as the initiator and solvent, respectively. The polymerized sample was precipitated in excess methanol and redissolved in benzene, and after this precipitation/redissolution process repeated twice, the sample was finally recovered from a benzene solution that was dried in high vacuum at 50 °C for 48 h. A small amount (∼0.02 wt % to the PI sample) of antioxidant, butylhydroxytoluene (BHT), was added to the final solution, and the dried sample was sealed in Ar and kept in a deep freezer until use. The PI 1.1M sample thus recovered was characterized with GPC (CO-8020 and DP-8020; Tosoh) equipped with a refractive index (RI) monitor (RI-8020, Tosoh) and low-angle laser light scattering (LALLS) detector (Viscotek 270, Malvern) connected in series: Its weight-average molecular weight Mw and polydispersity index Mw/Mn were calculated from the LALLS/RI signal ratio detected at respective 6068

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sections of elution volume. The elution solvent was tetrahydrofuran (THF), and monodisperse linear PI samples synthesized/characterized in our previous studies13,26,27 were utilized as the RI/LALLS standards. The remaining four lower-M samples shown in Table 1 were previously synthesized anionically and characterized through the similar RI/LALLS analysis as well as the elution volume calibration.13,26,27 (Those samples, containing a small amount of BHT, had been sealed in Ar and stored in a deep freezer. No degradation during the storage was confirmed with GPC.) For dielectric and linear viscoelastic measurements, blends of the PI 1.1M sample with one of the lower-M PI samples were prepared by dissolving prescribed masses of these PI samples in benzene at a concentration ∼5 wt % and then allowing benzene to thoroughly evaporate at 50 °C in high vacuum for 48 h. As done in our previous work,28 freshly baked glass containers were utilized in this preparation of the blend samples (as well as in the recovery of the pure PI 1.1M sample explained above) to reduce the content of ionic impurities in the samples through adsorption of those impurities on the glass wall. The sample mass in the benzene solution was adjusted so as to allow the dried blend to form a thin film of thickness = 0.5−1 mm in the container (which helped efficient evaporation of benzene), and the full evaporation was confirmed from the mass of the dried blend. 2.2. Measurements. For the blends of the PI 1.1M sample with the lower-M PI samples listed in Table 1 as well as for respective monodisperse samples, dynamic dielectric measurements were conducted with an impedance analyzer/dielectric interface system (1260 and 1296; Solartron). A homemade parallel-plate dielectric cell (with diameter of 30 mm) mountable on a laboratory rheometer (ARES; TA) was utilized, and the measurements were made at angular frequencies29 ω = 1.0 × 10−2−1.0 × 105 s−1 in the temperature control chamber of the rheometer at 25, 40, and 60 °C (ω = 2πf with f being the frequency in hertz). The dielectric loss ε″ was measured with uncertainty less than 20% (as judged from reproducibility in independent measurements). Those ε″ data obeyed the time− temperature superposition with the shift factor aT exactly the same as that for the viscoelastic data (described below). All those data were reduced/compared at a reference temperature, Tr = 40 °C. Linear viscoelastic measurements were conducted for the blends and monodisperse PI samples at ω = 4.0 × 10−4−1.0 × 102 s−1 with ARES at 25, 40, and 60 °C. A parallel plate fixture with the diameter of 8.0 mm was utilized. The storage and loss moduli, G′ and G″, were measured with uncertainty less than 10%. Those moduli data obeyed the time−temperature superposition with the shift factor aT reported previously,13 log aT = −4.13(T − Tr)/(150 + T − Tr) with Tr = 40 °C, and all data were reduced/compared at 40 °C.

Figure 1. Dielectric loss data of PI/PI blends and component PI samples at 40 °C. The volume fraction of the short component is υ1 = 0.1 for PI 21K, PI 43K, and PI 99K (panels a−c), and υ1 = 0.2 for PI 179K (panel d). Thick arrows show the peak frequency for the short component in the blend (evaluated in Figure 5). Black solid line indicates an empirical equation for the data of bulk PI 1.1M at ω > 0.1 s−1, ε2,m″(ω) = 4.0 × 10−3ω−0.25.

3. RESULTS AND DISCUSSION 3.1. Overview of Dielectric and Viscoelastic Data. Figures 1 shows the angular frequency (ω) dependence of the dielectric loss ε″(ω) of the PI/PI blends and bulk PI samples. The storage and loss moduli, G′(ω) and G″(ω), of those systems are presented in Figures 2 and 3, respectively. In the dielectric measurements, the direct current (dc) conduction due to ionic impurities began to significantly contribute to the ε″ data at ωaT < 0.1 s−1, although the impurity content was minimized with the method explained in the Experimental Section. Following the previous strategy,28 this study utilizes only the raw data (shown in Figure 1) at higher ω that were negligibly affected by the dc contribution. In Figure 1, the dashed curve shows the ε″ data of PI 179K shifted along the ω axis by a factor of (179/1120)3.5 (= relaxation time ratio expected for bulk PI 179K and PI 1.1M samples). The data of PI 1.1M, exhibiting the well-known power law behavior (ε″ ∝ ω−1/4) at high ω, agree well with this shifted curve at low ω. The PI 21K, PI 43K, PI 99K, and PI 179K samples have M1/ Me > 4 (Me = 5K for PI) and are entangled in respective

monodisperse bulk systems as well as in the blends. The viscoelastic and dielectric data of these monodisperse systems satisfied the full-dynamic tube dilation (full-DTD) relationship as shown in Figure S1 of the Supporting Information, which is in harmony with previous results.11−15 More importantly, those samples are rather dilute in the blends (volume fraction υ1 = 0.1 for PI 21K, PI 43K, and PI 99K, and υ1 = 0.2 for PI 179K), but their weak relaxation is clearly detected in Figures 1 and 3 as the peaks of the ε″(ω) and G″(ω) data of the blends at ω > 0.1 s−1. The storage modulus G′(ω) (Figure 2) is rather insensitive to such weak and fast relaxation, as can be easily understood from the expression of G′(ω) in terms of the relaxation spectrum.1 Thus, hereafter we mainly focus on the ε″(ω) and G″(ω) data to discuss the relaxation of the short components in the blends. In relation to this discussion, we should note that the ε″ data of the blends at ω higher than the ε″ peak frequency exhibit the power-law type ω dependence very similar to the dependence (ε″ ∝ ω−1/4) observed for the component PI (cf. Figure 1). This similarity indicates that the ε″ data shown in Figure 1 are of high quality and negligibly contributed from the dc 6069

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Figure 2. Storage modulus data of PI/PI blends and component PI samples at 40 °C. The volume fraction of the short component is υ1 = 0.1 for PI 21K, PI 43K, and PI 99K (panels a−c), and υ1 = 0.2 for PI 179K (panel d). Thick green curves indicate the shifted data of monodisperse PI 1.1M.

Figure 3. Loss modulus data of PI/PI blends and component PI samples at 40 °C. The volume fraction of the short component is υ1 = 0.1 for PI 21K, PI 43K, and PI 99K (panels a−c), and υ1 = 0.2 for PI 179K (panel d). Thick arrows show the peak frequency for the short component in the blend (evaluated in Figure 6). Solid green curves indicate the shifted data of monodisperse PI 1.1M.

conduction. (Note that the dc conduction (εdc″ ∝ ω−1), if any, always tends to enhance the decrease of the ε″ data with increasing ω. This study does not utilize such low-ω data affected by the dc conduction, as explained earlier.) In Figures 1 and 3, the thick vertical arrows indicate the peak frequencies of the ε″ and G″ data of the short component in the blends that are evaluated later in Figures 5 and 6 for discussion of the relaxation of this component. (That discussion does not utilize the high-ω peak frequency of the blend data because overlapping of the high-ω and low-ω peak tails of those data could result in an artificial shift of the of highω peak,30 as analyzed/explained in Supporting Information.) Clearly, the relaxation of the short component is slower in the blend than in the monodisperse system.31 This result allows us to conclude that the CR mechanism accelerates both dielectric and viscoelastic relaxation of monodisperse linear PI, and thus suppression of CR (and DTD) on blending with much longer PI 1.1M retards the relaxation in the blends. Consequently, refinement is necessary for the recent modeling19 that neglects the CR contribution to the dielectric relaxation of monodisperse PI as well as for the recent single-chain slip-link

simulation21 that underestimates this CR contribution (as explained in the Supporting Information). Now, we attempt to examine, in more detail, the CR effect(s) on the dielectric and viscoelastic relaxation in the monodisperse system. From Figures 1 and 3, we note that the viscoelastic relaxation of the short component is more significantly retarded than the dielectric relaxation on suppression of CR due to blending. In addition, the dielectric mode distribution of the short component hardly changes on suppression of CR (as can be noted from the similarity of the ω dependence of the ε″ data of the monodisperse system and blend at ω > ωpeak), whereas the viscoelastic mode distribution becomes considerably narrower on this suppression (as suggested by the steeper decrease of the G″ data of the blend at ω > ωpeak compared to the monodisperse system). These features are to be discussed more quantitatively for the dielectric and viscoelastic data of the short component in the blends, not for the data of the blend as a whole. Thus, in the remaining part of this paper, we first evaluate the component data and then discuss those features. 6070

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3.2.2. Evaluation Method for G 1,b ″(ω). The DTD mechanism decreases the orientational anisotropy and stress of the entanglement segments, thereby affecting the viscoelastic mode distribution.11−15 For this reason, the long component exhibits two-step relaxation in the blend, with the fast step occurring together with (and being activated by) the full relaxation of the short component. The relaxation time and intensity of the slow step of the long component, τ[G] 2,b and G[slow] N2 , change with Mi and υi in a nontrivial way according to the DTD contribution.13 Thus, in general, G2,b″(ω) of the long component in the entire range of ω cannot be simply expressed in terms of G2,m″(ω). Nevertheless, for our blends having small υ1 and exhibiting widely separated relaxation times of the two components (cf. Figure 3), the short component negligibly contributes to Gb″(ω) of the blend at sufficiently low ω in the terminal relaxation zone. For this case, G2,b″(ω) at such low ω can be related to G2,m″(ω) as

3.2. Method of Evaluation of Data for Low-M Components in Blends. The εb″(ω) and Gb″(ω) data of the blends, with the subscript “b” hereafter representing quantities in the blend, are expressed, in general, as3,10,13 εb″(ω) = υ1ε1,b″(ω) + υ2ε2,b″(ω)

(1)

G b″(ω) = υ1G1,b″(ω) + υ2G2,b″(ω)

(2)

Here, εi,b″(ω) and Gi,b″(ω) are the dielectric and mechanical losses of the component i in the blend, and υi is its volume fraction; i = 1 and 2 for the short and long components. In eqs 1 and 2, the contribution of the component i to the blend data is represented as υiεi,b″(ω) and υiGi,b″(ω). Consequently, εi,b″(ω) and Gi,b″(ω) are defined as the quantities normalized to the unit volume fraction of the component i, and their relaxation intensities, Δεi = (2/π)∫ ∞ −∞εi″(ω) d ln ω and GNi = (2/π)∫ ∞ G ″(ω) d ln ω, are independent of υi and Mi and −∞ i coincide with those in the monodisperse system. Specifically, GNi is identical to the entanglement plateau modulus GN. Equations 1 and 2 are deduced from the well-established feature3,8 that the electrical polarization (for eq 1) and orientational anisotropy (for eq 2) of polymeric blends, the basic quantities underlying the dielectric and viscoelastic relaxation, are additively contributed from the components therein. εi,b″(ω) and Gi,b″(ω) appearing in eqs 1 and 2 reflect the component dynamics in the blend and are different, in general, from εi,m″(ω) and Gi,m″(ω) of the component in its monodisperse system (with the subscript “m” hereafter standing for the monodisperse system). Nevertheless, eqs 1 and 2 still allow us to estimate, with satisfactory accuracy, ε1,b″(ω) and G1,b″(ω) of the short component in the blend from the εb″(ω) and Gb″(ω) data of the blend and the ε2,m″(ω) and G2,m″(ω) data of the long component in the monodisperse state, as explained below. 3.2.1. Evaluation Method for ε1,b″(ω). It is well-known that the CR/DTD mechanism for the long component is enhanced on blending with shorter components but hardly affects the dielectric mode distribution of the long component.10,13 Similarly, the local Rouse dynamics within the entanglement segment does not affect this distribution. For these reasons, ε2,b″(ω) of the long component appearing in eq 1 has the mode distribution close to that of ε2,m″(ω), as noted previously10,13a (see, for example, Figures 4 and 19 of ref 13a and also the Supporting Information of this paper). In addition, the short component is rather dilute in our blends (υ1 = 0.1 or 0.2) and thus accelerates the dielectric relaxation of the long component only moderately. These observations allow us to safely replace [ε] [ε] [ε] [ε] ε2,b″(ω) in eq 1 by ε2,m″(λ[ε] 2 ω), where λ2 = τ2,b /τ2,m with τ2,b [ε] and τ2,m being the dielectric relaxation time of this component in the blend and in its monodisperse system, respectively.13 Thus, ε1,b″(ω) of the short component in the blend can be evaluated as ε1,b″(ω) = υ1−1{εb″(ω) − υ2ε2,m″(λ 2[ε]ω)}

G2,b″(ω) = υ2−1G b″(ω) = I2G2,m″(λ 2[G]ω)

at low ω (4)

[slow] Here, I2 (= G[slow] N2 /GN2 = GN2 /GN) represents a fraction of the viscoelastic intensity of the long component that relaxes [G] [G] through the slow step, and λ[G] 2 (= τ2,b /τ2,m) is the viscoelastic relaxation time ratio of this component in the blend and monodisperse system. Both I2 and λ[G] 2 can be determined from comparison of the Gb″ and G2,m″ data at low ω, as explained later for Table 2.

Table 2. Parameters I2 and λ[G] 2 Obtained from Shift of G2,m′(ω) and G2,m″(ω) Data in Figures 2 and 3a blend PI PI PI PI

21K/PI 1.1M 43K/PI 1.1M 99K/PI 1.1M 179K/PI 1.1M

I2

λ[G] 2

I2,full‑DTD

[G] λ2,full−DTD

0.86 0.89 0.89 0.74

0.89 0.89 0.89 0.94

0.87 0.87 0.87 0.75

0.81 0.81 0.81 0.65

For comparison, the values of I2,full‑DTD and λ[G] 2,full−DTD for the simplest case of reptation/CLF of the long component along fully dilated tube are also shown. a

For the long component, the fraction I2 of its viscoelastic intensity relaxes through the slow step, as represented by eq 4. Thus, the remaining fraction, 1 − I2, should relax during the fast step. Because this fast step occurs together with the full relaxation of the short component, G2,b″(ω) of the long component in this step should be close to G1,b″(ω) of the short component, except for the relaxation intensity being smaller for the former by the factor of 1 − I2. Then, the contribution of the fast relaxation of the two components to the Gb″(ω) data of the blend can be safely approximated as υ1G1,b″(ω) + υ2(1 − I2) G1,b″(ω) = (1 − I2υ2)G1,b″(ω). From eqs 2 and 4 combined with this expression of the fast relaxation contribution, G1,b″(ω) of the short component at ω > 0.1 s−1 (where this component fully relaxes; cf. Figure 3) is evaluated experimentally as

(3)

G1,b″(ω) ≅ (1 − I2υ2)−1{G b″(ω) − I2υ2G2,m″(λ 2[G]ω)}

Previous study for PI/PI blends revealed that υ2 dependence [G] is very similar for τ[ε] 2,b and τ2,b (cf. Figure 7 of ref13a and Figure 4 of ref 13b), which means that λ[ε] coincides with the 2 corresponding viscoelastic relaxation time ratio, λ[G] 2 . On the [G] basis of this observation, we replace λ[ε] by λ in our later 2 2 analysis (1 > λ[G] 2 > 0.89 in the blends examined in this study, as explained later). 13

at ω > 0.1 s−1

(5)

It should be noted that loosening of the entanglement due to the relaxation of the short component (CR/DTD effect for the long component) is a prerequisite for the long component to exhibit the slow relaxation expressed by the υ2I2G2,m″(λ[G] 2 ω)term in eq 5. G2,m″(ω) of the long component in the 6071

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monodisperse system decreases with increasing ω in the relaxation regime for the short component (ω > 0.1 s−1; cf. Figure 3). Because of this decrease of G2,m″(ω), the [G] υ2I2G2,m″(λ[G] 2 ω) term utilizing the λ2 value determined after completion of the short component relaxation (at ω < 0.1 s−1) overestimates the contribution of the slow process to the value Gb″(ω) data at ω > 0.1 s−1, and eq 5 with this λ[G] 2 underestimates G1,b″(ω). In contrast, if we set λ[G] 2 = 1 in eq 5 by assuming that the slow process does not start until the short component fully relaxes, we overestimate G1,b″(ω). Thus, eq 5 just specifies the lower bound (for the experimentally [G] determined λ[G] = 1) for the real 2 ) and upper bound (for λ2 G1,b″(ω) values. Nevertheless, G1,b″(ω) is satisfactorily obtained with the aid of eq 5 if these lower and upper bound values are numerically close to each other. This feature of eq 5 is successfully utilized later in the evaluation of G1,b″(ω). Equation 4 indicates that the G2,m″(ω) data shifted double logarithmically along the modulus and frequency axes by factors of I2 and 1/λ[G] 2 , respectively, are superposed on the Gb″(ω) data of the blend in the terminal zone. The same superposition should be valid also for the data of storage moduli, Gb′(ω) and G2,m′(ω). (The superposition should work better for the storage moduli being insensitive to fast relaxation.) In fact, the superposition is well achieved at low ω as shown with the thick green curves in Figures 2 and 3, which confirms validity of eq 4. For all blends examined, Table 2 summarizes the I2 and λ[G] 2 values thus determined from the shift required for the superposition. 3.3. Comment for Relaxation of Long Component in Blends. The I2 and λ[G] 2 data summarized in Table 2 reflect the relaxation behavior of the long component (PI 1.1M) in the blend. If the short component relaxes much faster than the long component to behave as a solvent for the latter (as considered in the full-DTD picture), the effective entanglement molecular weight for the long component at low ω is given by Me,full−DTD 13,32 = υ−d and the corresponding I2 is expressed as 2 Me (d ≅ 1.3), I2,full‑DTD = Me/Me,full−DTD = υd2. Furthermore, in the simplest case of reptation/CLF of the long component along the axis of α [G] fully dilated tube, λ[G] 2 is given by λ2,full−DTD = (Me/Me,full−DTD) αd = υ2 , where the exponent α is related to the M dependence of the terminal relaxation time τ of this component,3 τ ∼ τR(M/ Me)α ∝ M2+α/Mαe with τR being the Rouse relaxation time of that component: For PI, τ ∝ M3.5 (as shown later in Figure 9a,b) and thus α = 1.5. For the simplest case explained above, we evaluated I2,full‑DTD and λ[G] 2,full−DTD for respective blends from their υ2 values (0.9 or 0.8). The results are summarized in Table 2. The λ[G] 2,full−DTD value is smaller for the PI 179K/PI 1.1M blend than for the other three blends, because υ2 is smaller and the fully dilated tube (sustained only by the long component) is shorter for the former. For the PI/PI blends containing PI 21K, PI 43K, and PI 99K as the short component, I2,full‑DTD and λ[G] 2,full−DTD are close to the I2 and λ[G] 2 data (cf. Table 2) so that the terminal relaxation of the long component appears to be governed by the mechanism of reptation/CLF along the fully dilated tube. This simple result [G] can be related to the wide separation of τ[G] 2,b and τ1,b of the two [G] [G] components in those blends (τ2,b ≥ 700 τ1,b ; cf. Figure 3) that possibly allows the short component to actually behave as a solvent in the time scale of τ[G] 2,b . In contrast, for the blend containing PI 179K, I2,full‑DTD is close to the I2 data, but λ[G] 2,full−DTD is considerably smaller than data (relaxation slower than the expectation for the the λ[G] 2

simplest case explained above). This result suggests that the CR equilibration of the long component due to the short [G] component relaxation is not fast enough (τ[G] 2,b ≅ 110τ1,b ), thereby not allowing the long component to achieve the equilibration over the axis of the fully dilated tube (prerequisite for reptation/CLF along this axis). Even for this case, the long component can be still equilibrated up to the fully dilated tube diameter ( 0.1 s−1 where the short component fully relaxes, the ε2,m″(ω) data are excellently described by an empirical equation, ε2,m″(ω) = 4.0 × 10−3ω−0.25 (black solid line in Figure 1). We utilized this empirical equation to evaluate υ1ε1,b″(ω) at those ω (>0.1 s−1). As an example, υ1ε1,b″(ω) obtained for the PI 99K/PI 1.1M blend is shown in Figure 4a with small filled red circles. The εb″(ω) and ε2,m″(ω) data (green squares and black circles) and the above empirical equation (solid line) are also shown for comparison. The mode distribution of υ1ε1,b″(ω) thus evaluated coincides with that of the ε1,m″(ω) data of monodisperse PI 99K, as shown later in Figure 5. This coincidence confirms the validity of eq 3. For the viscoelastic G1,b″(ω), the upper and lower bound evaluation is to be made, as explained in the previous section. For completeness, we can artificially set λ[ε] 2 = 1 in eq 3 to evaluate the upper bound of υ1ε1,b″(ω). (Note that υ1ε1,b″(ω) calculated from eq 3 with the aid of above empirical equation increases with increasing λ[ε] 2 .) The upper bound evaluated in this way is shown with large unfilled red circles. Close agreement is noted for the two sets of ε2,m″(ω) evaluated for [G] [ε] λ[ε] 2 = λ2 and λ2 = 1 (because our blends have small υ1 and [ε] λ2 is close to unity in any case), indicating that eq 3 gave ε2,m″(ω) with high accuracy. Similar results were obtained also for the other blends examined. We utilized eq 5 to evaluate (1 − I2υ2)G1,b″(ω). At ω > 0.1 s−1 where the short component fully relaxes, the G2,m″(ω) data included in eq 5 are excellently described by an empirical [R] [e] equation, G2,m″(ω) = G[e] 2,m″(ω) + G2,m″(ω) with G2,m″(ω) = 6072

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Figure 5. Comparison of ε1,b″ (plots) and ε1,m″ (dashed curves) of the short component in the blends and monodisperse systems at 40 °C. The solid curves show ε1,m″ shifted along the ω-axis to make superposition with ε1,b″ at low ω. The amount of shift gives the relaxation time ratio of the short component in the blend and [ε] monodisperse system, τ[ε] 1,b /τ1,m.

Figure 4. Evaluation of ε1,b″(ω) and G1,b″(ω) of the short component in the PI 99K/PI 1.1M blend. The solid line and solid curve in panels a and b, respectively, indicate empirical equations for the ε2,m″ and G2,m″data of bulk PI 1.1M, ε2,m″(ω) = 4.0 × 10−3ω−0.25 and G2,m″(ω) [R] [e] 4 −0.22 and G[R] = G[e] 2,m″(ω) + G2,m″(ω) with G2,m″(ω) = 4.7 × 10 ω 2,m″(ω) = 1.1 × 102ω0.68, both being valid at ω > 0.1 s−1. 2 0.68 4.7 × 104 ω−0.22 and G[R] , as shown in 2,m″(ω) = 1.1 × 10 ω Figure 4b with the solid curve. This empirical equation is composed of two terms: G[e] 2,m″(ω) describing the power-law increase of G 2,m ″(ω) with decreasing ω due to the entanglement relaxation and G[R] 2,m″(ω) describing the powerlaw decrease of G2,m″(ω) due to the local Rouse-like relaxation. (Note that the local relaxation does not contribute to the dielectric data, and thus the empirical equation for ε2,m″(ω) includes no term corresponding to G[R] 2,m″(ω).) Because the local [R] relaxation represented by G2,m ″(ω) does not change on blending, we replaced the I2υ2G2,m″(λ[G] 2 ω) term in eq 5 by [G] [R] I2υ2G[e] 2,m″(λ2 ω) + υ2G2,m″(ω). As an example, (1 − I2υ2) G1,b″(ω) thus obtained for the PI 99K/PI 1.1M blend is shown in Figure 4b. The small filled and large unfilled red circles indicate lower and upper bound values of G1,b″(ω) obtained [G] with the λ[G] 2 data (Table 2) and with λ2 = 1, respectively. The Gb″(ω) and G2,m″(ω) data (green squares and black circles) are also shown for comparison. Good agreement is noted for the upper and lower bound values of G1,b″(ω) (again because λ[G] 2 is close to unity), indicating that G1,b″(ω) was satisfactorily evaluated with eq 5. Similar results were obtained also for the other blends examined. Figure 5 compares the ε1,b″ data evaluated from eq 3 with λ[ε] 2 = λ[G] 2 (plots) and the ε1,m″ data (dashed curves) of the short component in the monodisperse system. Clearly, the relaxation is retarded on blending, confirming the CR/DTD effect on the dielectric τ[ε] 1,m in the monodisperse system concluded for Figure 1. Furthermore, the ε1,m″ curve can be superimposed on the ε1,b″ data at low ω after the shift along the ω-axis, as shown by the solid curves. (This superposition confirms that the dielectric mode distribution of the short component hardly changes on blending.) Figure 6 compares the viscoelastic G1,b″ (evaluated from eq 5 with the λ[G] 2 and I2 values being summarized in Table 2; plots) with G1,m″ (dashed curves). On blending, the viscoelastic relaxation of the short component is retarded and its mode distribution becomes narrower, as already noted qualitatively in Figure 3. (The relaxation intensity of this component, GN1 = (2/π)∫ ∞ −∞G1″(ω) dlnω, agreed in the blend and monodisperse system within experimental uncertainty. The narrowing of the

Figure 6. Comparison of G1,b″ (plots) and G1,m″ (dashed curves) of the short component in the blends and monodisperse systems at 40 °C.

mode distribution on blending resulted in a moderate increase of the G1,b″ peak height, as noted in Figure 6.) These effect of blending on G1,b″ is much more significant than that observed for the dielectric ε1,b″ (Figure 5), which confirms the stronger CR/DTD effect on the viscoelastic relaxation discussed for Figures 1−3. It is informative to directly compare the G1,b″ and ε1,b″ data in the blends (Figures 5 and 6). This comparison is made in Figure 7, where ε1,b″ (small filled symbols) is multiplied by an

Figure 7. Comparison of G1,b″ (large unfilled symbols) and ε1,b″ (small filled symbols connected with curves) of the short component in the blends at 40 °C.

appropriate factor A to match its peak height with that of the G1,b″ data (large unfilled symbols). For PI 21K and PI 43K, the G1,b″ and ε1,b″ data agree with each other for both relaxation time and mode distribution. This agreement is characteristic to the relaxation in fixed entanglements,10,11 indicating that the CR/DTD effect vanishes for these short components: The viscoelastic relaxation time τ[G] 2,b of the long component (PI 1.1M) is orders of magnitude longer than τ[G] 1,b of those short [G] components in the blends (τ[G] /τ ≥ 1.0 × 104; cf. Figure 3), 2,b 1,b which naturally allows the CR/DTD mechanism to be fully suppressed for the short components. (As explained earlier, our 6073

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evaluation of G1,b″ (eq 5) included an approximation, G2,b″(ω) ≅ G1,b″(ω), during the full relaxation of the short component. The agreement of G1,b″ and ε1,b″, which should be attained for [G] the CR/DTD-free case of τ[G] 2,b /τ1,b ≫ 1, was indeed observed for G1,b″ thus evaluated from eq 5. This result in turn suggests acceptable accuracy of the approximation.) For PI 99K, the G1,b″ and ε1,b″ data are still close to each other, but a hint of deviation is also noted (cf. Figure 7). For [G] this case, τ[G] 2,b /τ1,b ≅ 700 and the entanglement due to the long component seems to begin loosened (i.e., the CR/DTD effect for PI 99K begins to emerge) in the time scale of the global relaxation of PI 99K. Finally, for PI 179K in the blend having [G] τ[G] 2,b /τ1,b ≅ 110, the viscoelastic relaxation is faster and its mode distribution is broader compared to the dielectric relaxation, possibly because the motion of the long component is not slow enough compared to PI 179K and the CR/DTD mechanism considerably affects the motion of PI 179K. (Note that in the full-DTD limit the terminal viscoelastic relaxation is approximately twice faster and considerably broader than the dielectric relaxation, as explained previously10,11,13 and also demonstrated in the Supporting Information.) The magnitude of the CR/DTD effect in the monodisperse [ε] [G] [G] system can be characterized by the τ[ε] 1,b /τ1,m and τ1,b /τ1,m ratios of the short component chain in the blend and monodisperse system. We evaluated the relaxation times of this component, [ε] [G] [G] τ[ε] 1,b , τ1,m, τ1,b , and τ1,m, as a reciprocal of the angular frequencies ωpeak at the peaks of the ε1,b″, ε1,m″, G1,b″, and G1,m″ data shown in Figures 5 and 6 to determine those ratios. (The ωpeak in the blend has been shown in Figures 1 and 3 with thick arrows.) [ε] The dielectric τ[ε] 1,b /τ1,m ratio thus obtained agreed with the shift factor utilized to superpose the ε1,m″ data on the ε1,b″ data in [ε] [G] [G] Figure 5. In Figures 8a and 8b, the τ[ε] 1,b /τ1,m and τ1,b /τ1,m ratios are plotted against the number of entanglements per short component chain, M1/Me. (Note that the full scale for the vertical axis in Figure 8b is twice of that of Figure 8a.) The dashed curves indicate the ratios expected from a naive model explained later for eq 8, and the solid curve indicates the ratios deduced from a simple but more sophisticated DTD analysis (eq 12). [ε] As noted in Figure 8a, the τ[ε] 1,b /τ1,m ratio decreases gradually with increasing M1/Me so that the CR/DTD effect on the dielectric relaxation time in the monodisperse system, characterized by this ratio, may asymptotically vanish for very large M1/Me. However, this asymptotic limit is not achieved even for M1/Me ∼ 100, as suggested from the slow decrease of [ε] the τ[ε] 1,b /τ1,m ratio seen in Figure 8a. This behavior is later discussed with the aid of a naive model (eq 8) and a simple DTD analysis (eq 12). [G] The viscoelastic τ[G] 1,b /τ1,m ratio exhibits a qualitatively similar decrease, as noted in Figure 8b. However, this ratio does not appear to fully decay to unity. This lack of full decay can be related to the empirical relationship valid for the monodisperse [ε] systems,11,12 τ[G] 1,m ≅ τ1,m/2. When the CR/DTD contribution to the dynamics of the short component in the blend is [ε] completely quenched, this component should have τ[G] 1,b = τ1,b (as noted for PI 21K and PI 43K in Figure 7). Thus, the [G] increase of M1/Me allows the τ[G] 1,b /τ1,m ratio to decay only to [ε] [ε] 2τ1,b /τ1,m, i.e., only to 2 even in the high-M1/Me limit where the [ε] [ε] CR/DTD effect on τ[ε] 1,m may vanish to give τ1,b /τ1,m = 1. This argument, being in harmony with the result seen in Figure 8b, is made more quantitatively later in relation to the DTD analysis. 3.5. Molecular Weight Dependence of Relaxation Time in CR/DTD-Free Environment. Figure 8 has

[ε] [G] Figure 8. Changes of dielectric τ[ε] 1,b /τ1,m ratio (a) and viscoelastic τ1,b / [G] τ1,m ratio (b) for short component with the number of entanglement per component chain, M1/Me. Note that the full scale for the vertical axis in panel b is twice of that of panel a. The dashed curves indicate ratios expected from a naive model (eq 8), and the solid curves indicate the ratios deduced from simple DTD analysis (eqs 12 and 13).

demonstrated the CR/DTD-induced acceleration of the dielectric and viscoelastic relaxation in monodisperse bulk PI systems, the main subject of this paper. In relation to this acceleration, it could be informative to examine the molecular weight dependence of the dielectric and viscoelastic relaxation [G] times, τ[ε] 1 and τ1 , of the short component in the PI/PI blends and the monodisperse system, although this dependence is not the main but secondary subject of this paper. [G] The τ[ε] 1 and τ1 data have been evaluated for the discussion in Figure 8. In Figures 9a and 9b, these data are plotted against M1/Me (black unfilled and red filled circles). Small unfilled circles show the literature data for monodisperse PI summarized in the Supporting Information. Reviewers for [G] this paper suggested us to also examine the τ[G] 1,b and τ1,m data of linear polybutadiene (PB) in the blends and monodisperse system reported by Liu et al.25 (PB has no type A dipole, and thus no dielectric τ[ε] data are available.) Those data are 1 reproduced in Figure 9c. [G] We first focus on our data for PI. The τ[ε] 1,m and τ1,m data of PI in the monodisperse system are proportional to M13.5, as noted in Figures 9a and 9b (cf. unfilled circles). The M1 dependence [G] of the τ[ε] 1,b and τ1,b data of the short component PI in the blends (red filled circles) is similar to that in the monodisperse system, except for PI-179K having the highest-M1 examined: The τ[ε] 1,b and τ[G] 1,b values for PI-179K are smaller than those extrapolated from the data points of lower-M1 PI (dashed line). If we [G] attempt to describe the M1 dependence of all τ[ε] 1,b and τ1,b data, including the data for PI-179K, with a power-law relationship, τ ∝ Mα′, we find the exponent α′ to be considerably smaller than 3.5 (α′ ≅ 3.2). From this result, one might argue that the 6074

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only for PI 21K and PI 43K (and PI 99K), without including the data for PI 179K. Figures 9a and 9b demonstrate that this real dependence is not significantly different from the M3.5 1 dependence (compare black and red lines in the range of M1/ Me < 10), although minor weakening should have occurred for the dependence in the blend, as noted from the gradual [ε] [G] [G] decrease of the τ[ε] 1,b /τ1,m and τ1,b /τ1,m ratios (Figure 8). This result strongly suggests that the CR/DTD mechanism in the monodisperse system affects the M1 dependence of τ only moderately, lending support to a frequently made argument that the M3.5 1 dependence in that system mainly results from the combination of reptation and CLF.3,4 At the same time, we should emphasize that the CR/DTD mechanism does accelerate the relaxation in the monodisperse systems, as evidenced from the difference of the τ values in the blend and monodisperse system. [G] Now, we turn our attention to the τ[G] 1,b and τ1,m data for PB/ PB blends reported by Liu et al.25 The molecular weights of the long and short PB components were reported to be 1.2M and 13.2K−163K, and the volume fraction of the latter was υ1 = 0.1. Figure 6b of ref 25 includes two data points for PI/PI blends and one data point for polystyrene (PS) blend in addition to seven data points for PB/PB blends. However, a ratio of the viscoelastic CR time to the observed relaxation time in [G] monodisperse system, τ[G] CR,m/τm , is considerably different for PI and PS having the same M/Me value, as reported previously26 and also shown in the Supporting Information. (This ratio is smaller for PI than for PS.) Thus, the CR contribution to the relaxation should be different for PS and PB (the latter being chemically similar to PI), and comparison of [G] the τ[G] 1,b and τ1,m data of PS with those of PB and PI could introduce some uncertainty. For this reason, Figure 9c (reproducing Figure 6b of ref 25) does not include the data for PS. For PB in the monodisperse system, τ[G] 1,m is proportional to 3.4 M1 as claimed in ref 25 and also confirmed in Figure 9c. (This exponent of 3.4 is indistinguishable from 3.5 within the scatter 25 of the data points.) For τ[G] 1,b in the blend, Liu et al. utilized all data points (filled red symbols) to deduce a power-law [G] relationship in the CR/DTD-free condition, τ1,b ∝ M13.1 shown with the black dashed line. However, as explained for Figures 9a and 9b, the test of real power-law relationship requires us to check if the CR/DTD contribution to the relaxation is negligible for those data points. Unfortunately, PB has no type A dipole, and its global motion is dielectrically inert. For this reason, the method utilized for PI, comparison of the ε1,b″ and G1,b″ data (Figure 7), cannot be applied to PB. However, we may still apply the [G] CR/DTD-free criterion obtained for the PI/PI blend, τ[G] 2,b /τ1,b > 700, to the chemically similar PB/PB blends. The viscoelastic relaxation times of the long and short PB components in the [G] blends, τ[G] 2,b and τ1,b , can be estimated in Figure 1b of ref 25, [G] and we evaluated the τ[G] 2,b /τ1,b ratio for each data point shown [G] in Figure 9c. It turned out that τ[G] 2,b /τ1,b ≅ 50 for the highest[G] [G] M1 data point (M1/Me = 104), τ2,b /τ1,b ≅ 300 for the second [G] and third highest-M1 data points (M1/Me ≅ 60), and τ[G] 2,b /τ1,b 3 ≥ 6 × 10 for the remaining four data points (red squares [G] located at M1/Me < 30). The τ[G] 2,b /τ1,b values for the data points at M1/Me ≥ 60 are significantly smaller than 700, and those data points do not appear to satisfy the CR/DTD-free criterion. Then, the M1 dependence of τ[G] 1,b for the PB/PB blends under the CR/DTD-free condition is best deduced from the four data 3.4 points at M1/Me < 30. The power-law relationship τ[G] 1,b ∝ M1

Figure 9. M1/Me dependence of the dielectric and viscoelastic relaxation times determined for the short PI component in the blends and monodisperse systems (panels a and b). Panel c shows literature data25 for the viscoelastic relaxation time of PB (squares) and PI (circles). Data in panel c redrawn with permission from ref 25.

suppression of CR/DTD significantly weakens the M dependence of τ. However, this argument is incomplete because the CR/DTD contribution to the relaxation is dif ferent for the four short component PI in the blends: In the blends, the CR/DTD mechanism is fully suppressed for PI 21K and PI 43K, begins to operate for PI 99K, and operates considerably for PI 179K, as already noted from comparison of the ε1,b″ and G1,b″ data (Figure 7). It is obvious that the τ data can be cast in the real power-law relationship only when those data reflect the same relaxation mechanism. In other words, it is physically unreasonable to use the power-law for description of the τ data of all PI components that are differently affected by the CR/DTD mechanism. (As explained for Figure 7, we can focus on the viscoelastic relaxation time ratio of the long and short components in the blends to cast a CR/DTD-free criterion for [G] the short component as τ[G] 2,b /τ1,b > 700, with 700 being the [G] τ[G] /τ ratio for PI 99K for which the CR/DTD mechanism 2,b 1,b begins to operate.) [G] Thus, the real power-law M1 dependence of τ[ε] 1,b and τ1,b in the CR/DTD-free condition is to be deduced from the data 6075

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3.6.1. Non-Coarse-Grained View of CR. The fundamental molecular function describing the dielectric relaxation of linear PI is the local correlation function defined by3,10,11,33

with the exponent identical to that for the monodisperse system, shown with the red line in Figure 9c, appears to reasonably describe those four data points, although a minor decrease of the exponent should have occurred in the blend (as explained for Figures 9a and 9b). This result is in harmony with that obtained for our PI/PI blends, although a delicate chemical difference between PI and PB might result in some difference in the relaxation behavior of the short components in the blends. This relaxation behavior in the blend deserves further attention. 3.6. CR/DTD Analysis of Dielectric Relaxation Time in Monodisperse Systems. For entangled monodisperse PI and PI/PI blends, extensive experiments10−13,26 have been made to establish empirical equations for the terminal viscoelastic and [ε] dielectric relaxation times τ[G] m and τm as well as the longest CR 26 time in the monodisperse system (see Supporting Information for a summary of the literature data):

C(n , t ; m) =

(6)

[G] [ε] τCR,m /s = 1.5 × 10−25M 5 , τCR,m /s = 3.0 × 10−25M 5

for monodisperes PI at 40 °C

(7)

The τm/τCR,m ratio decreases with increasing M to have a very [G] small value for well-entangled PI. For example, τ[G] m /τCR,m ≅ 0.04 for PI 99K (having M/Me ≅ 20). For the monodisperse system, one may assume independence (lack of coupling) of the competing relaxation mechanisms therein to formulate a naive but frequently utilized model for the dielectric and viscoelastic relaxation times, τ[ε] m and τ[G] m : 1 τm[x]

=

1 [x] τCR,m

τ[x] CR,m

+

1 [x] τrep,m

(9)

Here, u(n,t) is the bond vector of nth submolecule at time t, a is the average size of this submolecule, and ⟨...⟩eq stands for the ensemble average at equilibrium. For discussion/description of the relaxation of entangled PI, the entanglement segment is conveniently utilized as the submolecule. For ordinary PI having noninverted type A dipoles, the normalized dielectric relaxation function Φ(t) is simply related to the local correlation function as Φ(t) = N−1∫ N0 C(n,t;m) dn dm with N = M/Me.3,10,11,33 For the pure reptation mechanism and exactly Rouse-type CR mechanism, the time evolution of C(n,t;m) can be easily calculated analytically. For both mechanisms, the eigenfunctions of C(n,t;m) are sinusoidal and expressed as f p(n) = sin(pπn/N) (p = 1, 2, ...). These eigenfunctions serve as the base for the expansion, C(n,t;m) = (2/N)∑p f p(n)f p(m) exp(−t/τp) with τp being the relaxation time of pth eigenmode. (Note that C(n,t;m) satisfies a time evolution equation ∂C(n,t;m)/∂t = 3 (n)C(n,t;m) with 3 (n) being an operator specific to the motional mechanism, and f p(n) satisfies the corresponding eigenvalue equation, 3 (n)f p (n) = −τp−1f p(n).3,11,33,34) If the entangled PI chain relaxes through competition of pure reptation and Rouse−CR mechanisms, pth-order reptation eigenmode for C(n,t;m) couples only with CR eigenmode of the same order p, because these mechanisms have the same functional form of 3 (= (N/π)2τ1−1 ∂2/∂n2)). For this case, the eigenfunction does not change its functional form on this competition, and the longest dielectric relaxation time is given by the harmonic average of the reptation and −1 longest CR times, τ[ε] = {1/τrep + 1/τ[ε] (which is equivalent CR} to eq 8). For this case, the dielectric mode distribution is not affected by the competition at all, and the Rouse−CR mechanism negligibly affects τ[ε] for well-entangled PI, as deduced from the naive model. The dielectric mode distribution of actual PI chains is nearly the same in different entanglement environments, i.e., in the monodisperse system and blends, as observed previously3,10,11,13 and also noted in Figure 5. This feature is superficially consistent with the above discussion for the pure reptation and Rouse−CR mechanisms. However, the actual eigenfunctions of C(n,t;m) deviate from the sinusoidal eigenfunctions for those mechanisms, as noted from dielectric data of specially designed PI chains having asymmetrically once-inverted type A dipoles.33−35 (This nonsinusoidal feature of the eigenfunctions is attributable to enhanced relaxation near the chain ends.33−35) Thus, a delicate non-Rouse feature of actual CR mechanism has been noted.33,35,36 The PI chains relax through competition of several mechanisms that include reptation (combined with CLF) and CR, and the blending in the high-M matrix suppresses the CR contribution. The nonsinusoidal eigenfunctions for respective mechanisms do not necessarily coincide with each other. In fact, some delicate difference has been observed experimentally in the monodisperse system and blend.33,34 This difference of the eigenfunctions of the competing mechanisms results in coupling of all eigenmodes of those mechanisms. Consequently, the slowest eigenmode of reptation/CLF is coupled not only with the slowest CR eigenmode but also with higher

τm[G]/s = 2.0 × 10−19M3.5 , τm[ε]/s = 4.2 × 10−19M3.5 for monodisperse PI at 40 °C

1 ⟨u(n , t ) ·u(m , 0)⟩eq a2

with x = ε , G (8)

τ[x] rep,m

Here, is the CR relaxation time (cf. eq 7), and is the relaxation time for reptation/CLF occurring in the f ixed [x] (undilated) tube. The τ[x] rep,m/τm ratio given by this model [x] corresponds to the τ[x] /τ ratio examined in Figure 8 (for the 1,b 1,m case that the CR/DTD contribution is quenched for the short component in the blend). Utilizing empirical eqs 6 and 7 in eq −1 [x] [x] 8, we evaluated τ[x] and further the rep,m = {1/τm − 1/τCR,m} [x] [x] [x] τ[x] /τ ratio. The τ /τ ratio thus obtained is shown in rep,m m rep,m m Figure 8 with the dashed curves. Clearly, the naive model gives [x] negligible CR/DTD effect on the τ[x] 1,b/τ1,m ratio for large M1/Me [x] (>10) and significantly underestimates the τ[x] 1,b/τ1,m data. This failure of the naive model results from its assumption of no coupling between the CR/DTD and reptation/CLF mechanisms: Lack of coupling leads to competition of just the lowest order eigenmodes of the CR/DTD and reptation/CLF mechanisms, which results in the harmonic average shown in eq 8. Thus, the coupling of the CR/DTD and reptation/CLF mechanisms appears to be essential for description of the τ[ε] 1,b / [G] [G] τ[ε] 1,m and τ1,b /τ1,m ratios. This coupling results in competition of the lowest order reptation/CLF eigenmodes with all orders of the CR eigenmodes (including fast, higher order eigenmodes) [x] to give the τ[x] 1,b/τ1,m ratio larger than that calculated from eq 8. In the remaining part of this paper, we consider this point, analyze the higher order CR eigenmodes and the corresponding DTD picture, and offer a simple DTD model for the τ[ε] 1,b / [G] [G] τ[ε] 1,m and τ1,b /τ1,m ratios based on this analysis. 6076

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Article [ε] relaxation time τ[ε] m and the dielectric CR time τCR,m of linear PI in the monodisperse system. In this test, we consider the partially dilated tube composed of N/β* dilated segments each having the size a* = β*1/2a, as depicted in Figure 10. Here, β* is the number of entanglement segments involved in each dilated segment, and N = M/Me. The reptation/CLF along this partially dilated tube requires the chain tension along the tube axis to be equilibrated through the CR mechanism.3,13b This tension equilibration involves all (N) entanglement segments of the chain except those sustaining the stress (i.e., remaining to be independent) in the time scale of terminal relaxation.13b,38 This situation would be easily understood if we consider a blend of short and much longer chains where the long chains are mutually entangled with each other to reptate along the tube formed by themselves. This reptation of the long chain occurs when all long−short entanglements are fully CR-equilibrated, and the tube-forming long−long entanglements are not involved in this equilibration.13b,32b In the monodisperse systems, the number of the entanglement segments that sustain the stress in the time scale of τ[ε] m is equivalent to the number of dilated segments, N/ β*. Consequently, we may consider that the number g of the entanglement segments involved in the CR-equilibration of the tension is given by g = N − N/β* + 1, where the extra factor of “1” has been introduced to guarantee g = N for β* = N (dilation involving all segments) and g = 1 for β* = 1 (dilation limited within one entanglement segment, i.e., no dilation). For the reptation/CLF to actually occur along the partially dilated tube in the time scale of τ[ε] m , the chain should satisfy a condition (prerequisite) that the time τ* required for the CR-equilibration of those g entanglement segments does not exceed τ[ε] m . This condition allows us to determine the g and β* values with the aid of the Rouse−CR model, as explained below. (Although a delicate non-Rouse feature of the eigenfunction is noted for actual CR mechanism, the ratio of the relaxation times of actual CR eigenmodes are close to those deduced from the Rouse− CR model.11,33,34 For this reason, the following analysis of the mode relaxation time utilizes this model.) For the chain composed of N entanglement segments, the characteristic time for pth Rouse−CR mode is given by5,39

order CR eigenmodes, which results in invalidity of the naive model (eq 8) explained earlier. The CR effect on the relaxation time can be further analyzed in relation to the eigenfunction if its functional form is known for all orders of eigenmodes. However, unfortunately, the functional form has been experimentally specified only for the lowest three eigenmodes,33,34 which does not allow us to make this analysis. Nevertheless, we may switch to a coarse-grained view of CR, namely, the DTD molecular picture, to further discuss this effect. This discussion is summarized below. 3.6.2. Coarse-Grained View of CR (DTD). The full-DTD relationship between the normalized relaxation modulus μ(t) (= G(t)/GN) and the dielectrically obtained tube survival fraction φ′(t), μ(t) = {φ′(t)}1+d (d ≅ 1.3 for PI), is valid for entangled linear PI in the monodisperse system, as reported in the previous papers10−13 and also demonstrated in the Supporting Information. Specifically, the terminal relaxation occurs in a fully dilated tube of the diameter afull‑DTD = β1/2 full−DTDa (≅ 1.7a), as illustrated in Figure 10. Here, βfull−DTD

Figure 10. Schematic illustration of DTD/reptation process. The chain moves (reptates) along the partly dilated tube (with the diameter a*) that wriggles in the fully dilated tube (with the diameter afull‑DTD).

(= Me,full−DTD/Me) is the number of the entanglement segments that are CR-equilibrated to effectively behave as a dilated entanglement segment (stress-sustaining unit) in the time scale τ[G] m of the terminal viscoelastic relaxation. Its value, βfull−DTD ≅ 3, has been specified from the analysis of the dielectric data (see Figure S2). From the validity of the full-DTD relationship between μ(t) and φ′(t), one might argue that the monodisperse PI chains exhibit reptation/CLF along the axis of the fully dilated tube. In this simplest case, the dielectric relaxation of those chains in the monodisperse system is accelerated by a factor of λ1 = (Me/ Me,full−DTD)1.5 = βfull−DTD−1.5 ≅ 0.2, and thus blending in the long matrix (PI 1.1M) should retard the relaxation by the factor of λ1−1 ≅ 5. However, the experimentally observed dielectric [ε] retardation is much less significant (τ[ε] 1,b /τ1,m < 2 as noted in Figure 8a). This fact strongly suggests that reptation/CLF in the monodisperse system occurs along the axis of a partially dilated tube that wriggles in the fully dilated tube of the diameter afull‑DTD, as illustrated in Figure 10. This wriggling motion allows the CR-equilibration over the length scale of afull‑DTD to give the full-DTD relationship between μ(t) and φ′(t), but the relaxation time is determined by the length of the partially dilated tube, not of the fully dilated tube. (This molecular picture is similar to “reptation along skinny tube wriggling in super tube” proposed by Viovy et al.,37 although their molecular picture did not consider the partial dilation for the skinny tube.) The above molecular picture can be straightforwardly tested with the aid of the empirical eqs 6 and 7 for the dielectric

⎛ pπ ⎞ ⎟ τp = τ ° sin−2⎜ ⎝ 2N ⎠

(p = 1, 2, ..., N − 1)

(10a)

with ⎛ π ⎞ [ε] ⎟ τ ° = τCR,m sin 2⎜ ⎝ 2N ⎠

(10b)

(We have adopted the discrete Rouse expression in eq 10 because p is not necessary small.) The unit time for the local CR motion, τ°, is related to τ[ε] CR,m of the slowest CR mode as shown in eq 10b. The pth CR mode involves cooperative motion of N/p entanglement segments. Thus, the entanglement segment number of our interest, g, can be estimated as the equilibrated segment number for a particular (p*th) CR mode, N/p*, that satisfies the condition explained above, τ*(g) = τp* ≤ τ[ε] m . The number g should basically coincide with N/p*, but a delicate difference due to the finiteness of the chain length may exist. For this reason, we adopted a relationship, N/p* = Ag + B (with g = N − N/β* + 1), and determined the coefficients A and B from two conditions, p* = 1 for g = N (the lowest CR mode involving all entanglement segments) and p* 6077

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Article 1.5 [ε] CLF time (τ ∝ M3.5/Me1.5) as τ[ε] = β*1.5, 1,b /τ1,m = (Me*/Me) where Me* is the effective entanglement molecular weight corresponding to a*. This estimate, obtained from the β* value in Figure 11, is shown with the solid curve in Figure 8a. This curve is surprisingly close to the data points, lending support for our molecular picture of reptation/CLF along the partially dilated tube. Now, we turn our attention to the retardation of the viscoelastic relaxation on blending (Figure 8b). The empirical [ε] relationship τ[G] 1,m ≅ τ1,m/2, being consistent with the full-DTD picture, is valid for the viscoelastic and dielectric relaxation times in the monodisperse system, as reported previsouly11,12 and also noted from empirical eqs 6 and 7. In contract, τ[G] 1,b = τ[ε] if the CR/DTD effect on the dynamics of the short 1,b component vanishes in the blend with much longer component. These results allows us to express, in a semiempirical sense (by use of the above empirical relationship), the viscoelastic retardation factor on blending as

= N − 1 (the highest CR mode) for g = 1.40 The result is summarized as 1 N−2 1 g+ = 2 p* (N − 1) (N − 1)2

(11)

Equation 11 reduces to the simplest relationship, g = N/p*, for large N. From eqs 10 and 11, the condition (prerequisite) for the chain in the monodisperse system to actually exhibit reptation/ CLF along the partially dilated tube can be expressed as ⎧ τ [ε] ⎫ sin 2 CR,m ⎪ * =⎪ ⎬ ⎨ [ε] ⎪ τm[ε] ⎪ ⎩ τm ⎭ sin 2 τp

( 2πN ) p*π 2N

( )

π [ε] ⎫ ⎧ sin 2 2N ⎪ τCR,m ⎪ ⎬ =⎨ [ε] ⎪ ⎪ ⎡ ⎩ τm ⎭ sin 2⎧ ⎨ π ⎢ N − 22 g + ⎩ 2N ⎣ (N − 1)

( ) ⎤−1⎫ ⎬ 2⎥ (N − 1) ⎦ ⎭

≤1

1

(12)

(Note that eq 12 can be approximated as (g /N ≤ τ[ε] m if N ≫ 1 and g ≫ 1.) [ε] Utilizing the τ[ε] CR,m/τm data (obtained from empirical eqs 6 and 7) in eq 12, we determined the critical value (maximum possible value) of g that corresponds to the maximum possible τp* and minimum possible p*. In Figure 11, this critical value, 2

2

)τ[ε] CR,m

[G] τ1,b [G] τ1,m

≅

[ε] 2τ1,b [ε] τ1,m

= 2β*1.5 (13)

[G] We evaluated this τ[G] 1,b /τ1,m ratio from β* shown in Figure 11. The result, shown in Figure 8b with the solid curve, is again [G] surprisingly close to the τ[G] 1,b /τ1,m data, suggesting that the retardation of the viscoelastic and dielectric relaxation on blending is consistently and satisfactorily explained from the simple partial-DTD picture depicted in Figure 10. Here, a comment needs to be made for the values of β*, g*, and βfull‑DTD (≅ 3 in the terminal relaxation zone; cf. Supporting Information). As can be noted in Figure 11, β* is smaller than βfull‑DTD, and thus the partially dilated tube wriggles in a wider, fully dilated tube. From this result, one might pose a question why the tube (reptation path) dilates only partially to have β* < βfull‑DTD, despite the validity of the full-DTD relationship between the viscoelastic relaxation function μ(t) and the tube survival fraction φ′(t), μ(t) = {φ′(t)}1+d (d ≅ 1.3). For this question, we should remember that βfull‑DTD (= {φ′(t)}−d) just represents the size of the stress-sustaining unit (dilated segment) determined by competition of several relaxation mechanisms. Consequently, βfull‑DTD does not necessarily specify the reptation path length. (Note also that the fullDTD relationship mentioned above does not specify the time dependence of φ′(t), which reflects this feature of βfull‑DTD.) This path length is equivalent to the length of the wriggling, partially dilated tube that is determined by CR-equilibration of the chain tension. This tension equilibration involves g* entanglement segments, and thus we have to compare βfull‑DTD with g*, not with β*. As can be noted in Figure 11, g* is close to βfull‑DTD (≅3) for small M/Me but gradually increases with M/Me to become considerably larger than βfull‑DTD for large M/ Me.41 In fact, this result, g* ≥ βfull‑DTD, ensures the chain to explore, on time, all conformations within the fully dilated tube diameter afull‑DTD (= aβfull−DTD1/2), thereby validating the fullDTD relationship between the dielectric and viscoelastic data. Finally, we should emphasize that the above analysis is based on the empirical equations (eqs 6 and 7) and the empirical [ε] relationship τ[G] 1,m ≅ τ1,m /2 obtained for monodisperse PI with M/Me < 130 (M ≤ 626k; cf. Supporting Information). Thus, the results of the analysis allows us to conclude the gradual decrease of β* and gradual increase of g* for the partly dilated tube in this range of M/Me. However, it is still uncertain whether β* further decreases to unity on a further increase of

Figure 11. M/Me dependence of the numbers β* (specifying the partially dilated tube diameter for reptation in the time scale of τ[ε] m ) and the number g* of the entanglement segments that are involved in the CR-equilibration in that time scale. The partially dilated tube wriggles in the fully dilated tube characterized by βfull‑DTD ≅ 3.

hereafter denoted as g*, and the corresponding β* (= N/(N − g* + 1)) are plotted against M/Me (= N). As noted in the top panel, β* specifying the diameter a* of the partially dilated tube along which the chain reptates in the time scale of τ[ε] m (a* = β*1/2a; cf. Figure 10) decreases with increasing M/Me. However, this decrease is considerably slow, and reptation appears to occur along the partially dilated tube even in a wellentangled system with M/Me = 100. Thus, the dielectric relaxation should be retarded on suppression of CR/DTD in the blend with much longer component. For the short chain component examined in Figure 8a, the magnitude of retardation of the dielectric relaxation on blending is estimated from an expression for the reptation/ 6078

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M/Me or levels off at a value >1. In addition, the above analysis just focuses on the β* and g* values at the terminal relaxation time and does not specify the evolution of β* and g* at shorter times. These issues are interesting subjects of future work.

ASSOCIATED CONTENT

S Supporting Information *

Validity of full-DTD relationship for viscoelastic and dielectric data of monodisperse linear PI, comparison of dielectric data of PI/PI blends with single-chain slip-link simulation, summary of extensive viscoelastic and dielectric data of PI/PI blends, and artificial shift of dielectric and viscoelastic loss peaks on blending. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

(1) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (2) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986. (3) Watanabe, H. Prog. Polym. Sci. 1999, 24, 1253. (4) McLeish, T. C. B. Adv. Phys. 2002, 51, 1379. (5) Likhtman, A. E. Viscoelasticity and molecular rheology. In Polymer Science: A Comprehensive Reference, 1st ed.; Möeller, M., Matyjaszewski, K., Eds.; Elsevier: Amsterdam, 2012; Vol. 1. (6) (a) Tapadia, P.; Ravindranath, S.; Wang, S. Q. Phys. Rev. Lett. 2006, 96, 196001. (b) Wang, S. Q.; Ravindranath, S.; Wang, Y. Y.; Boukany, P. Y. J. Chem. Phys. 2007, 127, 064903. (7) Richter, D.; Monkenbusch, M.; Arbe, A.; Colmenero, J. Adv. Polym. Sci. 2005, 174, 1. (8) Adachi, K.; Kotaka, T. Prog. Polym. Sci. 1993, 18, 585. (9) Boese, D.; Kremer, F. Macromolecules 1990, 23, 829. (10) Watanabe, H. Macromol. Rapid Commun. 2001, 22, 127. (11) Watanabe, H. Polym. J. 2009, 41, 929. (12) Watanabe, H.; Matsumiya, Y.; Inoue, T. Macromolecules 2002, 35, 2339. (13) (a) Watanabe, H.; Ishida, S.; Matsumiya, Y.; Inoue, T. Macromolecules 2004, 37, 1937. (b) Watanabe, H.; Ishida, S.; Matsumiya, Y.; Inoue, T. Macromolecules 2004, 37, 6619. (14) Watanabe, H.; Matsumiya, Y.; van Ruymbeke, E.; Vlassopoulos, D.; Hadjichristidis, N. Macromolecules 2008, 41, 6110. (15) Matsumiya, Y. Nihon Reoroji Gakakishi (J. Soc. Rheol. Jpn.) 2011, 39, 197. (16) Watanabe, H. Nihon Reoroji Gakakishi (J. Soc. Rheol. Jpn.) 2012, 40, 209. (17) Marrucci, G. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 159. (18) Milner, S. T.; McLeish, T. C. B. Phys. Rev. Lett. 1998, 81, 725. (19) Glomann, T.; Schneider, G. J.; Bras, A. R.; Pyckhout-Hintzen, W.; Wischnewski, A.; Zorn, R.; Allgaier, J.; Richter, D. Macromolecules 2011, 44, 7430. (20) Likhtman, A. E.; McLeish, T. C. B. Macromolecules 2002, 35, 6332. (21) Pilyugina, E.; Andreev, M.; Schieber, J. D. Macromolecules 2012, 45, 5728. (22) Green, P. F.; Mills, P. J.; Palmstrom, C. J.; Mayer, J. W.; Kramer, E. J. Phys. Rev. Lett. 1984, 53, 2145. (23) Green, P. F.; Kramer, E. J. Macromolecules 1986, 19, 1108. (24) Liu, C. Y.; Keunings, R.; Bailly, C. Phys. Rev. Lett. 2006, 97, 246001. (25) Liu, C. Y.; Halasa, A. F.; Keunings, R.; Bailly, C. Macromolecules 2006, 39, 7415. (26) Sawada, T.; Qiao, X.; Watanabe, H. Nihon Reoroji Gakakishi (J. Soc. Rheol. Jpn.) 2007, 35, 11. (27) Watanabe, H.; Chen, Q.; Kawasaki, Y.; Matsumiya, Y.; Inoue, T.; Urakawa, O. Macromolecules 2011, 44, 1570. (28) Matsumiya, Y.; Uno, A.; Watanabe, H.; Inoue, T.; Urakawa, O. Macromolecules 2011, 44, 4355. (29) A reviewer for this paper suggested to use “rad s−1”, rather than “s−1”, as the unit of the angular frequency ω. However, we prefer to use “s−1” for the following simple reason. Many dynamic quantities are related to each other through the angular frequency: For example, the loss modulus G″ and the dynamic viscosity η′ are related as η′ = G″/ω. The quantities in both sides of this relationship must have the same physical unit. Because G″ and η′ have the unit of “Pa” and “Pa s”, ω is to be expressed in the unit of “s−1” rather than “rad s−1”, although “rad” is the dimensionless unit similar to “%”. (Note that η′ = G″/ω is never expressed in the unit of “Pa s rad−1” that is directly deduced if ω is expressed in the unit of “rad s−1”.) (30) A reviewer for this paper posed a question about an artificial shift of the peaks in the εb″ and Gb″ data for the blends due to overlapping of the tails of the high-ω and low-ω peaks. This question was addressed on the basis of eq 1, and the results of a simple analysis are summarized in the Supporting Information: It turned out that the artificial shift toward the lower-ω side occurs for the high-ω peak of

4. CONCLUDING REMARKS We have examined dielectric and viscoelastic behavior of PI/PI blends containing rather dilute short components (M1 = 21K− 179K and υ1 = 0.1 and/or 0.2) in highly entangled long component (M2 = 1.1M). The dielectric relaxation time τ[ε] 1 of the short components was found to be longer in the blend than in the monodisperse system even for M1 = 179K (M1/Me = 36), and the viscoelastic τ[G] 1 of this component exhibited even stronger retardation on blending. Thus, both dielectric and viscoelastic relaxation processes of the short component in the monodisperse system are accelerated by the CR/DTD mechanism, and suppression of this mechanism on blending [G] increases τ[ε] 1 and τ1 . This fact in turn indicates necessity of refinement for the recent model,19 assuming no CR/DTD effect for the dielectric relaxation in monodisperse systems as well as for the recent single-chain slip-link simulation21 that appears to underestimate the CR/DTD effect. A naive model, just based on the comparison of the terminal relaxation time τm and the longest CR time τCR,m in the monodisperse system, cannot consistently explain the considerable CR/DTD effect observed for the dielectric and viscoelastic relaxation in well-entangled monodisperse PI systems. Nonsinusoidal features of eigenfunctions noted for dipole-inverted PI chains11,33,34 suggested that all orders of reptation/CLF and CR eigenmodes are coupled with each other, and thus the comparison of τm and τCR,m (of the lowest order modes) does not allow us to estimate the magnitude of the CR/DTD effect. In fact, a simple DTD analysis utilizing the empirical equations for τm and τCR,m suggested that the chain in the monodisperse system exhibits reptation/CLF along the partially dilated tube that wriggles in the fully dilated tube. The experimentally observed retardation on blending is quantitatively described as a result of suppression of this partial tube dilation in the blend.

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (H.W.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partly supported by the Grant-in-Aid for Scientific Research (A) from MEXT, Japan (Grant No. 24245045), Grant-in-Aid for Scientific Research (C) from JSPS, Japan (Grant No. 24550135), and Collaborative Research Program of ICR, Kyoto University (Grant No. 2013-33). 6079

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the blend data unless the relaxation times of the components in the blends are widely separated (by a factor ≥ 104) and that the artificial shift is much less prominent (and almost negligible) for the low-ω peak of the blend data. Thus, we face to a delicate problem when the relaxation time of the dilute short component (in particular, the viscoelastic τ[G] 1,b ) is to be evaluated from the high-ω peak frequency of the blend data. For this reason, in this study, the relaxation time of the short component was evaluated from its ε1,b″(ω) and G1,b″(ω) data, not from the peaks in the data for the blend as a whole. (31) (a) Retardation on blending is noted even for short chains exhibiting no entanglement effect.31b However, the retardation observed for our PI/PI blends having the component molecular weights Mi > 4Me appears to be exclusively attributed to suppression of CR/DTD because the local (segmental) dynamics is insensitive to Mi if Mi is sufficiently larger than Me. (b) Harmandaris, V. A.; Angelopoulou, D.; Mavrantzas, V. G.; Theodorou, D. N. J. Chem. Phys. 2002, 116, 7656. (32) (a) Recently, van Ruymbeke et al.32b analyzed the viscoelastic and dielectric data in the literature13 to suggest that the theoretical tube dilation exponent is unity but an extra relaxation due to tension equilibration effectively increases this exponent to 1.3 if the long and short components have widely separated relaxation times. In contrast, experimentally, the dilation exponent evaluated from the data in the terminal regime is close to 1.3.13 In this paper, we adopt this experimental view to utilize the dilation exponent of 1.3. (b) E. van Ruymbeke, E.; Masubuchi, Y.; Watanabe, H. Macromolecules 2012, 45, 2085. (33) (a) Watanabe, H.; Urakawa, O.; Kotaka, T. Macromolecules 1994, 27, 3525. (b) Watanabe, H.; Urakawa, O.; Kotaka, T. Macromolecules 1993, 26, 5073. (34) Matsumiya, Y.; Watanabe, H.; Osaki, K.; Yao, M. L. Macromolecules 1998, 31, 7528. (35) Watanabe, H.; Matsumiya, Y.; Osaki, K.; Yao, M. L. Macromolecules 1998, 31, 7538. (36) Watanabe, H.; Yao, M. L.; Osaki, K. Macromolecules 1996, 29, 97. (37) Viovy, J. L.; Rubinstein, M.; Colby, R. H. Macromolecules 1991, 24, 3587. (38) In the early review,3 the tension equilibration involving all entanglement segments was considered to be a prerequisite for the chain in the monodisperse systems to reptate along the dilated tube. However, this requirement is too strong, and the chain can actually exhibit reptation/CLF along the partially dilated tube in a time scale of terminal relaxation where the tension is CR-equilibrated for the segments except those sustaining the elasticity,13,32b i.e., for g = N − N/β* + 1 segments as described in the text. (39) Graessley, W. W. Adv. Polym. Sci. 1982, 47, 67. (40) The results of the analysis shown in Figures 8 and 11 (solid curves) hardly change even if the condition utilized to determine the coefficients A and B, p* = N − 1 (= index of the fastest CR mode) for g = 1, is replaced by an artificial condition, p* = N for g = 1. (41) As observed in Figure 11, g* increases with increasing N (= M/ Me). g* is expressed in terms of N and β* as g* = N(1 − 1/β*) + 1, and the change of g* with N is determined by balance of the increase of the front factor N and the decrease of the 1 − 1/β* term (due to the decrease of β* on the increase of N). The increase of the front factor overwhelms the decrease of the 1 − 1/β* term to result in the observed increase of g*.

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