Dielectric and Viscoelastic Behavior of Star-Branched Polyisoprene

Article pubs.acs.org/Macromolecules

Dielectric and Viscoelastic Behavior of Star-Branched Polyisoprene: Two Coarse-Grained Length Scales in Dynamic Tube Dilation Yumi Matsumiya,† Yuichi Masubuchi,† Tadashi Inoue,‡ Osamu Urakawa,‡ Chen-Yang Liu,§ Evelyne van Ruymbeke,∥ 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 § CAS Key Laboratory of Engineering Plastics, Institute of Chemistry, The Chinese Academy of Sciences, Beijing 100190, P. R. China ∥ Bio and Soft Matter, Institute on Condensed Matter and Nano-science, Université Catholique de Louvain, Louvain-la-Neuve, Belgium ‡

S Supporting Information *

ABSTRACT: cis-Polyisoprene (PI) chain has the type A dipole parallel along the backbone so that its large-scale (global) motion results in not only viscoelastic but also dielectric relaxation. Utilizing this feature of PI, this paper examined dielectric and viscoelastic behavior of star PI probe chains (arm molecular weight 10−3Ma = 9.5−23.5, volume fraction υ1 = 0.1) blended in a matrix of long linear PI (M = 1.12 × 106). The constraint release (CR)/dynamic tube dilation (DTD) mechanism was quenched for those dilute probes entangled with the much longer matrix, as evidenced from coincidence of the frequency dependence of the dielectric and viscoelastic losses of the probe in the blend. Comparison of the probe data in the blend and in monodisperse bulk revealed that the star probe relaxation is retarded and broadened on blending and the retardation/broadening is enhanced exponentially with Ma. This result in turn demonstrates significant CR/DTD contribution to the dynamics of star PI in bulk. The magnitude of retardation was quantitatively analyzed within the context of the tube model, with the aid of the dielectrically evaluated survival fraction of the dilated tube, φ′(t), and the literature data of CR time, τCR. In the conventional molecular picture of partial-DTD, the tube is assumed to dilate laterally, but not coherently along the chain backbone. The corresponding lateral partial-DTD relationship between φ′(t) and the normalized viscoelastic relaxation function μ(t), μ(t) = φ′(t)/β(t) with β(t) being the number of entanglement segments per laterally dilated segment (that was evaluated from the φ′(t) and τCR data), held for the μ(t) and φ′(t) data of star PI in bulk. Nevertheless, the observed retardation of the star probe relaxation on blending was less signif icant compared to the retardation expected for the arm motion (retraction) along the laterally dilated tube in bulk PI. This result suggests that the relaxation time of the probe in bulk is governed by the longitudinal partial-DTD that occurs coherently along the chain backbone. In fact, the magnitude of retardation evaluated from the φ′(t) and τCR data on the basis of this longitudinal partial-DTD picture was close to the observation. These results strongly suggest that the star PI chains in monodisperse bulk have two different coarse-grained length scales: the diameter of laterally dilated tube that determines the modulus level and the diameter of longitudinally dilated tube that reflects the path length for the arm retraction and determines the relaxation time. Thus, the star PI chains in bulk appear to move along the longitudinally dilated tube that wriggles in the laterally dilated tube. This molecular scenario is consistent with the previous finding for bulk linear PI [Matsumiya et al. Macromolecules 2013, 46, 6067].

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INTRODUCTION

the multichain dynamics in real systems into the single chain dynamics influenced by the topological constraint, and represents this constraint as an uncrossable tube surrounding a focused chain. The equilibrium chain motion along the tube competes with the tube motion activated by the motion of surrounding (tube-forming) chains. The viscoelastic

Long flexible polymer chains in concentrated systems deeply penetrate with each other to mutually constrain their large-scale (global) motion. This topological constraint, referred to as the “entanglement”, has been one of the central subjects of polymer physics.1,2 Among several molecular models, the tube model(s) appears to well describe the entanglement effect on the chain motion and the corresponding viscoelastic relaxation, in particular at equilibrium (in the linear viscoelastic regime):2−5 Adopting a dynamic mean-field view, the tube model simplifies © XXXX American Chemical Society

Received: July 30, 2014 Revised: September 20, 2014

A

dx.doi.org/10.1021/ma501561y | Macromolecules XXXX, XXX, XXX−XXX

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properties calculated for this competition agree well with experiments. 3−5 The tube motion can be described as the constraint release (CR) that occurs through accumulation of local hopping of entanglement segments activated by motion of the tube-forming chains.4−6 This process can be well approximated as the Rouse process with its local relaxation (hopping) time being determined by motion of the tube-forming chain, although delicate non-Rouse features of the actual CR process7,8 have been also noted experimentally. The CR process, accounting for the multichain effect within the dynamic mean-field view, is important in monodisperse systems, because the motion of the tube-forming chain in those systems is identical to the motion of the chain constrained in the tube so that the tube unequivocally moves in the time scale of the global chain motion. Focusing on this importance, we recently studied the CR process9 cast in a form of dynamic tube dilation (DTD)10−13 for monodisperse linear chains. That study revealed two types of DTD processes determining the modulus level and relaxation time of the linear chains. Features of those processes are first summarized below as the background of the current work for star-branched chains, and then the target of the current work is explained. Readers familiar with this background can skip the next section and directly proceed to the “target” section. (For convenience for readers, the basic parameters appearing in the following sections are listed in the Appendix.) Background (for Linear Chain).9 The topological constraint loosens with time t through the CR mechanism, thereby allowing consecutive entanglement segments (each having the molecular weight Me and diameter a) to be mutually equilibrated and together behave as a dilated stress-sustaining segment.4,5 Thus, the CR process is often recast as the dynamic tube dilation (DTD) process10−13 that results in increases of the number β(t) of entanglement segments per dilated segment and the dilated tube diameter a′(t) = a{β(t)}1/2 with t. For this DTD process, an initial, enlarged tubelike envelope having the diameter a′ and surrounding the chain at t = 0 is utilized to define the survival fraction φ′(t) of the dilated tube: φ′(t) is defined as the fraction of the entanglement segments at time t that still remain within this initial envelope.4,10,14 (This φ′(t) is to be distinguished from the survival fraction of undilated tube,3−5 φ(t).) The normalized viscoelastic relaxation function, μ(t) ≡ G(t)/GN with G(t) and GN being the relaxation modulus and the entanglement plateau modulus, is related to β(t) and φ′(t) as8,10,14

μ(t ) =

φ′(t ) β (t )

Φ(t ) = φ′(t ) − = φ′(t ) −

⎤2 1 ⎡ a′(t ) − 1⎥ ⎢ ⎦ 4N ⎣ a

1 [{β(t )}1/2 − 1]2 4N

for linear PI

(2)

In the partial-DTD molecular picture considering self-consistent coarse-graining of the time and length scales (the latter being defined in the direction lateral to the chain backbone), β(t) is specified as8,14 β(t ) = min[βf‐DTD(t ), βCR (t )]

(3a)

with βf‐DTD(t ) = {φ′(t )}−d ,

d ≅ 1.3 for PI14

(3b)

Here, d is an experimentally obtained dilation exponent. βf‑DTD(t) is the number of entanglement segments per dilated segment for the case of f ull-DTD where the relaxed portions of the chain behave as a “solvent”.10 (In this case, φ′(t) is equivalent to the polymer concentration in such a solvent.4,10) βCR(t) denotes the maximum number of the entanglement segments that can be CR-equilibrated in time: βCR(t) is determined by the dielectric CR time τCR. Combining eqs 2 and 3 with the available data of Φ(t) and τCR,8,9,14,18 we determined β(t) and φ′(t) in a purely empirical way to test eq 1 for monodisperse linear PI. It turned out that β(t) coincides with βf‑DTD(t), and the viscoelastic μ(t) data obey the full-DTD relationship, μ(t) = {φ′(t)}1+d (eq 1 with β(t) = {φ′(t)}−d).8,9,14 This validity of full-DTD relationship reflects a rather sharp distribution of relaxation modes of linear PI that allows φ′(t) to decay only moderately even at the terminal viscoelastic relaxation time τG:9 βf‑DTD(τG) = {φ′(τG)}−d ≅ 3 for monodisperse linear PI. From this result, one may naively expect that monodisperse linear PI reptates along the fully dilated tube having the diameter af−DTD′(t) = a{βf‑DTD(t)}1/2. In fact, most of tube models are based on this expectation. We tested this expectation for blends of short linear PI chains (probe) entangled with much longer matrix PI.9 The CR/DTD mechanism was quenched for the probe in the blends. Then, the terminal relaxation of the probe should be retarded on blending by a factor of {af‑DTD′(τG)/a}3 = {βf‑DTD(τG)}3/2 ≅ 5 if the above naive expectation is valid. (Note that the reptation time τrep scales as τrep ∝ M3.5/Me,eff1.5 ∝ M3.5/a′3 with Me,eff being the effective entanglement molecular weight that increases due to DTD.) However, the experiment9 showed that the dielectric relaxation of probe PI was retarded on blending much less significantly compared to the expected factor of ≅5. Thus, monodisperse linear PI does not reptate along the fully dilated tube, despite the validity of the full-DTD relationship μ(t) = {φ′(t)}1+d. This finding led us to focus on a prerequisite for reptation along the dilated tube, the CR-activated tension equilibration along the dilated tube.9,19,20 Namely, the coherent/cooperative motion of all dilated segments of the chain (i.e., reptation along the dilated tube) is allowed only after this tension equilibration. Equations 1−3 specify the DTD process occurring not coherently along the chain backbone but locally in the direction lateral to the chain backbone. This lateral DTD just determines the modulus level (cf. eq 1). Correspondingly, the lateral full-DTD relationship valid for monodisperse linear PI, μ(t) = {φ′(t)}1+d, just describes the relationship between the dielectrically evaluated φ′(t) and the viscoelastic μ(t) data, without specifying the relaxation time. On the basis of this conclusion, we introduced two coarsegrained length scales:9 the diameter of the laterally dilated tube,

(1)

μ(t) decays by a factor of 1/β(t) when no entanglement segment escapes from the initial envelope (φ′(t) = 1). Namely, β(t) simply specifies the modulus decay due only to CR/DTD. Because of this simplicity, most of the current tube models adopt the molecular picture of DTD. For linear cis-polyisoprene (PI) chains having noninverted type A dipoles parallel along the backbone, φ′(t) and β(t) are straightforwardly evaluated from dielectric data so that eq 1 can be tested in a purely empirical way.8,14−16 Specifically, for linear PI having N entanglement segments per chain, the normalized dielectric relaxation function Φ(t) (identical to the autocorrelation function of the end-to-end vector4,8,17) is related to φ′(t) and β(t) as8,14 B


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a′(t) = {β(t)}1/2a with β(t) specified by eq 3, and the diameter of longitudinally dilated tube, a*(t): Differing from a′(t), a*(t) is not a straightforwardly defined tube diameter but is a length scale recast from the effective reptation path length, L*(t) = ⟨R2⟩eq/a*(t). [The Viovy−Rubinstein−Colby model21 would be helpful for capturing the meaning of a*(t). For an extreme case of reptation along the undilated tube that wriggles in the fully dilated tube, this model gives a*(t) = a and a′(t) = {βf‑DTD(t)}1/2a.] Despite this nonstraightforward definition, the analysis described below was much more conveniently achieved for a*(t) rather than for L*(t), because the terminal dielectric relaxation times of the probe in the blend and monodisperse [ε] bulk, τ[ε] 1,b and τ1,m, are simply related to each other via a*(t), [ε] [ε] τ1,b = {a*(τ1,m)/a}3τ[ε] 1,m. Considering this relationship for dielectric τ[ε] 1 , we analyzed the CR mode that can be activated within the observed τ[ε] 1,m to evaluate the number of entanglement segments per longitudinally 2 [ε] dilated segment in monodisperse bulk,9 β*(τ[ε] 1,m) = {a*(τ1,m)/a} . [ε] The corresponding increase of τ1 on blending (by the factor of 1.5 {β*(τ[ε] 1,m)} ) was close to the observation. In addition, an increase of the viscoelastic relaxation time, deduced from the molecular picture of longitudinal partial-DTD combined with the lateral full-DTD relationship valid for monodisperse linear chains (μ(t) = {φ′(t)}1+d), was also close to the observation.9 These results lent support to the molecular concept of two coarse-grained length scales, a′(t) = {β(t)}1/2a and a*(t) = {β*(t)}1/2a for the lateral and longitudinal partial-DTD processes, the former describing the modulus level and the latter, the relaxation time.9 Target of Current Work (for Star Chain). For starbranched PI, the lateral partial-DTD relationship is also specified by eqs 1 and 3, but the survival fraction of the dilated tube φ′(t) included in this relationship is related to the dielectric Φ(t) data through the following equation (instead of eq 2):15 Φ(t ) = φ′(t ) − = φ′(t ) −

retraction quite possibly occurs along the longitudinally and partially dilated tube that wriggles in the laterally and partially dilated tube, which well corresponds to the molecular scenario for linear PI in bulk (reptation along longitudinally and partially dilated tube that wriggles in the laterally and fully dilated tube9). Details of these results are presented in this paper.

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

Table 1. Molecular Characteristics of PI Samples 6-arm star PI code (9k)6a

PI PI (13k)6 PI (18k)6 PI (24k)6b PI 1.1Mc

for star PI

linear PI (precursor of arm)

10−3Mw

Mw/Mn

10−3Mw

Mw/Mn

54.2 77.7 101 144

1.05 1.02 1.02 1.03

9.5 13.2 17.7 23.5 1120

1.07 1.02 1.02 1.05 1.13

a Synthesized/characterized in ref 23. bSynthesized/characterized in ref 15. cUtilized as linear matrix; synthesized/characterized in ref 9.

simplicity, those samples are referred to as monodisperse. The sample code number for the 6-arm star PI samples indicates the weight-average arm molecular weight Ma. These star samples have Ma ≥ 2Me (Me = 5 × 103 for PI) and are moderately entangled in respective monodisperse bulk well as in the blends with the long linear PI 1.1M. Two star PI samples, PI (13k)6 and PI (18k)6, were synthesized in this study via living anionic polymerization in vacuum utilizing secbutyllithium and benzene as the initiator and solvent, respectively. After recovering an aliquot of the arm anion (linear precursor anion) for the arm characterization, the remaining majority of the arm anion was coupled with 1,2-bis(trichlorosilyl)ethane (bTCSE) to obtain the 6-arm star PI. The amount of bTCSE utilized in this coupling reaction was set to be 90% equimolar to the arm anions so as to ensure full 6-arm coupling for the star PI samples. The crude products, obtained after terminating the excess arm anions with methanol, were repeatedly fractionated from solutions in benzene/methanol mixed solvents to recover pure 6-arm star PI samples, PI (13k)6 and PI (18k)6. These pure samples were finally dissolved in benzene, and those benzene solutions were freeze-dried at room temperature for 24 h and then further 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 benzene solutions, and the dried sample was sealed in Ar and kept in a deep freezer until use. The PI (13k)6 and PI (18k)6 samples were characterized with GPC (CO-8020 and DP-8020; Tosoh) equipped with a refractive index (RI) monitor (RI-8020, Tosoh) and a low-angle laser light scattering (LALLS) detector (Viscotek 270, Malvern) connected in series. The elution solvent was tetrahydrofuran (THF), and monodisperse linear PI samples synthesized/characterized in our previous studies14,18 were utilized as the RI/LALLS standards as well as the elution volume standards. The weight-average molecular weight Mw and polydispersity index Mw/Mn of the PI (13k)6 and PI (18k)6 samples were calculated from the LALLS/RI signal ratio measured at respective sections of elution volume. A slightly larger polydispersity index was obtained from the elution volume calibration, and an average of the indices obtained from this calibration and the LALLS/RI signal ratio is shown in Table 1. For the linear precursor (arm), Table 1 shows the Mw and Mw/Mn values obtained from the elution volume calibration. The other two 6-arm star-PI samples, PI (9k)6 and PI (24k)6, were similarly synthesized, purified, and characterized in previous studies.15,23 The high molecular weight linear PI sample, PI 1.1M, was synthesized anionically and characterized with LALLS-GPC in a recent study.9 Blends of one of the star PI samples (probe; volume fraction υ1 = 0.1) and the linear PI 1.1M sample (matrix; υ2 = 0.9) were subjected to

⎤2 1 ⎡ a′(t ) − 1⎥ ⎢ ⎦ 8Na ⎣ a

1 [{β(t )}1/2 − 1]2 8Na

EXPERIMENTAL SECTION

(4)

Here, Na is the number of entanglement segments per arm, a is the diameter of undilated tube, and a′(t) is the diameter of laterally and partially dilated tube. (Φ(t) of star PI is equivalent to the autocorrelation function of the end-to-branching point vector of the arm, and analysis of the Gaussian arm conformation gave eq 4.15) Now, we note an important question for star PI. Previous experiments15,22 established that monodisperse star PI obeys the lateral partial-DTD relationship explained above (with β(t) in eq 3 agreeing with βCR(t) in the dominant range of time), not the lateral full-DTD relationship μ(t) = {φ′(t)}1+d deduced for the case of β(t) = {φ′(t)}−d. However, this experimental fact does not necessarily mean that the star chain motion (arm retraction) occurs along the laterally and partially dilated tube, as judged from the behavior of linear PI chains9 explained in the previous section. Focusing on this problem, we have conducted dielectric and viscoelastic experiments for blends of star PI (probe) and much longer linear PI (matrix) and analyzed the data in a way similar to that for the previously examined linear PI blends. It turned out that the star PI in monodisperse bulk has two coarse-grained length scales, a′(t) = {β(t)}1/2a and a*(t) = {β*(t)}1/2a for the lateral and longitudinal partial-DTD processes: The arm C


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dielectric and linear viscoelastic measurements. These blends were prepared by dissolving prescribed masses of the components in benzene at a concentration ∼5 wt % and then allowing benzene to thoroughly evaporate at 50 °C in high vacuum for 48 h. Freshly baked glass containers were utilized in the blend preparation to reduce the content of ionic impurities through adsorption on the fresh glass wall.24 For the sample concentration in the solution, ∼5 wt %, we recovered the dried blend films of small thickness 0.1 s−1, ε2,m″(ω) = 4.0 × 10−3ω−0.25.

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RESULTS AND DISCUSSION Overview of Data. For the blends of star PI probe (volume fraction υ1 = 0.1) and the linear PI 1.1M matrix (υ2 = 0.9), Figure 1 shows the angular frequency (ω) dependence of the dielectric loss ε″(ω) at 40 °C (large red circles). For comparison, the ε″(ω) data of the components in respective monodisperse bulk are also shown. The storage and loss moduli, G′(ω) and G″(ω), of those blends and components are shown in Figures 2 and 3, respectively. At first, a comment needs to be made for the ε″ data (Figure 1). Although ionic impurities were largely removed with the method (adsorption on a fresh glass wall24) explained in the Experimental Section, the direct current (dc) conduction due to those impurities began to significantly contribute to the ε″(ω) data of the blends and high-M linear PI matrix at low ωaT < 0.1 s−1, as similar to the previous observation.9 This study does not utilize such low-ω data (not shown in Figure 1), following the strategy9,24 of just focusing on the raw ε″ data at high ωaT > 0.1 s−1 for which no subtraction of the dc contribution is needed. This exclusive use of the high-ω raw data did not disturb our analysis, as explained below. In Figure 1, the dashed curve shows expectation for ε″ of bulk PI 1.1M (matrix) reported in the previous study.9 This expected ε″ curve was obtained by shifting the ε″ data of a linear PI 179k sample (M = 17.9 × 104) along the ω-axis by a factor of (179/1120)3.5 (= relaxation time ratio for bulk PI 179k and PI

1.1M samples). The data for bulk PI 1.1M (black dots) available at ωaT > 0.1 s−1 agree with this expected curve and exhibit the well-known power-law behavior at high ω, ε″ ∝ ω−1/4. Furthermore, the ε″ data of the blend become very close to the data of PI 1.1M on a decrease of ω within our experimental window, which indicates that the probe relaxation of our interest occurred in that window. Thus, the exclusive use of high-ω raw data still allowed us to analyze the probe relaxation behavior satisfactorily, as shown later in detail. The star PI samples have the arm molecular weight Ma between 1.9Me and 4.7Me (Me = 5 × 103 for PI)1 and are moderately entangled in respective monodisperse bulk as well as in the blends. Previous studies8,15,22 revealed that the normalized viscoelastic relaxation function μ(t) and the survival fraction of the dilated tube φ′(t) of monodisperse star PI obey the lateral partial-DTD relationship, eqs 1 and 3 combined with eq 4, but not the lateral full-DTD relationship, μ(t) = {φ′(t)}1+d. The validity and failure of these relationships were confirmed also for the star PI samples utilized in this study (cf. Figure S1 in Supporting Information). More importantly, the star PI probes in the blends are dilute (volume fraction υ1 = 0.1), but their weak relaxation is still D


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Figure 2. Storage modulus data of star PI/linear PI 1.1M blends and component PI samples at 40 °C. The volume fraction of star PI (probe) in the blends is υ1 = 0.1. Thick green curves indicate the shifted data of bulk PI 1.1M.

Figure 3. Loss modulus data of star PI/linear PI 1.1M blends and component PI samples at 40 °C. The volume fraction of the star PI (probe) in the blends is υ1 = 0.1. Thick green curves indicate the shifted data of monodisperse PI 1.1M.

detected in Figures 1 and 3 as the ε″ and G″ data of the blends (red circles) being larger than the data of pure matrix at ωaT > 1 s−1. The storage modulus G′ (Figure 2) is insensitive to such weak and fast relaxation, as well known from its expression with respect to the relaxation spectrum.9 Thus, in the remaining part of this paper, we mainly focus on the ε″ and G″ data to discuss the relaxation of the star PI probe in the blends. It should be noted that the star PI probes exhibit a clear ε″peak in bulk but just a very broad shoulder in the blends at ωaT > 1 s−1 (compare blue triangles and red circles in Figure 1). This result suggests that the relaxation of star PI is significantly broadened (and retarded) on blending, which makes a contrast from the behavior of linear PI probe: The dielectric mode distribution of linear PI probe is insensitive to blending (see Figures 1 and 5 of ref 9). The broadening/retardation for the star probe is examined later more quantitatively for the probe data extracted from the blend data. Evaluation of Data for Star Probe in Blends. In general, the electrical polarization and orientational anisotropy of polymeric blends, the basic quantities underlying the εb″(ω) and Gb″(ω) data, are additively contributed from the components

therein.4,17 Consequently, the εb″(ω) and Gb″(ω) data of the blends, with the subscript “b” hereafter representing quantities in the blend, can be expressed as9,14,17 εb″(ω) = υ1ε1,b″(ω) + υ2ε2,b″(ω)

(5)

G b″(ω) = υ1G1,b″(ω) + υ2G2,b″(ω)

(6)

In eqs 5 and 6, εi,b″(ω) and Gi,b″(ω) are defined as the dielectric and mechanical losses of the component i in the blend normalized by respective volume fractions υi (i = 1 and 2 for the probe and matrix), and the terms υiεi,b″(ω) and υiGi,b″(ω) represent actual contribution of that component to the blend data. The relaxation intensities of those normalized quantities, εi,b″(ω) and Gi,b″(ω), are independent of υi and Mi and coincide with those in the monodisperse bulk.9 εi,b″(ω) and Gi,b″(ω) 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 bulk (with the subscript “m” standing for monodisperse bulk). Nevertheless, we can still evaluate empirically ε1,b″(ω) and G1,b″(ω) of the probe in the E


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For this case, the probe negligibly contributes to Gb*(ω) of the blend at sufficiently low ω, and G2,b*(ω) at such low ω can be related to G2,m*(ω) as9

blend on the basis of eqs 5 and 6 with satisfactory accuracy, as explained previously9 in detail. A brief summary of this evaluation method is given below. Evaluation of ε1,b″(ω). The CR/DTD mechanism for the long linear matrix (component 2), being enhanced on blending with the faster probe (component 1), hardly changes the dielectric mode distribution of the matrix.9,14,17 For this reason, ε2,b″(ω) appearing in eq 5 has the mode distribution close to that of ε2,m″(ω), as also noted previously (see, for example, Figures 4 and 19 of ref 14a and also Supporting Information of ref 9). Consequently, we may safely replace ε2,b″(ω) in eq 5 [ε] [ε] [ε] [ε] [ε] by ε2,m″(λ[ε] 2 ω), where λ2 = τ2,b/τ2,m with τ2,b and τ2,m being the dielectric relaxation time of the matrix in the blend and in its monodisperse bulk, respectively. Then, ε1,b″(ω) of the probe in the blend, our target quantity, can be evaluated from the εb″ and ε2,m″ data 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 (8)

Here, I2 represents a fraction of the viscoelastic intensity of the long matrix that relaxes through the slow step, and λ[G] 2 [G] (= τ[G] 2,b /τ2,m) is the viscoelastic acceleration factor of the matrix on blending. Equation 8 indicates that the G2,m*(ω) data of the matrix in its monodisperse bulk (black dots in Figures 2 and 3), being shifted horizontally and vertically in the double-logarithmic scale by appropriate factors λ[G] 2 and υ2I2, can be superposed on the Gb* data of the blend at low ω. In fact, this superposition was satisfactorily achieved, as shown in Figures 2 and 3 with the green curves (shifted G2,m* curves). The factors I2 and λ[G] 2 appearing in eq 8 (common for G2,m′ and G2,m″) were obtained from those shift factors. I2 and λ[G] 2 thus obtained were numerically close to the factors υ2d and υ21.5d (d = 1.3) expected for the case that the short probe behaves as a solvent for the long matrix at long times. A comment needs to be added for this solution-like behavior of the long linear matrix chains. Those chains in blends do not always behave as in a real solution:20 For example, reptation of the matrix in the blends is slower than that in a real low-M solvent by a factor representing a time necessary for the CR equilibration of the matrix chain tension activated by motion of the short chain component in the blend. This factor becomes considerably large on a decrease of the matrix volume fraction υ2 (below 0.5)20 but is negligible for the PI 1.1M matrix chains having large υ2 (= 0.9) in our blends. For this reason, I2 and λ[G] 2 for the PI 1.1M matrix were close to υ2d and υ21.5d, as explained above. From the fraction I2 of the viscoelastic intensity of the matrix relaxing through the slow step, the fraction for the fast step, 1 − I2, is straightforwardly obtained. This fast-step relaxation of the matrix is attributable to the lateral DTD activated by the probe motion and thus proceeds together with the full relaxation of the probe. We may assume rather safely that the relative relaxation mode distribution of the matrix for the fast step is close to that of the probe because the tube for the matrix should dilate laterally with time to the same extent as that for the probe to activate nearly the same type of relaxation for the matrix and probe. Then, we can approximate the modulus for the fast step as υ1G1,b″(ω) + υ2(1 − I2)G1,b″(ω) (= (1 − I2υ2)G1,b″(ω)), with the first and second terms showing the contributions from the probe and matrix, respectively. (Note that the intensity fraction 1 − I2 is included in the matrix contribution.) This fast-step modulus should be obtained by subtracting the matrix modulus for the slow step, υ2G2,b″(ω) = I2υ2G2,m″(λ[G] 2 ω) (cf. eq 8), from the Gb″(ω) data of the blend at high ω where the fast-step relaxation occurs. Thus, G1,b″(ω) of the probe in the blend is experimentally evaluated as9,25

(7)

For our blends, the star probe is dilute (υ1 = 0.1) so that the dielectric acceleration factor for the matrix, λ[ε] 2 , is close to unity in any case. Thus, eq 7 allowed us to evaluate ε1,b″(ω) of the probe in the blend with satisfactory accuracy, as explained in more detail later for Figure 4.

Figure 4. Evaluation of ε1,b″(ω) and G1,b″(ω) of the PI (18k)6 star probe in the blend. The solid line in (a) and solid curve in (b) respectively indicate the empirical equations describing the ε2,m″ and G2,m″ data (black dots) of bulk PI 1.1M at ω > 0.1 s−1; ε2,m″(ω) = 4.0 × 10−3ω−0.25 [R] [e] 4 −0.22 and G2,m″(ω) = G[e] 2,m″(ω) + G2,m″(ω) with G2,m″(ω) = 4.7 × 10 ω 1 0.67 (Pa) and G[R] ″(ω) = 8.3 × 10 ω (Pa). These empirical equations are 2,m utilized in eqs 7 and 9 to evaluate ε1,b″ and G1,b″ of the star probe in the blends.

Evaluation of G1,b″(ω). The DTD mechanism decreases the orientational anisotropy/stress of the entanglement segments.11−15 In general, the long matrix in the blend exhibits two-step relaxation due to this DTD effect, and the fast-step relaxation occurs together with the full relaxation of the short probe. The relaxation time and intensity of the slow step are affected by the fast step, so that G2,b*(ω) (= G2,b′(ω) + iG2,b″(ω)) of the long matrix in the blend cannot be simply expressed in terms of G2,m*(ω) in the entire range of ω. Nevertheless, our blends have small υ1 (= 0.1) and widely separated relaxation times of the matrix and probe (cf. Figure 3).

G1,b″(ω) ≅ (1 − I2υ2)−1{G b″(ω) − I2υ2G2,m″(λ 2[G]ω)} at ω > 0.1 s−1

(9)

λ[G] 2

A comment needs to be made for eq 9. The factor appearing in eq 9 represents the acceleration of the terminal relaxation of the matrix after completion of the fast step (full relaxation of star probe and partial relaxation of the matrix). The use of this λ[G] 2 in eq 9 overestimates the contribution of the slow-step relaxation to the Gb″(ω) data of the blend during the fast step, thereby underestimating G1,b″(ω), as fully discussed previously.9 F


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Article [G] [R] by I2υ2G[e] 2,m″(λ2 ω) + υ2G2,m″(ω) to evaluate (1 − I2υ2)G1,b″(ω). As an example, Figure 4b shows the result of this evaluation for the PI (18k)6 star probe. The small filled and large unfilled red circles indicate lower and upper bounds explained for eq 9. Good agreement is noted for the upper and lower bounds, again because λ[G] 2 is close to unity in any case. This result suggests that G1,b″(ω) is also satisfactorily evaluated (although eq 9 utilized in this evaluation is based on the assumption that the fast relaxation of the matrix and probe proceed simultaneously to have the same viscoelastic mode distribution). G1,b″(ω) was similarly obtained also for the other probes in the blends examined. For the four star PI probes, Figure 5 compares the dielectric [G] ε1,b″ in the blends (obtained from eq 7 with λ[ε] 2 = λ2 ; unfilled

In contrast, if we artificially assume that the slow process is not = 1 in eq 9, we accelerated by the probe motion to set λ[G] 2 overestimate G1,b″(ω). Thus, eq 9 just specifies the lower or upper bound for G1,b″(ω) if the factor λ[G] therein is experimentally 2 determined from the Gb*(ω) and G2,m*(ω) data at low ω (Figures 2 and 3) or set to be unity, respectively. Nevertheless, we were able to evaluate G1,b″(ω) satisfactorily because the upper and lower bounds were numerically close to each other, as explained in more detail later for Figure 4. Comparison of Probe Data in Blends and Monodisperse Bulk. Evaluation of ε1,b″(ω) of the star probe in the blends requires us to know the dielectric acceleration factor for [ε] [ε] [ε] the matrix, λ[ε] 2 (= τ2,b /τ2,m) appearing in eq 7. In principle, λ2 should be evaluated from the terminal dielectric relaxation data of the matrix in the blend and monodisperse bulk measured at very low ω < 10−2 s−1 (cf. Figure 1). However, the dielectric data at such low ω, being contributed from the dc conduction as explained earlier, are not utilized in this study. Nevertheless, we can utilize the viscoelastic λ[G] 2 (giving the superposition shown with the thick green curves in Figures 2 and 3) as the dielectric λ[ε] 2 because the terminal relaxation mechanism (reptation/CLF) of the long linear matrix in the blend with small υ1 (= 0.1) should be very similar to that in monodisperse bulk. Then, the dielectric to viscoelastic relaxation time ratio should be common in the [G] [ε] [G] [ε] [G] blend and bulk (τ[ε] (see 2,b /τ2,b = τ2,m/τ2,m) to give λ2 = λ2 sections 3.2.1 and 3.4 of ref 9 for further details). Thus, we utilized λ[G] 2 in eq 7 to evaluate υ1ε1,b″(ω) = εb″(ω) − υ2ε2,m″(λ[G] 2 ω). The subtraction in this evaluation was made with the aid of an empirical equation, ε2,m″(ω) = 4.0 × 10−3ω−0.25 (black solid line in Figure 1), that well describes the ε2,m″(ω) data of bulk matrix at ω > 0.1 s−1 where the star probe fully relaxes. As an example, Figure 4a shows υ1ε1,b″(ω) thus evaluated for the PI (18k)6 star probe in the blend (cf. small filled red circles). The subtraction εb″(ω) − υ2ε2,m″(λ[G] 2 ω) results in a large numerical uncertainty if εb″(ω) and υ2ε2,m″(λ[G] 2 ω) are too close to each other. For this reason, we limited ourselves to make the subtraction and evaluate υ1ε1,b″(ω) only in a range of ω where εb″(ω) ≥ 1.3υ2ε2,m″(λ[G] 2 ω), with the factor of 1.3 being chosen according to the maximum uncertainty of the εb″(ω) data (30%) mentioned earlier. [G] The υ1ε1,b″(ω) obtained from eq 7 with λ[ε] cor2 = λ2 responds, as a matter of form, to the lower bound of the viscoelastic G1,b″(ω) obtained from eq 9 utilizing the same λ[G] 2 factor. For completeness, we artificially set λ[ε] 2 = 1 in eq 7 to evaluate υ1ε1,b″(ω) as a hypothetical upper bound, although eq 7 [G] with λ[ε] 2 = λ2 has the sound basis to correctly give υ1ε1,b″(ω), as explained earlier. This hypothetical upper bound is shown in Figure 4a with large unfilled red circles. Close agreement between the two sets of υ1ε1,b″(ω) (filled and unfilled circles) confirms high accuracy of υ1ε1,b″(ω) obtained from eq 7 with [G] λ[ε] 2 = λ2 . Similar results were obtained also for the other blends examined. For evaluation of G1,b″(ω) of the probe (eq 9), we adopted an empirical equation that well describes the G2,m″(ω) data at ω > 0.1 s−1 (where the star probe fully relaxes), G2,m″(ω) = [R] [e] 4 −0.22 (Pa) G[e] 2,m″(ω) + G2,m″(ω) with G2,m″(ω) = 4.7 × 10 ω 26 [R] 1 0.67 (Pa) (see solid curve in and G2,m″(ω) = 8.3 × 10 ω [R] Figure 4b). The G[e] 2,m″(ω) and G2,m″(ω) terms represent the entanglement relaxation and the local Rouse-like relaxation, respectively. (The local relaxation term is absent in the empirical equation for ε2,m″(ω) because the dielectric relaxation exclusively detects the global motion.) The local relaxation does not change on blending, so that we replaced the I2υ2G2,m″(λ[G] 2 ω) term in eq 9

Figure 5. Comparison of ε1,b″ (large unfilled circles) and ε1,m″ (small triangles) of the star PI probes in the blends and in monodisperse bulk at 40 °C.

circles) with the ε1,m″ data in monodisperse bulk (small triangles). Figure 6 compares the viscoelastic G1,b″ (eq 9 with λ[G] and I2 being determined from the shift/superposition in 2 Figures 2 and 3) with the G1,m″ data. Remarkable broadening and retardation on blending (on quenching CR/DTD) are noted for both dielectric and viscoelastic relaxation, confirming that the star dynamics in monodisperse bulk is significantly contributed from the CR/DTD mechanism. It should be noted that these features of the star PI probe are totally different from those noted previously for the linear PI probes: The retardation on blending is much less significant for the linear PI probes9 (as shown more quantitatively later in Figures 8 and 9). Furthermore, for the linear PI probes, the dielectric relaxation mode distribution negligibly changes whereas the viscoelastic mode distribution narrows (not broadens) G


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Figure 6. Comparison of G1,b″ (large unfilled circles) and G1,m″ (small triangles) of the star PI probes in the blends and in monodisperse bulk at 40 °C.

Figure 7. Comparison of G1,b″ (green squares) and ε1,b″ (red circles) of the star PI probes in the blends at 40 °C. The ε1,b″ data are multiplied by appropriate factors A to achieve the best superposition on the G1,b″ data. The solid curves indicate ε1,b″ recalculated from spectrum analysis, and the arrows represent the corresponding terminal relaxation frequency. For further details, see text.

on blending (see Figures 5 and 6 in ref 9). These differences reflect a difference in the dominant relaxation mechanisms reptation for the linear probes and the arm retraction for the star probes. The viscoelastic G1,b″ and dielectric ε1,b″ data of the probes in the blends (Figures 5 and 6) are directly compared with each other in Figure 7, where ε1,b″ (red circles) is multiplied by an appropriate factor A to achieve the best superposition on the G1,b″ data (green squares). The solid curves indicate results of the spectrum analysis explained later. Good superposition of G1,b″ and Aε1,b″ is noted for all probes within the scatter of data, except at high ω where the local Rouse relaxation contributes to G1,b″ but not to ε1,b″. The normalized dielectric and viscoelastic relaxation functions agree with each other in the absence of the CR/DTD effect, irrespective of the details of the relaxation mechanism.4,8 Thus, the superposition seen in Figure 7 strongly suggests that the CR/DTD mechanism for the star probe is quenched in the much longer matrix: In our blends, the matrix [G] and probe have the relaxation time ratio τ[G] 2,b /τ1,b > 300 (and this ratio increases with decreasing Ma of the probe). This range of [G] τ[G] 2,b /τ1,b for the absence of the CR/DTD effect on the star probe relaxation is not identical but close to that noted previously [G] for the linear PI probe,9 τ[G] 2,b /τ1,b ≥ 700. Keeping this lack of the CR/DTD effect on the star probe relaxation in the blends in our mind, we further analyze the probe relaxation behavior in the remaining part of this paper.

Relaxation Time of Star Probe. For linear PI probes in the blends examined previously,9 G1,b″ and ε1,b″ exhibited sharp peaks immediately followed by respective low-ω power-law asymptotes (G1,b″ ∝ ω and ε1,b″ ∝ ω). For those linear probes, the viscoelastic and dielectric terminal relaxation times were successfully evaluated as reciprocal of the angular frequencies ωpeak for those G1,b″ and ε1,b″ peaks. However, this method cannot apply to the star PI probes in the blends because the peaks for those probes are quite broad and the low-ω asymptotes emerges at ω ≪ ωpeak (see Figure 7). Thus, in this study, we adopt the following method based on the rigid phenomenological framework of linear viscoelastic and dielectric relaxation. Within this framework, the second-moment average relaxation times (often utilized as the terminal relaxation time) are defined [X] 2 [X] [X] as ⟨τ[X]⟩w ≡ ∑ph[X] p {τp } /∑php τp with X = G and ε for the viscoelastic and dielectric times.4,17,27,28 Here, h[X] and τp[X] p denote the intensity and characteristic time of pth viscoelastic and/or dielectric relaxation mode. Experimentally, the viscoelastic ⟨τ[G]⟩w is straightforwardly evaluated from the low-ω asymptotes of G′ (∝ ω2) and G″ (∝ ω), and the dielectric ⟨τ[ε]⟩w, from the asymptotes of the decrease of the dynamic dielectric constant Δε′(ω) ≡ ε′(0) − ε′(ω) (∝ ω2) and the dielectric loss ε″(ω) (∝ ω).17 Indeed, these low-ω asymptotes were well detected for the star PI samples in bulk (cf. Figures 1−3; the H


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Δε′(ω) data are not shown but were similar to those previously reported for the other star PI samples8,29,30), and their ⟨τ[X] 1,m⟩w were obtained as ⎡ G1,m′ ⎤ [G] ⎥ ⟨τ1,m ⟩w = ⎢ , ⎢⎣ ωG1,m″ ⎥⎦ ω→0

⎡ Δε1,m′ ⎤ [ε] ⎥ ⟨τ1,m ⟩w = ⎢ ⎢⎣ ωε1,m″ ⎥⎦ ω→0 (10)

Unfortunately, this phenomenological method cannot directly apply to the star PI probe in the blend because the G′(ω) (and Δε′(ω)) data of the blend are insensitive to fast and weak relaxation of the probe as explained earlier for Figure 2, and the subtraction explained for eqs 7 and 9 did not give G1,b′ (and Δε1,b′) of the probe with acceptable accuracy. Thus, in this study, we adopted a method that is still based on the rigid framework of linear relaxation but utilizes just the G1,b″ and ε1,b″ data shown in Figure 7. Namely, we utilized the iteration method explained previously29 to calculate the viscoelastic and dielectric relaxation [X] spectra {h[X] p , τp } (with X = G and ε) from those data and evaluated the second-moment average relaxation time according [X] 2 [X] [X] to its definition, ⟨τ[X]⟩w ≡ ∑ph[X] p {τp } /∑php τp . To complete the spectrum calculation, we need to have the data down to the low-ω asymptotic regime where G1,b″ ∝ ω and ε1,b″ ∝ ω. As noted in Figure 7, we have the G1,b″ data down to this asymptotic regime whereas the ε1,b″ data are not available. However, these two sets of data exhibit the same ω dependence at low ω. On the basis of this observation, we assumed the proportionality [ε] between the dielectric and viscoelastic intensities, h[G] p and hp , [ε] [G] and coincidence of the dielectric τp and viscoelastic τp for loworder relaxation modes, thereby compensating the lack of the ε1,b″ data in the low-ω asymptotic regime. In Figure 7, the ε1,b″ recalculated from the dielectric spectrum {hp[ε], τ[ε] p } thus [ε] obtained with the aid of the G1,b″ data, ε1,b″ = ∑ph[ε] p ωτp / [ε] 2 [1 + {ωτp } ], is shown with the solid curves. These recalculated curves well describe the ε1,b″ data, suggesting that the spectrum was obtained satisfactorily. In fact, the terminal dielectric relaxation frequency 1/⟨τ[ε] 1,b ⟩w (that was indistinguishable from the viscoelastic 1/⟨τ[G] 1,b ⟩w), shown with the arrows, well specifies the onset of the low-ω asymptotic regime. We further checked the reliability of our method based on the spectrum calculation by applying this method to the PI star [ε] samples in bulk to evaluate their ⟨τ[G] 1,m⟩w and ⟨τ1,m⟩w. The results agreed well (within ∼10%) with those directly obtained from the G1,m* and ε1,m* data (eq 10). However, for the star PI probe in [G] the blends, the uncertainty in ⟨τ[ε] 1,b ⟩w and ⟨τ1,b ⟩w was larger because of the scatter of the ε1,b″ and G1,b″ data (cf. Figure 7). [X] Moderate changes of {h[X] p , τp } still described those data within the scatter, and those changes were equivalent to ∼40% changes [G] (uncertainty) of ⟨τ[ε] 1,b ⟩w and ⟨τ1,b ⟩w of the probe in the blend. Nevertheless, this uncertainty was considerably smaller than the order of magnitude increase of the probe relaxation time on blending, in particular for the probe with large Ma. Thus, our method allowed quantitative discussion of the retardation of the probe relaxation on blending. The second-moment average relaxation time is the average heavily weighing on slow modes and is close to the longest relaxation time.4,17,27 Thus, ratios of those times in the blend and monodisperse bulk can be utilized as the ratios of the longest [ε] [ε] [G] [G] relaxation times, τ1,b /τ1,m and τ1,b /τ1,m , that quantify the retardation of the dielectric and viscoelastic relaxation of the star PI probe on blending (on quenching the CR/DTD effect). In Figure 8, these ratios are plotted against the number of entanglement segments per star arm, Na = Ma/Me (see circles).

[ε] Figure 8. Changes of dielectric τ[ε] 1,b /τ1,m ratio (panel a) and visco[G] [G] elastic τ1,b /τ1,m ratio (panel b) for star PI probe with the number of entanglements per star arm, Ma/Me. The curves indicate results of simple DTD analyses. For further details, see text.

[ε] Figure 9. Changes of (a) dielectric τ[ε] 1,b /τ1,m ratio and (b) viscoelastic [G] [G] τ1,b /τ1,m ratio for linear probe with the number of entanglement per probe, M/Me. The data for M/Me > 4 and M/Me < 3 were obtained in refs 9 and 31, respectively. Thick red curves indicate the ratios deduced from the molecular picture of longitudinal partial-DTD.9

[ε] [G] [G] For comparison, Figure 9 shows the τ[ε] 1,b /τ1,m and τ1,b /τ1,m ratios 9,31 of linear PI probe obtained in the previous studies. The curves in Figure 8 and the thick curve in Figure 9 indicate results of simple DTD analyses explained later. (For convenience of future

I


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data. Thus, in the followings, we first focus on the Ball−McLeish full-DTD model12 and then borrow its method of formulating the relaxation time to analyze those data. Ball−McLeish Model.12 The Ball−McLeish (BM) model starts from the harmonic entropic barrier F(z) for the arm retraction over the contour distance z along the nondilated (skinny) tube, as illustrated in Figure 10. An increment dF(z) of

[X] Table 2. τ[X] 1,b /τ1,m Ratios for the Star Probe in the Blend and Monodisperse Bulk

10−3Ma

9.5

13.2

4.8 lateral partial-DTD longitudinal partial-DTD

2.8 1.9

[G] τ[G] 1,b /τ1,m data 7.9 17.4 [G] calculated τ[G] 1,b /τ1,m 9.6 31.2 5.1 12.5

viscoelastic 5.6 lateral partial-DTD longitudinal partial-DTD

17.1

[ε] τ[ε] 1,b /τ1,m data 6.8 13.8 [ε] calculated τ[ε] 1,b /τ1,m 8.4 29.1 4.5 11.7

dielectric

3.6 2.4

23.5 50.1 225 65.8

64.6 261 76.3

[X] Table 3. τ[X] 1,b /τ1,m Ratios for the Linear Probe in the Blend and Monodisperse Bulk

10−3M

7.8a

dielectric 1.15 − viscoelastic

14.4a

21.4b

43.2b

98.5b

179b

[ε] τ[ε] 1,b /τ1,m data 1.62 1.78 1.58 1.35 1.23 [ε] calculated τ[ε] 1,b /τ1,m (longitudinal partial-DTD) − 2.05 1.57 1.29 1.18

[G] τ[G] 1,b /τ1,m data 1.15 2.75 3.71 3.23 2.63 2.24 [G] calculated τ[G] 1,b /τ1,m (longitudinal partial-DTD) − − 4.09 3.13 2.57 2.36

a [X] Examined in ref 31 (no DTD calculation of the τ[X] 1,b /τ1,m ratios for short linear PI with 10−3M ≤ 14.4). bExamined in ref 9.

[X] analysis, the τ[X] 1,b /τ1,m data and the partial-DTD calculation shown in Figures 8 and 9 are summarized in Tables 2 and 3, respectively.) Comparison of Figures 8 and 9 demonstrates clear differences between the star and linear probes. First of all, the retardation on blending is much more significant for the star probe than for the linear probe (note the difference of the full scale of the vertical axis in Figures 8 and 9). Furthermore, the retardation becomes exponentially enhanced for the star probe with increasing Na > 2 whereas the retardation is suppressed for the linear probe with [X] increasing N (= M/Me) > 3. Figure 9 also shows that the τ[X] 1,b /τ1,m ratio for the linear probe decreases to unity when N approaches unity and the entanglement effect vanishes, as naturally expected. The star probe should also exhibit this decrease of the ratio as the entanglement number 2Na for two arms (the longest span in the star molecule) approaches unity,32 although the data do not cover such a low-Na regime. The differences between the star and linear probes (Figures 8 and 9) suggest that the CR/DTD effect on the star chain dynamics in monodisperse bulk is much stronger than that for the linear chain and is enhanced exponentially with increasing Na. This feature of the star chain dynamics is further analyzed below on the basis of the DTD concept. DTD Analysis. Because the CR/DTD mechanism for the star probe dynamics is quenched in the blends, the data of the [X] τ[X] 1,b /τ1,m ratio (Figure 8) serve as an experimental basis for analyzing the contribution of that mechanism to the star dynamics in monodisperse bulk. Although experiments8,15,22,29 have established that the full-DTD model assuming the arm retraction along the fully dilated tube is not valid for monodisperse star PI, it is informative to revisit the formulation of [X] this model to find a clue for the analysis of the τ[X] 1,b /τ1,m ratio

Figure 10. Schematic illustration of arm retraction process.

this barrier on an infinitesimal increase of z is expressed in terms of the number of entanglement segments per star arm, Na, and a normalized retraction distance, z/Leq, with Leq being the arm contour length (= length of the skinny tube) at equilibrium, as ⎧ z ⎫ ⎧ z ⎫ d F (z ) = 2νGNa ⎨ ⎬ d⎨ ⎬ kBT ⎩ Leq ⎭ ⎩ Leq ⎭ ⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

with vG = 15/8 (11)

Here, kB and T denote the Boltzmann constant and absolute temperature, respectively, and νG (= 15/8) is the intrinsic coefficient for the Gaussian arm. The BM model assumes that the relaxed portion of the arm behaves as a solvent for the unrelaxed portion (full-DTD assumption) to replace Na appearing in eq 11 by ⎧ z ⎫ ⎬ Na′ = Naφ′ = Na ⎨1 − Leq ⎭ ⎩ ⎪

⎪

⎪

⎪

(12)

Here, φ′ = 1 − z/Leq is the survival fraction of the dilated tube on the arm retraction over the distance z. (The BM model adopts the dilation exponent d = 1.12) Utilizing the corresponding barrier increment, dFBM(z)/kBT = 2vGNa′{z/Leq}d{z/Leq} = 2vGNa{1 − z/Leq}{z/Leq}d{z/Leq}, the BM model relates the J


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relaxation time τ(z + dz) of the arm segment at z + dz to τ(z) at z as ⎛ d F (z ) ⎞ τ(z + dz) = τ(z) exp⎜ BM ⎟ ⎝ kBT ⎠

calculate the relaxation time (cf. eq 13). The factor {z/Leq} involved in this increment is equivalent to the nonsurvival f raction of the dilated tube, θ(t) = 1 − φ′(t). Thus, in our partial-DTD analysis, we consider eq 14 to express the increment of arm retraction barrier in terms of the known β(t) and θ(t) data (shown later in Figure 12) as

(13a)

or, equivalently, d{ln τ(z)} 1 dFBM(z) = dz kBT dz

dFp ‐ DTD kBT (13b)

Na β (t )

Na θ (t ) d θ (t ) β (t )

with vG = 15/8

(15)

Integrating this dFp‑DTD with the aid of those data, we obtain the dielectric relaxation time of the star PI in bulk as τ[ε] 1,m = τ0 exp{2vGNa∫ 10(θ/β) dθ}, where τ0 is a reference relaxation time defined for an entanglement segment. The dielectric relaxation time of the star PI in the blend is calculated as the relaxation time in the absence of DTD, τ[ε] 1,b = τ0 exp(vGNa). Thus, the dielectric relaxation time ratio deduced from our partial-DTD analysis is summarized as

Equation 13 can be straightforwardly integrated to give the dielectric relaxation time of the star probe in its monodisperse bulk, τ[ε] 1,m = τ(Leq) = τ0 exp(vGNa/3) with τ0 being a reference relaxation time defined for an entanglement segment. In the high-M linear matrix, the probe relaxation is not contributed from the DTD mechanism, and its dielectric relaxation time is expressed as τ[ε] 1,b = τ0 exp(vGNa). Thus, the BM model gives [ε] [G] τ[ε] /τ = exp(2ν 1,b 1,m GNa/3) = exp(5Na/4). The viscoelastic τ1,m [ε] in monodisperse bulk is close to τ1,m/2 if the full-DTD [ε] assumption is valid, whereas τ[G] 1,b = τ1,b in those blends. Thus, [G] [ε] [ε] a relationship τ[G] /τ ≅ 2τ /τ = 2 exp(5Na/4) is deduced 1,b 1,m 1,b 1,m from the BM model. (A moderately different relationship, 12 [G] τ[G] 1,b /τ1,m = exp(5Na/4), was obtained in the original BM paper focusing only on the viscoelastic relaxation. However, the above [ε] [G] [G] relationship τ[ε] 1,b /τ1,m = exp(5Na/4) ≅ τ1,b /2τ1,m seems to be more reasonable if we consider both dielectric and viscoelastic relaxation processes.) The dashed curves in Figures 8a and 8b respectively show the re[ε] [G] [G] lationships τ[ε] 1,b/τ1,m = exp(5Na/4) and τ1,b /τ1,m ≅ 2 exp(5Na/4), thus deduced from the full-DTD BM model. These curves [X] significantly overestimate the τ[X] 1,b /τ1,m ratio of the star PI probe, as naturally expected from the failure of the full-DTD relationship (μ(t) = {φ′(t)}1+d) for monodisperse star PI. Although not shown in Figure 8, the Milner−McLeish (MM) full-DTD [X] model13 also overestimated the τ[X] 1,b /τ1,m ratio of the star PI probe. The tube for star PI in monodisperse bulk dilates only partially, so that the reduction of the arm retraction barrier in actual bulk systems is less significant than considered in the BM [X] and MM models. For this reason, the τ[X] 1,b /τ1,m ratio deduced from those full-DTD models is larger than the data. (Note that a [G] considerable discrepancy still remains for the τ[G] 1,b /τ1,m ratio even if [G] [G] we utilize the original BM relationship, τ1,b /τ1,m = exp(5Na/4).) [X] [X] Method of Calculating τ1,b /τ1,m Ratio for Partial-DTD Picture. Despite the significant deviation of the BM model cal[X] culation from the τ[X] 1,b /τ1,m ratio data explained for Figure 8, the basic idea in the model is useful for our analysis based on the lateral and longitudinal partial-DTD pictures. The number of the entanglement segments per dilated segment, β, and the survival fraction of the dilated tube, φ′, play the key role in the analysis, as explained below. [ε] [ε] Dielectric τ1,b /τ1,m Ratio. The BM model replaces Na appearing in eq 11 by the number of dilated segments per star arm, Na′ = Naφ′, where φ′ (= 1 − z/Leq in the model) is equivalent to 1/β(t) under the full-DTD assumption adopted in the model (with the dilation exponent d = 1). Thus, in our partial-DTD analysis, Na in eq 11 is replaced by Na′(t ) =

= 2νG

[ε] τ1,b [ε] τ1,m

⎛ ⎧ = exp⎜νGNa ⎨1 − 2 ⎩ ⎝

∫0

1

⎫⎞ 1 θ dθ ⎬⎟ β ⎭⎠

(16)

[ε] Note that eq 16 gives τ[ε] 1,b /τ1,m = 1 if β = 1 in the entire range of θ (i.e., for a hypothetical case of no DTD for bulk star PI). The number β of the entanglement segments per dilated segment, included in the integral in eq 16, increases as the relaxation proceeds to increase the nonsurvival fraction θ from 0 to 1. However, β has a physically determined upper limit, βmax. Obviously, β cannot exceed Na (βmax = Na for this case). In addition, β should stay even at a smaller value if the (longitudinal) partial-DTD analysis gives such a small upper bound value, βmax < Na. Considering this feature, we rewrite eq 16 as

[ε] τ1,b [ε] τ1,m

⎛ ⎧ = exp⎜⎜νGNa ⎨1 − 2 ⎩ ⎝ ⎪

⎪

∫0

θmax

⎫⎞ 1 1 (1 − θmax 2)⎬⎟⎟ θ dθ − β βmax ⎭⎠ ⎪

⎪

with vG = 15/8

(17)

where θmax specifies the θ value where β increases to βmax. Thus, [ε] the τ[ε] 1,b /τ1,m ratio for the cases of lateral and/or longitudinal partial-DTD is calculated by eq 17 with the aid of the β(t) and θ(t) data (shown later in Figure 12). [G] [G] [G] Viscoelastic τ[G] 1,b/τ1,m Ratio. The viscoelastic τ1,b /τ1,m ratio can [ε] [ε] be estimated by combining the dielectric τ1,b /τ1,m ratio (eq 17) and the lateral partial-DTD relationship known to be valid for μ(t) data of bulk star PI, μ(t) = φ′(t)/β(t) (eq 1) with β(t) = min[βf‑DTD(t), βCR(t)] (eq 3a). For all bulk star PI samples examined in this study, βf‑DTD(t) (= {φ′(t)}−d) was found to be larger than βCR(t) (= 1/ψCR(t) with ψCR(t) being the normalized Rouse-CR decay function given later in eq 19); see Supporting Information. Thus, for those samples, β(t) = 1/ψCR(t) and μ(t) = [G] φ′(t)ψCR(t). Correspondingly, the viscoelastic τ[G] 1,b /τ1,m ratio can [ε] [ε] be estimated from the dielectric τ1,b /τ1,m ratio (eq 17), the known dielectric CR time τ[ε] CR (terminal relaxation time of ψCR(t)), and ] the known terminal relaxation time of φ′(t), τ[φ 1,m′ , as [G] τ1,b [G] τ1,m

(14)

] τ[φ 1,m′

=

[ε] τ1,b [G] τ1,m

[ε] ⎞ ⎛ τ [ε] ⎞⎛ τ [ε] τ1,m 1,b 1,m ≅ ⎜⎜ [ε] ⎟⎟⎜⎜ [φ ′] + [ε] ⎟⎟ τCR ⎠ ⎝ τ1,m ⎠⎝ τ1,m

(18)

τ[ε] 1,m

is 10−30% longer than the dielectric data for star PI samples examined (see Figure S2 in Supporting Information). In eq 18, we have considered the fact that the CR/DTD mechanism is quenched for the probe star PI in the blends (cf. Figure 7) to

(This Na′(t) does not depend on z explicitly.) We also note that the BM model integrates the increment of arm retraction barrier dFBM(z)/kBT = 2vGNa′{z/Leq}d{z/Leq} to K


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Article [G] evaluated as τ[ε] CR = 2τCR (a relationship valid for the Rouse-CR process). The maximum CR-equilibration number, βCR(t), is given by reciprocal of the normalized Rouse-CR decay function ψCR(t),8,14,15,22 and this ψCR(t) is fully specified by the dielectric τ[ε] CR evaluated as above. For the star arm, ψCR(t) (and βCR(t)) can be expressed in the tethered Rouse-CR form:15,22

[ε] replace the viscoelastic τ[G] 1,b in the blend by the dielectric τ1,b . Note also that eq 18 is based on an approximation that focuses just on competition of the slowest modes of φ′(t) and ψCR(t) for the relaxation of μ(t) = φ′(t)ψCR(t): This approximation gives [φ ] [ε] the relationship 1/τ[G] 1,m ≅ 1/τ1,m′ + 1/τCR. In the following sections, we first evaluate the β(t) and θ(t) [ε] data. Then, we utilize those data to calculate the τ[ε] 1,b /τ1,m and [G] [G] τ1,b /τ1,m ratios (eqs 17 and 18) for the cases of lateral and longitudinal partial-DTD. Finally, those ratios are compared with the data to discuss the arm retraction path in relation to the CRactivated tension equilibration of the star arm. Evaluation of β(t) and θ(t). In the partial-DTD picture, the number of entanglement segments equilibrated in a dilated segment, β(t), is specified as β(t) = min[βf‑DTD(t), βCR(t)] (eq 3a) with βf‑DTD(t) (= {φ′(t)}−d with d ≅ 1.3; eq 3b). Thus, evaluation of β(t) first requires us to know the maximum number βCR(t) of entanglement segments that can be CR-equilibrated in a given time scale. For this purpose, we need to evaluate the longest dielectric CR time of star PI in bulk, τ[ε] CR. The basic data necessary for this evaluation, the data of the second-moment average viscoelastic CR time ⟨τ[G] CR ⟩w of star PI probes in star/ linear and star/star blends, have been obtained in previous studies.15,22 ⟨τ[G] CR ⟩w of monodisperse bulk star PI, obtained from extrapolation of those probe data in blends, is summarized in Figure 11 (unfilled circles). Unfilled squares indicate the terminal

ψCR (t ) =

1 1 = βCR (t ) Na

Na

⎛

rpt ⎞ ⎟ [ε] ⎟ ⎝ τCR ⎠

∑ exp⎜⎜− p=1

(19a)

with ⎛ {2p − 1}π ⎞ −2⎛ ⎞ π rp = sin 2⎜ ⎟ sin ⎜ ⎟ ⎝ 2{2Na + 1} ⎠ ⎝ 2{2Na + 1} ⎠

(19b)

rp is the ratio of the longest dielectric CR time τ[ε] CR to the characteristic time of pth CR mode. Here, a comment needs to be added for eq 19a. Strictly speaking, ψCR(t) includes the Rouse-CR part and the intrinsic contour length fluctuation (CLF) part.15,22 However, for the four star PI probes examined in this study, the CR time is just moderately longer (by a factor ≤2) than the observed relaxation time, as noted in Figure 11. For this reason, the characteristic time of the slowest CLF mode was found to be similar to that of higher order CR modes (typically the second to fourth modes). Thus, for those star PI probes, we have included the CLF part into the Rouse-CR part, as a harmless approximation, to express ψCR(t) in the pure Rouse-CR form (eq 19a). The survival fraction of the dilated tube, φ′(t), is related to the dielectric Φ(t) data and β(t) (cf. eq 4), whereas β(t) = min[βf‑DTD(t), βCR(t)] with βf‑DTD(t) = {φ′(t)}−d (eq 3) is determined by φ′(t) and ψCR(t) (= 1/βCR(t)). Thus, we utilized ψCR(t) calculated from the known CR time τ[ε] CR (eq 19) and the Φ(t) data to determine φ′(t) and β(t) self-consistently, as did in previous studies.15,22 It turned out that β(t) for our star PI in bulk coincided with βCR(t) (= 1/ψCR(t)) in the entire range of time and that φ′(t) was very close to Φ(t), except at the long time end of the relaxation (see Figures S2 and S3 in Supporting Information). As an example, Figure 12 shows time evolution

Figure 11. Terminal viscoelastic relaxation time ⟨τ[G] m ⟩w and CR time ⟨τ[G] CR ⟩w of monodisperse bulk star PI at 40 °C (unfilled symbols). Arrows specify the four star PI samples utilized in this study. Filled circles indicate ⟨τ[G] CR ⟩w of two PI samples (in those four) estimated from short interpolation of the available data (unfilled circles). Data are taken from refs 15 and 22.

viscoelastic relaxation time ⟨τ[G] m ⟩w (defined by eq 10) measured [G] for bulk star PI. ⟨τ[G] CR ⟩w is only moderately longer than ⟨τm ⟩w even for the star PI having the highest Ma examined (Ma = 80.1 × 103, Na = 16), which confirms the importance of the CR/DTD mechanism for the dynamics of bulk star chains. The arrows in Figure 11 specify the four star PI samples utilized in this study, and the CR time ⟨τ[G] CR ⟩w is known for two of them. ⟨τ[G] CR ⟩w for the remaining two samples (filled circles) were estimated from short interpolation of the available ⟨τ[G] CR ⟩w data (unfilled circles). From ⟨τ[G] CR ⟩w of those four samples, the longest viscoelastic CR time τ[G] CR was evaluated after a minor correction of the tethered Rouse-CR mode distribution (cf. eq 19): τ[G] CR = [ε] (π2/8)⟨τ[G] CR ⟩w. Finally, the longest dielectric CR time τCR was

Figure 12. Time evolution of the normalized number of the entanglement segments per dilated segment, β(t)/Na, and the nonsurvival fraction of the dilated tube, θ(t), obtained for bulk star PI (18k)6 sample. θc indicates the θ value where β = Na, and θ* indicates the θ value where β attains the maximum value β* deduced from the longitudinal partial-DTD analysis. For further details, see text.

of β(t) and φ′(t) thus determined for the bulk star PI (18k)6 sample: The β(t)/Na ratio and the nonsurvival fraction of the L


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dilated tube, θ(t) = 1 − φ′(t), are plotted against t, for convenience of the calculation with eq 17. As t approaches the terminal viscoelastic relaxation time ⟨τ[G] m ⟩w, the β(t)/Na ratio increases with t more abruptly compared to θ(t). Test of Lateral Partial-DTD Picture. Utilizing the β(t) and θ(t) data obtained in the previous section, we numerically conducted the integral in eq 17 (by regarding β as a function of θ). For all PI star samples examined, β(t) attained its maximum value for the case of lateral partial-DTD, βmax = Na, as t approaches the terminal viscoelastic relaxation time ⟨τ[G] m ⟩w (see Figure 12 as an example). Thus, Na and the corresponding θc (≅ 1 − φ′(⟨τ[G] m ⟩w); cf. Figure 12) were incorporated in eq 17 as βmax and θmax, respectively. [ε] The dielectric τ[ε] 1,b /τ1,m ratio thus evaluated from eq 17 and the [G] corresponding viscoelastic τ[G] 1,b /τ1,m ratio (eq 18) are shown in Figures 8a and 8b with the thin solid curves. These curves are closer to the data compared to the full-DTD BM curves because the arm retraction barrier in bulk is less reduced in the lateral partial-DTD picture than in the full-DTD picture. However, the thin solid curves still overestimate the data for stars having large Na. This result strongly suggests that the lateral partial-DTD picture still overestimates the reduction of the arm retraction barrier in bulk. This overestimation emerges possibly because this picture does not incorporate the tension equilibration of the dilated segments required for the coherent arm retraction along the dilated tube. This point is further discussed in the next section for the longitudinal partial-DTD picture. Test of Longitudinal Partial-DTD Picture. As similar to reptation of linear chains, the retraction of star arm is the coherent/cooperative motion of all dilated segments occurring along the arm backbone. Thus, the arm retraction along the dilated tube should require all dilated segments to equilibrate their tension through the CR motion, as explained in Introduction. This CR-equilibration process can be modeled as the longitudinal partial-DTD process,9 as illustrated in Figure 13.

CR equilibration of the tension increases with the total number g* of entanglement segments simultaneously/cooperatively involved in this equilibration process, and βmax should stay small so as to ensure this process to occur in time (within the observed relaxation time). In the remaining part of this section, we focus on the CR modes that can be activated in time to evaluate g* and βmax (with the same method as applied for linear PI9) and then test the validity of the longitudinal partial-DTD scenario through comparison of the data shown in Figure 8 and [X] the τ[X] 1,b /τ1,m ratio calculated from eqs 17 and 18 with this βmax. The prerequisite for longitudinal partial-DTD in bulk star PI can be cast as a condition for the lowest index p* of the CR mode available for the tension equilibration in the time scale of terminal dielectric relaxation, τ[ε] 1,m: [ε] τCR [ε] < τ1,m rp *

⎛ {2p* − 1}π ⎞ −2⎛ ⎞ π with rp * = sin 2⎜ ⎟sin ⎜ ⎟ ⎝ 2{2Na + 1} ⎠ ⎝ 2{2Na + 1} ⎠ (20)

The time τ[ε] CR/rp* appearing in eq 20 indicates the characteristic time of p*th CR mode expressed in terms of the longest dielectric CR time τ[ε] CR (of the first mode) and the relaxation time ratio of p*th and first modes, 1/rp* (cf. eq 19b). The lowest possible index p* thus specified by eq 20 (with the [ε] aid of τ[ε] CR and τ1,m data) can be converted to the total number g* of the entanglement segments simultaneously/cooperatively involved in the longitudinal partial-DTD (CR-equilibration of tension) in the time scale of τ[ε] 1,m. This conversion is straightforward for the linear chain (g* = N/p* with N being the entanglement segment number of the chain), but it requires delicate consideration for the star arm. The star arm is modeled as the tethered chain composed of Na segments. Then, its p*th CR eigenfunction is expressed as f p*(n) = sin(πn/Q) with n (= 0 − Na) being the segment index and Q = 2Na/(2p* − 1). This eigenfunction has p* nodes at n = 0, Q, 2Q, ..., (p* − 1)Q, and one free antinode at n = Na. g* corresponds to the number of segments in a span between successive nodes, Q, and also to the number of segments in a span between the node at n = (p*−1)Q and the free antinode, Q/2. Because of this duality of g* (not existing for the linear chain having no tethered end), it becomes delicate to uniquely determine g* for the tethered chain (star arm). As the simplest but most reasonable choice, we make an average for all those spans, p* − 1 spans between successive nodes and one span between the node at n = (p* − 1)Q and the free antinode. Then, we find g* =

Figure 13. Schematic illustration of arm retraction along the longitudinally and partially dilated tube that wriggles in the laterally and partially dilated tube.

N (p* − 1)Q + (Q /2) = a p* p*

(21)

Note that g* = Na and 1 for p* = 1 and Na, as naturally expected.33 The number g* thus obtained can be related to the number β* of entanglement segments per longitudinally dilated segment as

The longitudinally and partially dilated tube, having the diameter ap‑DTD* (= aβ*1/2), wriggles in the laterally and partially dilated tube having the diameter ap−DTD′ > ap‑DTD* (ap−DTD′ = aβ1/2 with β being evaluated experimentally; cf. Figure 12). This wriggling motion allows the μ(t) data of star PI in bulk to obey the partialDTD relationship μ(t) = φ′(t)/β(t) (eq 1), as observed experimentally. For the longitudinal partial-DTD scenario explained above, [X] the τ[X] 1,b /τ1,m ratios are calculated from eqs 17 and 18 with the aid of the β and θ data (Figure 12), as was the case also for the lateral partial-DTD scenario. However, the upper limit values βmax and θmax appearing in eq 17 are expected to be smaller for the longitudinal partial-DTD case because the time required for the

Na = Na − g * + 1 β*

(22)

The Na/β* ratio appearing in eq 22 represents the total number of dilated segments that are tension-equilibrated through the CR mechanism. Because the dilated segments work as the independent stress sustaining units, the total number of the dilated segments (Na/β*) is equivalent to the number of entanglement segments remaining independent (not being involved in the longitudinal partial-DTD), Na − g*,9 which leads to eq 22. An extra factor of “1” has been introduced in eq 22 to guarantee M


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β* = 1 for g* = 1 (no DTD) and β* = Na for g* = Na (DTD involving the whole arm backbone). The two relaxation times appearing in eq 20 are known: The dielectric relaxation time τ[ε] 1,m is measured for star PI in bulk, [G] whereas the dielectric CR time τ[ε] CR is evaluated from the ⟨τCR ⟩w [ε] data, as explained for Figure 12. Thus, we utilized these τ1,m and τ[ε] CR in eq 20 to evaluate p*, calculated g* from this p* (cf. eq 21), and then evaluated β* from this g* (eq 22). It should be noted that β* thus evaluated gives the diameter of longitudinally and partially dilated tube, ap‑DTD* = aβ*1/2 (cf. Figure 13), in the time scale of terminal relaxation. Namely, this β* serves as βmax explained for eq 17. Thus, we replaced βmax and θmax in eq 17 by β* and θ*, the latter being the value of θ data at a time where the β data increase to β* (see Figure 12), and conducted the integral therein with the aid of the β and θ data to [ε] evaluate the dielectric τ[ε] 1,b /τ1,m ratio for star PI. Furthermore, this dielectric ratio was utilized in eq 18 to estimate the viscoelastic [G] τ[G] 1,b /τ1,m ratio. [X] The τ[X] 1,b /τ1,m ratios thus obtained are shown in Figures 8a and 8b with the thick red curve. These curves are close to the data for large Na, suggesting the validity of the longitudinal partial-DTD scenario for well-entangled star chains in bulk. Thus, the arm retraction in bulk quite possibly occurs along the longitudinally and partially dilated tube, not along the laterally dilated tube, because of the prerequisite (CR-equilibration of tension of dilated segments) for the arm retraction along the dilated tube. The above results strongly suggest that the star chain in bulk has two coarse-grained length scales: the diameter ap‑DTD′ of laterally and partially dilated tube that determines the modulus level and the diameter ap‑DTD* of longitudinally and partially dilated tube: ap‑DTD* is a recast of the arm retraction path length and determines the relaxation time. These two length scales are essential when we attempt to describe the star chain dynamics with the coarse-grained view (within the framework of DTD). The importance of these two length scales is also known for the linear chain.9 (For monodisperse linear chains in bulk, ap‑DTD′ reduces to the fully dilated tube diameter.) Comments for Duality in Coarse-Grained View for Entanglement Relaxation. In relation to the importance of two coarse-grained length scales mentioned above, we note that most of the tube models so far proposed4,5,10−13,34−39 (but not all19) assume the chain motion along the laterally dilated tube having the diameter ap‑DTD′ and do not incorporate the CR-activated tension equilibration process (the process underlying the longitudinal partial-DTD). Those tube models successfully describe viscoelastic behavior of entangled polymers of various architectures. Nevertheless, further refinement of the model is desirable through incorporation of the CR-activated tension-equilibration process (or, equivalently, the longitudinal partial-DTD process). Here, some comments need to be made for this improvement. This and previous9 studies have demonstrated that the path for the chain motion, the longitudinally dilated tube, can be different from the laterally dilated tube that determines the modulus level. In this molecular scenario, the chain motion is associated with its intrinsic curvilinear friction, ζc. This point would be easily noted if we consider a hypothetical, entangled solution having an entanglement mesh size ap‑DTD* as a reference for bulk: For example, a linear chain in bulk exhibits reptation along the longitudinally dilated tube (of the diameter ap‑DTD*) that wriggles in the laterally dilated tube.9 The relaxation time for this reptative motion is identical to that in an iso-ζc solution having the mesh size ap‑DTD*. Namely, there is no delay in the

chain motion in bulk compared to the motion in this reference solution. Nevertheless, we also have the other route for describing the linear chain motion as the delayed reptation along the laterally dilated tube having the diameter ap‑DTD′ (= a{φ′(t)}−d/2 for the linear chain in monodisperse bulk). For this case, the reptative motion in bulk is slower, by a certain delay factor, than that in the other iso-ζc reference solution having the mesh size ap‑DTD′. This delay factor represents a ratio of the tension-equilibration times in the bulk and solution: The tension equilibration is achieved through the intrinsic Rouse motion in the solution and through the CR motion in bulk.20 In fact, this molecular scenario successfully describes the component relaxation times in binary blends of linear PI, as demonstrated in a recent study.20 Thus, we have two equivalent routes for description of the linear chain motion: reptation along longitudinally dilated tube without delay with respect to the first reference solution and the reptation along the laterally dilated tube with the delay with respect to the second reference solution. In the first route, we need to incorporate two coarse-grained length scales, ap‑DTD* and ap‑DTD′, for description of the viscoelastic behavior. In the second route, we need just one length scale, ap‑DTD′, but also need to incorporate the delay factor explained above. This duality should emerge also for the star chains. Namely, we may regard the arm retraction to proceed along the longitudinally dilated tube without an extra retardation (the molecular scenario adopted in this paper) or along the laterally dilated tube but with an extra retardation due to the CR-activated tension equilibration. Such a duality could be helpful for improvement of the tube models mentioned above.

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CONCLUDING REMARKS Dielectric and viscoelastic behavior has been examined for PI blends containing dilute star probes (10−3Ma = 9.5−23.5, and υ1 = 0.1) in a much longer linear matrix (10−6M2 = 1.12). The relaxation of the star probe was found to be retarded and broadened significantly on blending, and this effect of blending was exponentially enhanced on an increase of the arm molecular weight Ma. The retardation and broadening occurred because the CR/DTD mechanism was quenched in the long matrix (as evidenced from the coincidence of the frequency dependence of ε1,b″ and G1,b″ of the star probe in the blend). This observation in turn indicates that the star chain dynamics in monodisperse bulk is significantly affected by the CR/DTD mechanism. The retardation of the star probe relaxation was analyzed within the context of the tube model. The simplest molecular picture of full-DTD cannot explain the relationship between the viscoelastic μ(t) data and dielectrically evaluated φ′(t) of star PI in monodisperse bulk, as noted previously and also confirmed in this study. A refined molecular picture of lateral partial-DTD can describe this relationship but still failed to explain the magnitude of retardation of the star probe relaxation on blending. Thus, a further refinement was attempted by introducing a molecular picture of longitudinal partial-DTD that reflects the CR-activated tension equilibration required for coherent/cooperative motion of all dilated segments (arm retraction along the dilated tube). It turned out that the retardation on blending was satisfactorily described by this molecular picture. The above results strongly suggest that the star chain in bulk exhibits the arm retraction along the longitudinally and partially dilated tube that wriggles in the laterally and partially dilated tube. Thus, the star chain in bulk quite possibly has two coarsegrained length scales: the diameters of the longitudinally and N


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laterally dilated tubes, ap‑DTD* and ap−DTD′, the former being a recast of the retraction path length and determining the relaxation time, and the latter governing the modulus level. The existence of these two coarse-grained length scales is essentially important for description of the star chain dynamics (and of the linear chain dynamics) in the coarse-grained view. It is desirable to further refine the current tube model(s) through incorporation of those two length scales.

APPENDIX. LIST OF BASIC PARAMETERS μ(t): normalized viscoelastic relaxation function obtained experimentally (μ(t) data obey the lateral partial-DTD relationship (eq 1)) Φ(t): normalized dielectric relaxation function obtained experimentally φ′(t): survival fraction of laterally dilated tube for PI probe (evaluated from Φ(t) data and CR time data; cf. eqs 2−4) θ(t): nonsurvival fraction of laterally dilated tube for PI probe (θ(t) = 1− φ′(t); cf. Figure 12) β(t): number of entanglement segments per laterally dilated segment (evaluated from Φ(t) data and CR time data; cf. eqs 2−4 and Figure 12) βCR(t): β(t) due only to Rouse-CR process (βCR(t) = 1/ψCR(t) evaluated from CR time data; cf. eq 19) ψCR(t): normalized Rouse-CR decay function (evaluated from CR time data; cf. eq 19) βf‑DTD(t): β(t) for the case of lateral full-DTD (βf‑DTD(t) = {φ′(t)}−d with d ≅ 1.3; cf. eq 3b) af‑DTD′(t): diameter of laterally and fully dilated tube (af‑DTD′(t) = a{βf‑DTD(t)}1/2 = a{φ′(t)}−d/2) a′(t) in Introduction and ap‑DTD′(t) in DTD Analysis section: diameter of laterally and partially dilated tube (a′(t) = ap−DTD′(t) = a{β(t)}1/2; cf. Figure 13) β*: number of entanglement segments per longitudinally and partially dilated segment (cf. eqs 20−22) g*: total number of entanglement segments involved in longitudinal partial-DTD (cf. eq 21) a*(t) in Introduction and ap‑DTD*(t) in DTD Analysis section: diameter of longitudinally and partially dilated tube (a*(t) = ap‑DTD*(t) = a{β*(t)}1/2; cf. Figure 13) ASSOCIATED CONTENT

S Supporting Information *

Validity of lateral partial-DTD relationship for monodisperse star PI. 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) Graessley, W. W. Polymeric Liquids and Networks: Dynamics and Rheology; Garland Science: New York, 2004. (3) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986. (4) Watanabe, H. Prog. Polym. Sci. 1999, 24, 1253. (5) McLeish, T. C. B. Adv. Phys. 2002, 51, 1379. (6) Graessley, W. W. Adv. Polym. Sci. 1982, 47, 67. (7) (a) Watanabe, H.; Urakawa, O.; Kotaka, T. Macromolecules 1993, 26, 5073. (b) Watanabe, H.; Urakawa, O.; Kotaka, T. Macromolecules 1994, 27, 3525. (8) Watanabe, H. Polym. J. 2009, 41, 929. (9) Matsumiya, Y.; Kumazawa, K.; Nagao, M.; Urakawa, O.; Watanabe, H. Macromolecules 2013, 46, 6067. (10) Marrucci, G. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 159. (11) Milner, S. T.; McLeish, T. C. B. Phys. Rev. Lett. 1998, 81, 725. (12) Ball, R. C.; McLeish, T. C. B. Macromolecules 1989, 22, 1911. (13) Milner, S. T.; McLeish, T. C. B. Macromolecules 1997, 30, 2159. (14) (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. (15) Watanabe, H.; Sawada, T.; Matsumiya, Y. Macromolecules 2006, 39, 2553. (16) Watanabe, H.; Matsumiya, Y.; van Ruymbeke, E.; Vlassopoulos, D.; Hadjichristidis, N. Macromolecules 2008, 41, 6110. (17) Watanabe, H. Macromol. Rapid Commun. 2001, 22, 127. (18) Sawada, T.; Qiao, X.; Watanabe, H. Nihon Reoroji Gakakishi (J. Soc. Rheol. Jpn.) 2007, 35, 11. (19) van Ruymbeke, E.; Masubuchi, Y.; Watanabe, H. Macromolecules 2012, 45, 2085. (20) Watanabe, H.; Matsumiya, Y.; van Ruymbeke, E. Macromolecules 2013, 46, 9296. (21) Viovy, J. L.; Rubinstein, M.; Colby, R. H. Macromolecules 1991, 24, 3587. (22) Qiao, X.; Sawada, T.; Matsumiya, Y.; Watanabe, H. Macromolecules 2006, 39, 7333. (23) Yoshida, H.; Adachi, K.; Watanabe, H.; Kotaka, T. Polym. J. 1989, 21, 863. (24) Matsumiya, Y.; Uno, A.; Watanabe, H.; Inoue, T.; Urakawa, O. Macromolecules 2011, 44, 4355. (25) In principle, the storage modulus of the probe in the blend, G1,b′(ω), can be evaluated by an equation corresponding to eq 9. However, the Gb′(ω) data of the blends are insensitive to the fast and weak relaxation process. For this reason, we did not attempt to evaluate G1,b′(ω). (26) In this study, the Gb″ and G1,m″ data of the blends and bulk matrix were measured to higher ω (to lower T) compared to the previous study.9 Thus, for evaluation of G1,b″(ω) of the star probe (Figure 4b), we have modified slightly the previous empirical equation for the local 2 0.68 1 0.67 , as G[R] to viscoelastic relaxation, G[R] 2,m″ =1.1 × 10 ω 2,m″ = 8.3 × 10 ω cover such high-ω data as well. However, this modification gave negligible differences for G1,b″(ω) of the probe evaluated in the range of ω of our interest. (27) Graessley, W. W. Adv. Polym. Sci. 1974, 16, 1. (28) The second-moment average relaxation time ⟨τ⟩w is sometimes referred to as “weight-average” relaxation time because its definition is analogous to the definition of the weight-average molecular weight.27 The subscript “w” in the notation ⟨τ⟩w comes from this analogy. (29) Watanabe, H.; Matsumiya, Y.; Inoue, T. Macromolecules 2002, 35, 2339. (30) Horio, K.; Uneyama, T.; Matsumiya, Y.; Masubuchi, Y.; Watanabe, H. Macromolecules 2014, 47, 246. (31) Matsumiya, Y.; Watanabe, H. J. Soc. Rheol. Jpn. (Nihon Reoroji Gakkaishi) 2014, 42, 235. (32) The longest Rouse relaxation time is the same for the star chain having the arm molecular weight Ma and the linear chain having M = 2Ma. Thus, the entanglement effect for the star chain should fully vanish

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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 JSPS, Japan (Grant No. 24245045), Grant-inAid for Scientific Research (C) from JSPS, Japan (Grant No. 24550135), Collaborative Research Program of ICR, Kyoto University (Grant No. 2014-42), Fonds de la Recherche Scientifique (FNRS), and National Natural Science Foundation of China (No. 21174153). O


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[X] and the τ[X] 1,b /τ1,m ratio should approach unity as 2Ma approaches Me, as in the case for the linear chain with M → Me.31 (33) In the previous study for the linear chain,9 the highest CR mode index ph was set to be N − 1 so that a minor correction factor was introduced in the relationship between g* and p* to satisfy the requirement, g* = N and 1 for p* = 1 and ph. This minor correction is not needed in eq 21 because in this study ph for the star arm has been set to be Na (cf. eq 19a). (34) Mead, D. W.; Larson, R. G.; Doi, M. Macromolecules 1998, 31, 7895. (35) McLeish, T. C. B.; Larson, R. G. J. Rheol. 1998, 42, 81. (36) Larson, R. G. Macromolecules 2001, 34, 4556. (37) Kapnistos, M.; Vlassopoulos, D.; Roovers, J.; Leal, L. G. Macromolecules 2005, 38, 7852. (38) Das, C.; Inkson, N. J.; Read, D. J.; Kelmanson, M. A.; McLeish, T. C. B. J. Rheol. 2006, 50, 207. (39) van Ruymbeke, E.; Kapnistos, M.; Vlassopoulos, D.; Liu, C. Y.; Bailly, C. Macromolecules 2010, 43, 525.

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Dielectric and Viscoelastic Behavior of Star-Branched Polyisoprene

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