Entanglement Length in Miscible Blends of cis-Polyisoprene and Poly

Nov 2, 2015 - Abstract Image. The entanglement length a, being equivalent to the plateau modulus GN (∝Me–1 ∝ a–2), is one of the most basic pa...
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Entanglement Length in Miscible Blends of cis-Polyisoprene and Poly(p-tert-butylstyrene) Yumi Matsumiya,† Natthida Rakkapao,†,‡ and Hiroshi Watanabe*,† †

Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan Faculty of Science and Industrial Technology, Prince of Songkla University, Suratthani 84000, Thailand

‡

S Supporting Information *

ABSTRACT: The entanglement length a, being equivalent to the plateau modulus GN (∝Me−1 ∝ a−2), is one of the most basic parameters that determine the slow dynamics of high molecular weight (M) polymers. In miscible blends of chemically different chains, the components would/should have the common a value. However, changes of a with the blend composition have not been fully elucidated. For this problem, this study conducted linear viscoelastic tests for miscible blends of high-M cis-polyisoprene (PI) and poly(p-tert-butylstyrene) (PtBS) and analyzed the storage and loss moduli (G′ and G″) data in a purely empirical way, considering the very basic feature that unentangled and entangled blends having the same composition exhibit the same local relaxation. (From a molecular point of view, this local relaxation reflects the chain motion within the length scale of a.) On the basis of this feature, a series of barely entangled low-M PI/ PtBS blends having various component molecular weights and a given composition were utilized as references for well-entangled high-M PI/PtBS blends with the same composition, and the modulus data of the reference were subtracted from the data of the high-M blends. For an optimally chosen reference, the storage modulus of the high-M blends obtained after the subtraction (Gent′ = Ghigh‑M blend′ − Gref′) exhibited a clear plateau at high angular frequencies ω. The corresponding loss modulus Gent″ decreased in proportion to ω−1 at high ω, which characterized the short-time onset of the global entanglement relaxation: A mischoice of the reference gave no plateau of Ghigh‑M blend′ − Gref′ and no ω−1 dependence of Ghigh‑M blend″ − Gref″ at high ω, but a survey for various low-M PI/PtBS blends allowed us to find the optimum reference. With the aid of such optimum reference, the entanglement plateau modulus GN of the high-M PI/PtBS blends was accurately obtained as the high-ω plateau value of Gent′. GN thus obtained was well described by a linear mixing rule of the entanglement length a with the weighing factor being equated to the number fraction of Kuhn segments of the components, not by the reciprocal mixing rule utilizing the component volume fraction as the weighing factor. This result, not explained by a mean-field picture of entanglement (constant number of entanglement strands in a volume a3), is discussed in relation to local packing efficiency of bulky PtBS chains and skinny PI chains. “dynamic tube dilation”, has been incorporated in the tube model as an important mechanism of stress relaxation.4−13 In homopolymer systems, the entanglement length a is insensitive to the polymer molecular weight and its distribution, as long as all component chains are sufficiently long and mutually entangled.1−3 Thus, a in those systems can be regarded as the fundamental constant in molecular description of the entanglement dynamics. The situation is different in miscible blends of chemically different chains. All component chains in those blends would/should have the same a value,25−29 but this value should change with the blend composition. Consequently, the change of the entanglement length needs to be incorporated in molecular description/ understanding of the component dynamics.

1. INTRODUCTION In concentrated systems, high molecular weight (M) polymer chains are deeply penetrating with each other. Because those chains cannot cross each other, they mutually constrain the large-scale motion. This topological effect on the chain motion, referred to as entanglement, has been one of the central subjects in polymer physics, and extensive studies have been made, from both theoretical and experimental aspects, with an attempt of better describing/understanding the relaxation/motion of entangled polymers. 1−24 The basic parameters in this description include the entanglement length a, the relaxation time (or internal equilibration time) τa of the subchain having the size a, the number Z of those subchains per chain, and the number of chains penetrating into a pervaded volume (∼a3) of the subchain. This subchain serves as the basic stress-sustaining unit in a time scale t ∼ τa, and is referred to as the entanglement segment (or entanglement strand).2−5 In longer time scales, some entanglement segments of a chain in monodisperse bulk are mutually equilibrated to behave as a coarse-grained stresssustaining unit.4−6,14−16 This equilibration, referred to as © XXXX American Chemical Society

Received: August 24, 2015 Revised: October 19, 2015

A

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= ρRT/Me. Equation 6 is obtained from eq 3 combined with this expression and the above expression of GN,X. For the experimental test of eqs 4−6, we usually need to choose the component polymers A and B having chemically dissimilar structures, because chemically similar polymers have similar aX and/or Me,X values and eqs 4−6 give almost indistinguishable GN values for that case. Among the miscible blends so far examined, for example, polystyrene (PS)/ poly(vinyl methyl ether) (PVME) blends,29,32 poly(methyl methacrylate) (PMMA)/poly(ethylene oxide) (PEO) blends,33−35 cis-polyisoprene (PI)/polyvinylethylene (PVE) blends,25,30,36−38 and PI/poly(p-tert-butylstyrene) (PtBS) blends,26,27,39,40 the PI/PtBS blends have the largest asymmetry in both size aX and relaxation time τa,X of the entanglement segments of the components PI and PtBS in bulk (with the asymmetry of τa,X significantly contributed from a large difference of glass transition temperatures of those components), thereby serving as a good model system for testing eqs 4−6. A previous study27 for high-M PI/PtBS blends focused on the storage and loss moduli data in the terminal relaxation zone of the fast component (PI) to show that the data are described by a model based on eq 2, not on eqs 1 and 3. In addition, for low-M PtBS in low-M PI/PtBS blends that was judged to be unentangled and entangled according to eqs 1 and 2, respectively, the relaxation was slower than expected for the unentangled case.26,39 This finding also lent support to eq 2. Nevertheless, no direct test has been made for GN, i.e., for the modulus at short times before the entanglement relaxation starts, and a further study has been desired. Thus, we have examined linear viscoelastic behavior of highM, entangled PI/PtBS blends at high angular frequencies ω where the entanglement plateau should emerge. Experimentally, the storage and loss moduli, G′ and G″, at such high ω are always contributed from the fast, local relaxation within the entanglement segment (Rouse-like relaxation and faster glassy relaxation), and this contribution disturbs simple evaluation of GN from the raw data of G′: For example, GN is often equated to G′(ωmin) at the angular frequency ωmin where G″ and/or tan δ (=G″/G′) has the minimum value.1,20 However, this simple method does not give an accurate GN value, unless the material exhibits a very deep minimum of G″ and/or tan δ (that corresponds to a very wide separation of the relaxation times for the local and global relaxation processes), as discussed by Liu et al.20 Considering the above problem, we used a series of low-M PI/PtBS blends as reference materials for the high-M PI/PtBS blend, all having the same composition, to evaluate GN for the high-M PI/PtBS blend. Specifically, we regarded the Gref′ and Gref″ data of the low-M reference blend as the moduli Glocal′ and Glocal″ for the fast, local relaxation in the high-M blend, and attempted to evaluate G ent ′ and Gent″ for the global entanglement relaxation in the high-M blend by subtracting Gref′ and Gref″ from the G′ and G″ data of this blend. When the chosen reference had too small M, the differences G′ − Gref″ and G″ − Gref″ increased on an increase of ω toward the high-ω end of our experimental window. In contrast, when the chosen reference had too large M, both G′ − Gref′ and G″ − Gref″ exhibited unphysical decreases on this increase of ω (the decrease of G″ − Gref″ was stronger than the fastest possible decrease specified by Gent″ ∝ ω−1). These features of G′ − Gref′ and G″ − Gref″, if observed, unequivocally indicate a mischoice of the reference for evaluation of Gent′ and Gent″. Nevertheless, a wide survey of reference blends having various M enabled us

For miscible blends of components A and B, two simple mixing rules have been proposed for the entanglement length a:25−27

υ υ 1 = A + B a aA aB

(1)

a = nA aA + nBaB

(2)

Here, aX (X = A, B) is the entanglement length of the component X in its bulk system. υX and nX, respectively, denote the volume fraction of the component X and the number fraction of the Kuhn segments of this component in the A/B blend. A mixing rule similar (though not identical) to eq 1 has been proposed for the molecular weight M e of the entanglement segment:30,31 1

=

Me1/2

υA Me,A

1/2

+

υB Me,B1/2

(3)

Here, Me,X represents Me of the component X in its bulk system. Equations 1-3 have respective molecular meanings, as explained later in some detail in the Discussion. However, before such a molecular discussion, it is highly desired to test eqs 1−3 in a purely experimental way. For this purpose, eqs 1−3 can be rewritten as the mixing rules of the entanglement plateau modulus GN, a quantity that can be directly evaluated from viscoelastic data at a given temperature T: From eq 1: 2 ⎧ G w a 2ρ G N,BwBaB 2ρ ⎫⎧ υA υ ⎫ N,A A A ⎬⎨ + B ⎬ GN = ⎨ + aB ⎭ ρA ρB ⎭⎩ aA ⎩ ⎪

⎪

⎪

⎪

(4)

From eq 2: ⎧ G w a 2ρ ⎫2 G N,BwBaB 2ρ ⎫⎧ 1 N,A A A ⎬⎨ ⎬ GN = ⎨ + ρA ρB ⎭⎩ nA aA + nBaB ⎭ ⎩ ⎪

⎪

⎪

⎪

(5)

From eq 3: ⎧ G N,A wAMe,A ρ G N,BwBMe,Bρ ⎫ ⎬ + GN = ⎨ ρA ρB ⎭ ⎩ ⎪

⎪

⎪

⎪

⎧ υ ⎫2 υB ⎪ A ⎨ ⎬ ×⎪ + 1/2 Me,B1/2 ⎪ ⎩ Me,A ⎭ ⎪

(6)

Here, GN,X, ρX, and Me,X (X = A, B), respectively, denote the plateau modulus, density, and entanglement molecular weight of the component X in its bulk system, ρ is the density of the A/B blend, and wX is the weight fraction of the component X in the blend. υX and nX have the same meaning as in eqs 1−3. Equations 1 and 2 consider that the two components in the blend have the same a. Thus, the corresponding expression of the plateau modulus of the blend, GN = ΣX=A,B ρRTwX/{Me,Xa2/ aX2} with the factor {Me,Xa2/aX2} giving the entanglement molecular weight for the component X in the blend, and the expression of the plateau modulus for bulk components, GN,X = ρXRT/Me,X, are combined with eqs 1 and 2 to derive eqs 4 and 5, respectively. (Note that GN is defined in Ferry’s way1 without the Doi−Edwards factor2,3 of 4/5.) In contrast, eq 3 considers the two components in the blend to have the same Me, which leads to the expression of the plateau modulus of the blend, GN B

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Macromolecules to find an optimum reference that gave the plateau of G′ − Gref′ at high ω and the corresponding decrease of G″ − Gref″ (∝ ω−1). For this case, G′ − Gref′ and G″ − Gref″ can be taken to be Gent′ and Gent″ for the entanglement relaxation in the highM PI/PtBS blends, and GN is evaluated as the plateau value of G′ − Gref′. Namely, the survey of various reference blends was accompanied by a clear experimental criterion for judging if GN was satisfactorily obtained. The GN values thus obtained for the high-M PI/PtBS blends were well described by eq 5, not by eqs 4 and 6. Nevertheless, eq 2 underlying eq 5 is not explained by a mean-field picture of entanglement (fixed number of entanglement strands in a pervaded volume of the entanglement segment).20−22 This paper reports details of those results and discusses a possible origin of the validity of eq 2, an efficiency of local packing of bulky PtBS chains and skinny PI chains.

Table 2. Characteristics of Low-M Samples sample codea PI PI PI PI PI PI

PtBS PtBS PtBS PtBS

PtBS 219 PtBS 442 PtBS 729 a

Polyisoprene 321 Poly(p-tert-butylstyrene) 219 442 729

1.02 1.04 1.05 1.03

temperature for 12 h and then at 60 °C for 48 h. Finally, the dried blend was compression-molded in vacuum for ≅ 5 min into a disk that matched a viscoelastic measurement fixture (cone-and-plate), and was stored at room temperature in argon until use. The compression molding was made at temperatures where the blend was softened to easily flow under compression in short time (≅5 min), namely at ≅60, ≅90, and ≅160 °C for the blends with wPI = 75, 50, and 25 wt %, respectively. Lack of thermal degradation was confirmed with GPC for all blends recovered after the compression molding. (We attempted to also utilize a PI 1120 sample with higher Mw (=1.12 × 106)15,16 as the PI component in the blends. However, long time compression molding at high T (≅1 h at T > 160 °C) was necessary for blends containing this PI, and thermal degradation of PI unavoidably occurred during the molding. Thus, the PI 1120 sample could not be utilized as the component in the high-M PI/PtBS blends.) 2-2. Measurements. For the high-M and low-M PI/PtBS blends, dynamic viscoelastic measurement was conducted under nitrogen atmosphere with a laboratory rheometer (ARES-G2, TA Instruments Co. Ltd.) utilizing a cone−plate fixture (diameter = 7.9 mm and gap angle = 0.1 rad). The measurement was made in a range of angular frequency ω between 0.01 s−1 and 300 s−1. The oscillatory strain amplitude was kept small (γ0 = 0.05) so as to obtain the storage and loss moduli (G′ and G″) in the linear viscoelastic regime. The G′ and G″ data of the PI/PtBS blends do not obey the time− temperature superposition, because the temperature dependence of the segmental friction is different for the components PI and PtBS even though they are in the miscible state.26,27,39 Thus, a choice of the measurement temperature was crucial for evaluation of the entanglement plateau modulus GN of the high-M PI/PtBS blends. We made preliminary tests at several temperatures to find an optimum experimental temperature where the terminal relaxation of the fast component (PI321) in the high-M blends and the faster, local relaxation within the entanglement segment were simultaneously detected in our experimental window (ω = 0.01−300 s−1). It turned out that the optimum temperature was 40, 65, and 130 °C for our blends with wPI = 75, 50, and 25 wt %, respectively. Thus, for both high-M and low-M PI/PtBS blends, the measurements were made at those temperatures, and lack of thermal degradation was confirmed with GPC for all blends recovered after the measurements. For comparison, the dynamic viscoelastic measurement was conducted also for the components PI and PtBS in respective bulk state. The G′ and G″ data of those components, presented in Appendix A, were utilized to confirm the validity of our method of evaluating GN (use of the low-M reference materials). For all PI/PtBS blends examined in this study, differential scanning calorimetric (DSC) measurement was conducted with a laboratory calorimeter (DSC Q20, TA Instruments) at a heating rate of 15 K

Table 1. Characteristics of High-M Samples

PI 321

1.04 1.04 1.04 1.04 1.03 1.03

Sample code number indicates Mw in unit of 1000. bSynthesized/ characterized in ref 42. cSynthesized/characterized in ref 41. d Synthesized/characterized in ref 43. eSynthesized/characterized in this study. fSynthesized/characterized in ref 44. gSynthesized/ characterized in ref 45.

2-1. Materials. High-M and low-M linear polyisoprene samples were synthesized via living anionic polymerization in vacuum at room temperature. Benzene, sec-butyllithium, and methanol were utilized as the solvent, initiator, and terminator, respectively. High-M and low-M linear poly(p-tert-butylstyrene) (PtBS) samples were similarly synthesized anionically. Some of the as-synthesized high-M PtBS samples had a little broad molecular weight distribution (Mw/Mn ≅ 1.15). Thus, we repeatedly fractionated them from benzene/methanol mixed solvents to recover narrowly distributed samples. For simplicity, narrowly distributed samples are hereafter referred to as “monodisperse”. The monodisperse samples thus obtained 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 and PtBS samples synthesized/characterized in our previous studies26,27,41−45 were utilized as the RI/LALLS standards as well as the elution volume standards. Table 1 summarizes the weight-average molecular weight Mw and the polydispersity index Mw/Mn of the high-M PI and PtBS samples utilized in this study.

10−3Mw

27g 42f 53e 70g

Mw/Mn

Polyisoprene 13.7 17.6 21.4 35.0 42.7 53.4 Poly(p-tert-butylstyrene) 27.2 41.8 53.0 69.5

a

2. EXPERIMENTAL SECTION

sample codea

14b 18c 21d 35b 43e 53f

10−3Mw

Mw/Mn 1.04 1.04 1.01 1.06

Sample code number indicates Mw in unit of 1000.

We used low-M PI/PtBS blends as the reference materials for the high-M PI/PtBS blends of our interest. Low-M PI and PtBS samples synthesized/characterized in this and previous studies41−45 were utilized as the components of those low-M blends (and as the reference materials for high-M PI and PtBS samples in bulk). The characteristics of those low-M samples are summarized in Table 2. The materials subjected to linear viscoelastic measurements were high-M and low-M PI/PtBS blends with the PI content wPI = 25, 50, and 75 wt %. For preparation of those blends, prescribed masses of the component PI and PtBS samples were first dissolved in THF at the total polymer concentration of ≅5 wt %. Then, the solution was precipitated in a vigorously stirred methanol/acetone 3/1 (v/v) mixed solvent containing a small amount of an antioxidant, butylhydroxytoluene (≅ 0.001 wt % to the solvent), and the recovered precipitant (PI/PtBS blend) was thoroughly dried in vacuum, first at room C

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Macromolecules min−1 to characterize the glass transition of PI and PtBS therein. The results are summarized in Supporting Information.

the high-M blends having different MPtBS but the same wPI, as shown in Supporting Information. Correspondingly, for each series of the blends having the same wPI, the G′ and G″ data become independent of MPtBS and exhibit power-law like increases at high ω. (For wPI = 75 wt %, this increase is observed only for G″ because ω is not high enough to allow G′ to clearly exhibit the increase overwhelming its plateau.) This power-law like increase at high ω is attributable to the MPtBSand MPI-independent, local relaxation within the entanglement segment (Rouse-like relaxation and faster glassy relaxation), as noted in the previous study for similar blends.27 In contrast, at low ω, the blend behavior changes with MPtBS. For each wPI, the blends containing PtBS 729 and/or PtBS 442 (cf. red and green symbols in Figure 1) exhibit no terminal relaxation characterized by the power-law behavior, G′ ∝ ω2 and G″ ∝ ω, even at the lowest ω examined. In particular, for the PI 321/PtBS 442 blend with wPI = 25 wt % (Figure 1c), two-step relaxation is clearly noted, and the slow-step relaxation just begins at the low-ω side of our experimental window. For those blends, the terminal relaxation (of the slow step) should occur at sufficiently low ω (≪10−2 s−1; not covered in our experiments), as noted for the previous data for similar PI/PtBS blends.27 In contrast, for the blend containing the shorter PtBS 219 (cf. blue symbols in Figure 1), broad but almost single-step terminal relaxation occurs at the low-ω side of our experimental window. The above results suggest that PI 321 is the fast component in all blends examined, and the terminal relaxation time of the slow component, PtBS, significantly increases with MPtBS (because, for a given wPI, the number of entanglement per PtBS chain increases in proportion to MPtBS). This assignment of the fast and slow components is consistent with our previous study27 that exclusively detected the relaxation of PI in similar PI/PtBS blends with the dielectric method. (PI has so-called type-A dipole but PtBS does not, so that the slow dielectric relaxation was exclusively attributed to the large-scale motion of PI.27) It should be also noted that our PI/PtBS blends clearly exhibit the two-step relaxation only when the fast and slow components have widely separated relaxation times, as similar to the behavior of PI/PI binary blends.14−16 This two-step relaxation is an interesting subject of research, and its feature has been analyzed in detail for PI/PtBS blends27 similar to those examined in Figure 1b. However, the target of this study is the entanglement plateau modulus in the PI/PtBS blends that is observed before the onset of the entanglement relaxation of the fast component (PI 321). Thus, no further discussion is made here for the two-step relaxation. In this regard, we should emphasize again that the experimental temperatures for respective wPI were chosen according to that target, namely, in a way that the experimental window covers the local relaxation within the entanglement segment as well as the (onset of) terminal relaxation of the fast component. No attempt was made to cover the full relaxation of the slow component (PtBS). 3-2. Apparent Plateau Modulus. The high-M PI/PtBS blends are in the well-entangled state, as can be noted from the G′ and G″ data in Figure 1. In fact, the end-to-end distance of the PI and PtBS component chains therein is much larger than the entanglement length deduced from all mixing rules, eqs 1−3. Thus, those blends serve as good model systems for testing the composition dependence of the entanglement plateau modulus, GN.

3. RESULTS 3-1. Overview of Data. Figure 1 shows the storage and loss moduli (G′ and G″) and loss tangent (tan δ) measured for the high-M PI/PtBS blends at respective experimental temperatures, T = 40, 65, and 130 °C for the blends with the PI content wPI = 75, 50, and 25 wt %, respectively. The G′, G″, and tan δ data are double logarithmically plotted against the angular frequency ω. All blends commonly include PI 321 as one component, and the other component (PtBS) therein has several different molecular weights MPtBS. In the blends, the friction coefficients ζPI and ζPtBS of monomeric segments of PI and PtBS are differently dependent on T because of the difference in the glass transition temperatures Tg of PI and PtBS therein.26,27,39 Consequently, the viscoelastic data of PI/PtBS blends do not obey the time− temperature superposition.26,27,39 The coefficients ζPI and ζPtBS change with wPI as well. However, for a given set of T and wPI, those coefficients, determined by the local environment, should be independent of MPtBS because no chain-end effect on Tg is expected for the high-M component chains. In fact, DSC data reflecting the segmental dynamics were indistinguishable for

Figure 1. Storage and loss moduli (G′ and G″) and loss tangent (tan δ) measured for the high-M entangled PI/PtBS blends as indicated. Note that the experimental temperature is different for the blends with different PI content (wPI = 75, 50, and 25 wt %). D

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Macromolecules GN is often equated to G′(ωmin) at the angular frequency ωmin where G″ and/or tan δ has the minimum value,1,20 although this method is sufficiently accurate only when the entanglement relaxation has a narrow mode distribution and occurs much more slowly than the local relaxation within the entanglement segment.20 We first applied this simple method to our blend data to evaluate the apparent plateau modulus as Gapp N = G′(ωmin−tan δ), where ωmin‑tan δ stands for the minimum of tan δ: We utilized the G′ and tan δ data shown in Figure 1 to evaluate Gapp N at respective experimental temperatures, T = 40, 65, and 130 °C for wPI = 75, 50, and 25 wt %, and reduced those Gapp N values to a reference temperature Tr = 40 °C through a relationship obtained from the theory of rubber elasticity,1−3 ⎧ T ρ (T ) ⎫ G N(Tr) = ⎨ r r ⎬G N(T ) ⎩ Tρ(T ) ⎭

the parameters being evaluated in a way consistent with the evaluation of Gapp N for the blends, as explained below. GN,X (X = PI, PtBS) appearing in eqs 4-6 was evaluated from the moduli data of the highest-M PI321 and PtBS 729 samples (presented in Appendix A) as the G′ value at ωmin‑tan δ. Me,X was calculated from this GN,X as G N,X

(9)

where R and T are the gas constant and absolute temperature. (Note that Me,X is defined in Ferry’s way1 without the Doi− Edwards factor2,3,47 of 4/5.) The bulk density ρX was estimated from literature data48 (ρPI = 0.92 g cm−3 and ρPtBS = 1.05 g cm−3 at 40 °C, with the latter being calculated from the ρPtBS data at 200 °C and the estimated thermal expansion coefficient in the molten state, α = 6 × 10−4 K−1). The entanglement length in bulk, aX, was calculated from Me,X with the aid of the relationship between the average, unperturbed end-to-end distance ⟨R2⟩1/2 of a chain having the molecular weight M:48

(7)

Here, T and Tr are expressed in the absolute temperature scale (e.g., Tr = 313 K), and the mass density of the blend at T, ρ(T), is related to ρ(Tr) at Tr through a thermal expansion coefficient α: ρ(Tr) = 1 + α(T − Tr) ρ (T )

ρX RT

Me,X =

1/2

⟨R2⟩PI

= 8.24 × 10−2M1/2 (nm) for PI

1/2

⟨R2⟩PtBS

= 6.01 × 10−2M1/2 (nm) for PtBS

(10a) (10b)

(Substituting the Me,X value in M appearing in eq 10, we obtained the aX value.) The density ρ of the blend and the component volume fractions υX were evaluated from ρPI, ρPtBS, and wPI, under an assumption of no volume change on mixing. Finally, the number fractions of the Kuhn segments, nX, were calculated from wPI and the molecular weights of the Kuhn segments of PI and PtBS, MK,PI ≅ 130 for PI48 and MK,PtBS ≅ 1500 for PtBS.49 Figure 2 demonstrates that none of eqs 4−6 describes Gapp N accurately, but among them eq 4 does a fairly good job. From this result, one might tend to conclude that the entanglement length can be reasonably described by eq 1 underlying eq 4. However, we should emphasize that Gapp N is just the apparent plateau modulus evaluated as the G′ value at ωmin‑tan δ. It is necessary to test if this Gapp N agrees with the real entanglement plateau modulus GN. The simplest criterion for this test is deduced from the definition of GN, as explained below. GN is defined as the initial modulus for the entanglement relaxation, and the internally equilibrated entanglement segments sustain the modulus GN. Thus, the complex modulus of entangled systems can be generally expressed, irrespective of the details of the entanglement dynamics, as

(8)

In the molten state, most of polymers including PI and polystyrene (chemically not very different from PtBS) have α ≅ 6 × 10−4 K−1.46 Thus, this α value was used to reduce Gapp N at 40 °C. Note that the density reduction specified by eq 8 is minor, 1 ≤ ρ(Tr)/ρ(T) ≤ 1.05, compared to the temperature reduction, 1 ≤ T/Tr ≤ 1.29. Thus, the use of α ≅ 6 × 10−4 K−1 introduces just a small uncertainty: The ρ(Tr)/ρ(T) ratio changes only by a factor of 5% even if we use unrealistically large or small α values, 1.2 × 10−3 or 3 × 10−4 K−1 (double or half of the above α value). In Figure 2, Gapp N thus reduced at 40 °C is plotted against wPI (unfilled symbols). The plot should have uncertainty less than 20% (vertical bar), as judged from the accuracy of the data (within 10% error) and the uncertainty in the ρ(Tr)/ρ(T) ratio (less than 5%). The curves indicate predictions of eqs 4-6, with

G*(ω) = Gent*(ω) + G local*(ω)

(11)

Here, Gent*(ω) is the complex modulus associated only to the entanglement relaxation, and Glocal*(ω) stands for the local relaxation within the entanglement segment. Equation 11 has an experimental basis, as explained below. (A delicate issue related to the Glocal*(ω) term in eq 11 is later discussed in section 3-5.) Inoue, Osaki, and co-workers50,51 conducted extensive rheooptical experiments to show that G*(ω) of homopolymers (such as polystyrene50 and PI51) is expressed as a sum of the moduli Grubbery*(ω) and Gglass*(ω) for the rubbery and glassy relaxation processes. They also showed that Grubbery*(ω) can be expressed as a sum of the moduli GRouse*(ω) and Gent*(ω) for the local Rouse relaxation (within the entanglement segment) and the global entanglement relaxation.51 Considering these results, eq 11 expresses G*(ω) as a sum of Gent*(ω) and

Figure 2. Apparent plateau modulus Gapp N of high-M PI/PtBS blends evaluated as the G′ value at the angular frequency ωmin‑tan δ where tan δ has the minimum value. E

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Macromolecules Glocal*(ω), the latter being given by Gglass*(ω) + GRouse*(ω) for the case of homopolymers. For the PI/PtBS blends of our interest, a previous study revealed that the local rubbery relaxation within the entanglement segment occurs cooperatively for PI and PtBS (namely, PtBS retards the rubbery relaxation of PI within this segment),27 and this relaxation proceeds with “Rouse-like” dynamics but not necessarily with the rigorous Rouse dynamics because of the cooperativity. Nevertheless, eq 11 should be still valid even for that case, because it just separates the global and local relaxation processes and the modulus for the latter, Glocal*(ω), can include the Rouse-like (but not necessarily rigorous Rouse) relaxation together with the glassy relaxation. The entanglement relaxation should occur only after the local relaxation completes, so that the storage modulus for the entanglement relaxation, Gent′(ω), should be independent of ω and coincide with GN at high ω where the local relaxation has not completed. This feature, valid irrespective of details of the entanglement dynamics, allows us to rewrite eq 11 as

high-M PI/PtBS blends with wPI = 50 and 25 wt %. (For the blends with wPI = 75 wt %, the G′(ω) data are only a little larger than Gapp N even at the highest ω examined (cf. Figure 1a) so that the difference G′(ω) − Gapp was not accurately N evaluated. For this reason, no test was attempted for those blends.) Clearly, the difference G′(ω) − Gapp N changes with MPtBS and relaxes faster for smaller MPtBS. Thus, for our PI/ PtBS blends with wPI = 50 and 25 wt %, the above criterion indicates that Gapp N evaluated as G′(ωmin−tan δ) differs from the real GN. We also utilized the other simple method1,20 that evaluates the apparent plateau modulus as Gapp N = G′(ωmin−G″) with ωmin−G″ being the angular frequency for the minimum of G″. app The difference G′(ω) − Gapp N obtained from this GN was found to violate the above criterion (M-independence of G′(ω) − Gapp N ) more severely than noted in Figure 3. Thus, the method utilizing ωmin−tan δ and/or ωmin−G″ is inadequate for accurately evaluating GN of our PI/PtBS blends with wPI = 50 and 25 wt %, because those blends have broad relaxation mode distributions and the time scales of the local and global (entanglement) relaxation processes of the fast component (PI 321) are not widely separated.52 In fact, we can utilize the data of high-M and low-M monodisperse PI to demonstrate the problem of that method, significant underestimation of GN: The Gapp N /GN ratio can be as small as 0.6 (or even smaller) for the cases of narrowly separated local and global relaxation processes; see Appendix B. Of course, the problem vanishes for high-M monodisperse polymers having very widely separated local and global processes. However, this is simply not the case for the PI/PtBS blends with wPI = 50 and 25 wt %. Thus, for the high-M PI/PtBS blends, accurate evaluation of GN requires a method other than the above method utilizing ωmin. The method of integrating Gent″ (GN = (2/π)∫ ∞ −∞Gent″ d[ln ω]) requires us to obtain Gent″ by extrapolating the raw G″ data around G″-peak to higher ω,1,20 and this extrapolation unavoidably introduces significant uncertainties. In addition, for our blends containing PtBS 442 and PtBS 729, the G″ data do not cover the full entanglement relaxation process (PtBS did not fully relax in our experimental window; cf. Figure 1), and the integral for this full process cannot be obtained. For these reasons, the method of integrating Gent″ is not applicable to our blends. Considering the above difficulties, we attempted to evaluate GN of the high-M PI/PtBS blends with the aid of reference materials, a series of nonentangled and/or lightly entangled low-M PI/PtBS blends. This method allows us to find a reference blend that has the modulus Gref*(ω) being close to Glocal*(ω) in eq 11, and evaluate GN by subtracting this Gref*(ω) from the G*(ω) data of the high-M blend. In the following sections, we explain details of the method, evaluate GN, and then discuss the composition dependence of GN. (As a supplementary method, we also attempted to fit the G* data of the blend with data of two homopolymer samples (PI or PtBS) and estimate GN of the blend from the known GN data of those samples. This method is not conceptually rigorous, but it worked well in a practical sense when the glassy relaxation negligibly contributed to the blend data and the constraint release relaxation of the slow component was minor, as explained in detail in Appendix C and also summarized in section 3-4.) 3-3. Accurate Evaluation of Entanglement Plateau Modulus. 3-3-1. Blends with wPI = 75 wt %. Figure 4 shows the G* data at 40 °C measured for the high-M PI 321/PtBS

G′(ω) − G N = G local′(ω) at high ω in the local relaxation regime

(12)

The storage modulus for the glassy relaxation in the high-M PI/ PtBS blends should rapidly decay in proportion to ω2 in the frequency scale of local Rouse-like relaxation within the entanglement segment, as noted from the behavior of homopolymers.50,51 Thus, in that frequency scale, Glocal′(ω) of the blends should be mostly contributed from the local Rouse-like relaxation being insensitive to MPI and MPtBS. Equation 12 leads to a criterion that the difference G′(ω) − Gapp N at low ω (in its terminal relaxation zone) should not depend on the component molecular weights if G′(ω) − Gapp N agrees with the local modulus Glocal′(ω) thereby allowing Gapp N to coincide with the real GN, and vice versa. Following this criterion, Figure 3 tests the behavior of G′(ω) − Gapp N for the

Figure 3. Storage modulus for apparent local relaxation process, G′ − Gapp N , obtained for high-M PI/PtBS blends with wPI = 50 and 25 wt %. F

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high-M blend (dotted curves). At high ω where the local relaxation within the entanglement segment has not completed, Gent*(ω) for the global, entanglement relaxation should remain unrelaxed. This feature of Gent*(ω), noted also from rheooptical analysis for homopolymers,51 represents existence of an onset time of the entanglement relaxation. If the reference blend is adequately chosen, the Gref*(ω) data should match the local modulus Glocal*(ω) of the high-M blend and thus the difference G*(ω) − Gref*(ω) should coincide with Gent*(ω). Then, the difference G*(ω) − Gref*(ω) should exhibit the ω dependence characterizing the behavior before the onset of entanglement relaxation: G′(ω) − Gref ′(ω) → G N ∝ ω0 at high ω

(13a)

G″(ω) − Gref ″(ω) ∝ ω−1 at high ω

(13b)

tan δ ∝ ω−1 at high ω

(13c)

If eq 13 is valid at high ω, the entanglement relaxation has not significantly started at those ω and we can accurately evaluate GN according to eq 13a. Thus, eq 13 serves as an experimental criterion for judging if the reference blend is adequately chosen and the experiment covers sufficiently high ω where the entanglement plateau emerges. This point is further discussed later in section 3-5. When the PI 35/PtBS 42 blend is chosen as the reference, the storage modulus difference G′(ω) − Gref′(ω) decreases on an increase of ω above 100 s−1 and the loss modulus difference G″(ω) − Gref″(ω) and tan δ strongly decrease to negative values; see green circles in Figure 5. As judged from these features, the components in the PI 35/PtBS 42 blend have too large molecular weights and are lightly entangled, so that Gref*(ω) of the reference blend does not match Glocal*(ω) of the high-M blend appearing in eq 11. For this case, the reference data should be expressed as Gref*(ω) = Gl‑ent*(ω) + Glocal*(ω), where Gl‑ent*(ω) is the modulus for the light entanglement relaxation in the reference, and the difference G*(ω) − Gref*(ω) should coincide with Gent*(ω) − Gl‑ent*(ω). At high ω in the local relaxation regime (where Gent′(ω) → GN and Gent″(ω) ∝ ω−1), Gl−ent′(ω) should increase to its plateau value Gl−ent′(∞) that would be smaller than GN because of the “lightness” of entanglement, whereas Gl−ent″(ω) should decrease but still keep a relatively large value that corresponds to the increase of Gl−ent′(ω). Then, G′(ω) − Gref′(ω) (=Gent′(ω) − Gl−ent′(ω)) does not level off at GN but decreases to GN − Gl−ent′(∞) at high ω, whereas G″(ω) − Gref″(ω) (=Gent″(ω) − Gl−ent″(ω)) decreases to negative values because of the rapid decrease of Gent″(ω) (∝ω−1). Thus, the observed feature of G*(ω) − Gref*(ω) indicates oversubtraction of Gref*(ω) due to a mischoice of the reference (PI 35/PtBS 42 blend). In contrast, when the PI 14/PtBS 27 blend is chosen as the reference, G″(ω) − Gref″(ω) and tan δ stay positive and decrease with increasing ω > 0.1 s−1 (see blue circles in Figure 5), but this decrease is much weaker than specified by eqs 13b and 13c. Thus, the components in the PI 14/PtBS 27 blend have too low molecular weights to match Gref*(ω) with Glocal*(ω). For this case, Gref*(ω) should coincide with the modulus Glocal 0.03 s−1; cf. blue and purple symbols. Namely, Glocal* of high-M PtBS 729 was significantly overestimated for those cases. In contrast, for PtBS 42 utilized as the reference, G″ − Gref″ and tan δ stayed positive and decreased gradually with increasing ω > 0.1 s−1, and their ω dependence is slightly weaker than the ω−1 dependence specified by eqs 13b and 13c, meaning that the use of PtBS 42 slightly underestimates Glocal*. Thus, we utilized, in a phenomenological sense, an average of the G* data for PtBS 42 and PtBS 53 as the reference data, as explained for eq 14. It turned out that an average heavily weighing on PtBS 42, Gref‑ave*(ω) = 0.9GPtBS 42*(ω) + 0.1GPtBS 53*(ω), served as the optimum reference data to give the difference G*(ω) − Gref‑ave*(ω) satisfying eq 13; see red dots. The plateau of this G′(ω) − Gref‑ave′(ω) agrees well with the plateau of G′(ω) − GPtBS 42′(ω), the latter being shown with green squares. This agreement suggests satisfactorily high accuracy in evaluation of GN as the plateau value of G′(ω) − Gref‑ave′(ω). Figure 16 shows the modulus for entanglement relaxation of the three high-M PtBS samples, Gent*(ω) = G*(ω) − Gref‑ave*(ω). The same optimum reference data, Gref‑ave*(ω) = 0.9GPtBS 42*(ω) + 0.1GPtBS 53*(ω) (cf. Figure 15), were commonly utilized as the modulus Glocal*(ω) for the local

Figure 17. G* data of PI 1120 and PI 8 samples at 40 °C. O

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Macromolecules between 0.01 and 100 s−1. Thus, the local relaxation negligibly contributes to the G* data of PI 1120 at ω < 100 s−1, and our simple simulation utilizes those data as Gent* due only to the entanglement relaxation. The terminal relaxation time of Gent* is τent = 300 s. In contrast, the GPI 8*(ω) data of unentangled PI 8 can be utilized as Glocal* for the local relaxation. The terminal relaxation time of PI 8 is 1.0 × 10−5 s, but the relaxation time of Glocal*(ω) in our simulation can be tuned if we shift the GPI 8* data along the ω axis by an appropriate factor 1/λ. Namely, for Glocal*(ω) = GPI 8*(ωλ), the relaxation time is given by τlocal = 1.0 × 10−5λ s. We added the Gent*(ω) data at ω < 100 s−1 and the Glocal*(ω) data (= GPI 8*(ωλ)) explained above to evaluate their sum Gsum*(ω) = Gent*(ω) + Glocal*(ω) for several values of the τent/τlocal ratio ranging from 30000 to 100 (that corresponded to 10−3λ = 1−300). This “simulation” of the Gsum*(ω) value was conducted in a purely experimental way so that it is free from any uncertainty in the molecular argument of GN. Figure 18 compares Gent*(ω) (small blue dots connected with blue curves), Glocal*(ω) (green curves), and Gsum*(ω) (large red circles) thus obtained. The loss tangent of Gsum*(ω), tan δ = Gsum″/Gsum′, is also shown (black squares). A decrease of the τent/τlocal ratio broadens the relaxation mode distribution of Gsum*(ω) at low ω to decrease the apparent plateau modulus app Gapp N , as naturally expected. We evaluated GN as the Gsum′ value at the angular frequency ωmin for the minimum of tan δ and/or Gsum′, and compared it with the real entanglement plateau modulus known for PI 1120, GN = 0.48 Pa.16 The results are summarized in Figure 19. Figure 19 demonstrates that the apparent Gapp N is smaller than the real GN by 15−20% even for the case of τent/τlocal = 30000 (Figure 18a) where Gsum*(ω) and Gent*(ω) are almost indistinguishable at low ω ( 20 s−1.) The entanglement relaxation of the blend proceeds in two steps (cf. Figure 1). We may be able to utilize two PtBS samples of adequately chosen molecular weights to mimic values of the global relaxation times of the blend, τent,PI and τent,PtBS, and fit the two-step G* data of the blend with the data averaged for those two PtBS samples and further subjected to the above shift. Because the data of the two PtBS samples agree with each other in the local relaxation regime, we may be able to fit the blend data in the entire relaxation regime with the shifted average data of the two PtBS samples, given that the relationship ζPI = ζPtBS is valid. (The same argument holds for the use of two PI samples.) However, in actual PI/PtBS blends, ζPI differs from ζPtBS (ζPI < ζPtBS) in particular at temperatures not well above Tg,PtBS. Thus, unfortunately, the shifted homopolymer data cannot mimic all values of τglass,X and τa,X (X = PI and PtBS) simultaneously, so that we cannot rigorously fit the blend data in the entire relaxation regime with the homopolymer data.

C-1. Failure of Rigorous Fitting

+ G Rouse − like*(ω)

[bulk] τa,h ∼ ζh[bulk]ah 4 in bulk

τent,X /τa ,X = {ζX /ζPtBSa 4}FX(MPI , MPtBS , a(wPI)) (X = PI, PtBS) in blend

(C9)

For a homopolymer of molecular weight Mh, the ratio is expressed as15 (cf. eqs C7 and C8) [bulk] [bulk] τent,h /τa ,h ∼ Mh 3.5/ah 7

(C10)

Thus, utilizing two PI samples (or two PtBS samples) of adequate molecular weights, we may be able to mimic the τent,PI/τa,PI and τent,PtBS/τa,PtBS ratios in the blend with the ratios for those samples specified by eq C10. Then, we can utilize the

(C6) Q

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Macromolecules G1*(ω) and G2*(ω) data of the two samples at a given temperature to make an average, ϕG1*(ω) + (1 − ϕ)G2*(ω) with ϕ (=0−1) being a weighing factor, and tune the intensity and relaxation time of this average by factors of Q and λ to “simulate” the data for a hypothetical blend, Ghypo*(ω) = Q{ϕG1*(λω) + (1 − ϕ)G2*(λω)}. If this Ghypo*(ω) fits the G*(ω) data of the blend, GN of the blend is estimated from the factor Q and the known plateau modulus GN,h of the PI samples (or PtBS samples) as GN = QGN,h. Here, we should emphasize that the components in the blends (PI and PtBS) have the same characteristic time of the local Rouse-like relaxation within the entanglement segment; see eq C5. To mimic this situation, the above fitting procedure utilizes a pair of chemically identical samples that automatically have the same local Rouse relaxation time at a given temperature (cf. eq C7); no explicit evaluation of this time is needed for the fitting. Consequently, the fit cannot be readily attempted with a pair of chemically different samples (PI and PtBS), because these samples have different Rouse relaxation times unless the temperatures for those samples are finely tuned to ensure coincidence of τ[bulk] and τ[bulk] a,PI a,PtBS. (Practical difficulty of this fine tuning is found also for the case of nonnegligible contribution of the glassy relaxation to the blend data explained in the previous section. Thus, the fitting with a pair of chemically different samples cannot be readily attempted also for that case.) An additional comment should be made for the constraint release (CR) mechanism. In general, this mechanism contributes less significantly to the G* data of monodisperse samples than to the data of binary blends at intermediate ω where the partial CR relaxation of the slow component occurs together with the full relaxation of the fast component.14,43 The partial CR relaxation of the slow component has a Rouse-like, broad mode distribution not observed for monodisperse samples, as noted from extensive data for PI/PI blends.14,43 Thus, the fitting of the blend data with the data of two monodisperse samples (PI or PtBS) can be successfully achieved only when the partial CR relaxation intensity of the slow component is small. This minor CR relaxation is observed when the slow component is concentrated in the blend or when the terminal relaxation times of the fast and slow components are not so widely separated.14,43 Thus, the PI/PtBS blend data may be fitted with the homopolymer data only in limited cases of insignificant glassy relaxation contribution to the blend data and minor partial CR relaxation of the slow component. Nevertheless, the fitting is much simpler than the modulus subtraction explained for eq 13 and may serve as a supplementary method of estimating GN with acceptable accuracy. Thus, we examined the fitting method for representative PI/PtBS blends. The results are summarized below.

Figure 20. Estimation of GN of PI 321/PtBS 219 blend (wPI = 75 wt %) with the aid of data of monodisperse bulk PI 128 and PI 180.

at 40°C multiplied by a factor of Q = 0.853 and shifted along the ω axis by a factor of 1/λ = 0.0178, QGPI 128*(λω) and QGPI 180*(λω). The Q and λ factors are common for the two PI samples, so that the data of the two samples agree with each other at high ω where the entanglement-to-local Rouse transition is observed. More importantly, those factors allowed us to satisfactorily superimpose the PI data on the PI/PtBS blend data at high ω. In the bottom panel of Figure 20, the blend data (red circles) are compared with the hypothetical G h y p o *(ω) = Q{0.5GPI 128*(λω) + 0.5GPI 180*(λω)} (black solid curves). The curves well describe the PI 321/PtBS 219 blend data in the entire range of ω, suggesting that GN of this blend is estimated from the known GN,PI value of PI 128 and PI 180 (= 0.44 MPa) as GN = QGN,PI = 0.38 Pa (at 40°C). This GN is plotted in Figure 11; see the small and filled blue square at wPI = 0.75. For the PI 321/PtBS 219 blend (wPI = 50 wt %) and PI 321/ PtBS 442 blend (wPI = 25 wt %), we focused on various monodisperse PtBS samples utilized in this and previous studies,26,27,45 because PtBS is the major component in those blends. It turned out that the hypothetical Ghypo* for those blends was best simulated with the data of the monodisperse high-M PtBS samples utilized in this study. (The data of those samples are summarized in Appendix A.) As shown in Figure 21, the shifted moduli QGPtBS 219*(λω) and QGPtBS 442*(λω) of PtBS 219 and PtBS 442 at 180°C (with Q = 3.82 and 1/λ = 100) agree well with the data of the PI 321/PtBS 219 blend (wPI = 50 wt %) in the entanglement-tolocal Rouse transition zone at high ω (top panel), and the hypothetical G h y p o *(ω) = Q{0.35G P t B S 2 19 *(λω) + 0.65GPtBS 442*(λω)} satisfactorily describes the blend data (bottom panel). GN of this blend was estimated from the known GN,PtBS value of PtBS (= 0.081 MPa) as GN = QGN,PtBS = 0.29 Pa (at 40°C; after the temperature/density correction explained for Figures 2 and 11). This GN value is plotted in Figure 11; see the small and filled blue square at wPI = 0.5. For the PI 321/PtBS 442 blend (wPI = 25 wt %), the shifted moduli QGPtBS 219*(λω) and QGPtBS 442*(λω) of PtBS 219 and

C-3. Result of Approximate Fitting

In the PI 321/PtBS 219 blend with wPI = 75 wt %, PI is the major component. Thus, we focused on PI samples of various M utilized in this and previous studies,27,39,41−43,57 simulated the hypothetical Ghypo* by averaging the PI data, and compared Ghypo* with the data for the PI 321/PtBS 219 blend. It turned out that 1:1 average of the data of monodisperse PI 128 and PI 180 samples (M = 128k and 180k)27,57 was best compared with the data of the PI 321/PtBS 219 blend, as explained below. In the top panel of Figure 20, large red circles show the blend data at 40°C (identical to the data shown in Figure 1), and small green and blue circles indicate the data of two PI samples R

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shown in Figure 11; see the small and filled green triangle at wPI = 0.25. For the PI/PtBS blends with wPI = 75 and 50 wt %, the GN values estimated from the above fitting agree well with the accurate values obtained from the modulus subtraction explained for eq 13; see Figure 11. This result reflects validity of the prerequisite of the fitting method, negligible contribution of the glassy relaxation to the data of these blends and minor CR relaxation of the slow component (with the latter being consistent with the broad but almost single terminal relaxation of those blends). For such cases, the approximate fitting works well in a practical sense to give the GN value with satisfactory accuracy. We should also emphasize the limitation of the method noted for the blend with wPI = 25 wt % (that does not satisfy the above prerequisite). Despite this limitation, the fitting method would serve as an easy, supplementary method of estimating GN for miscible blends of various polymers.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01866. Glass transition in low-M and high-M PI/PtBS blends (PDF)

Figure 21. Estimation of GN of PI 321/PtBS 219 blend (wPI = 50 wt %) with the aid of data of monodisperse bulk PtBS 219 and PtBS 442.

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AUTHOR INFORMATION

Corresponding Author

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

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was partly supported by the Grant-in-Aid for Scientific Research (B) from MEXT, Japan (grant No. 15H03865), Grant-in-Aid for Scientific Research (C) from JSPS, Japan (grant No. 15K05519), Collaborative Research Program of ICR, Kyoto University (Grant No. 2015-89), and Grant Funds S&T550442S and SIT580584S from Prince of Songkla University, Thailand.

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REFERENCES

(1) Ferry, J. D. Viscoelastic Properties of Polymers; 3rd ed., Wiley: New York, 1980. (2) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: New York, 2003. (3) Graessley, W. W. Polymeric Liquids and Networks: Dynamics and Rheology; Garland Science: New York, 2004. (4) Watanabe, H. Prog. Polym. Sci. 1999, 24, 1253. (5) McLeish, T. C. B. Adv. Phys. 2002, 51, 1379. (6) Marrucci, G. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 159. (7) Park, S. J.; Larson, R. G. J. Rheol. 2003, 47, 199. (8) Wang, Z.; Chen, X.; Larson, R. G. J. Rheol. 2010, 54, 223. (9) van Ruymbeke, E.; Bailly, C.; Keunings, R.; Vlassopoulos, D. Macromolecules 2006, 39, 6248. (10) van Ruymbeke, E.; Kapnistos, M.; Vlassopoulos, D.; Liu, C. Y.; Bailly, C. Macromolecules 2010, 43, 525. (11) van Ruymbeke, E.; Masubuchi, Y.; Watanabe, H. Macromolecules 2012, 45, 2085. (12) Milner, S. T.; McLeish, T. C. B. Macromolecules 1997, 30, 2159. (13) Das, C.; Inkson, N. J.; Read, D. J.; Kelmanson, M. A.; McLeish, T. C. B. J. Rheol. 2006, 50, 207. (14) Watanabe, H. Polym. J. 2009, 41, 929.

Figure 22. Estimation of GN of PI 321/PtBS 442 blend (wPI = 25 wt %) with the aid of data of monodisperse bulk PtBS 219 and PtBS 729.

PtBS 729 at 180°C (with Q = 2.63 and 1/λ = 45.7) deviated from the blend data at high ω; see top panel of Figure 22. This deviation, noted for G″, can be partly attributed to the glassy relaxation contribution that enlarges G″ but hardly affects G′ at the high-ω side of our experimental window. Thus, the hypothetical G h yp o *(ω) = Q{0.76G P t BS 2 19 *(λω) + 0.24GPtBS 729*(λω)} could describe the blend data at low-tomiddle ω but not at high ω (bottom panel). A rough estimate of GN of the blend obtained from the fit, GN = QGN,PtBS = 0.18 MPa (at 40°C; after the temperature/density correction) is S

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DOI: 10.1021/acs.macromol.5b01866 Macromolecules XXXX, XXX, XXX−XXX