Component Relaxation Times in Entangled Binary Blends of Linear

Article pubs.acs.org/Macromolecules

Component Relaxation Times in Entangled Binary Blends of Linear Chains: Reptation/CLF along Partially or Fully Dilated Tube Hiroshi Watanabe,†,* Yumi Matsumiya,† and Evelyne van Ruymbeke‡ †

Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan Bio and Soft Matter, Institute on Condensed Matter and Nano-science, Université Catholique de Louvain, Louvain-la-Neuve, Belgium

‡

S Supporting Information *

ABSTRACT: Recent dielectric analysis suggested that entangled linear cis-polyisoprene (PI) chains in monodisperse bulk exhibit, in the terminal relaxation regime, reptation/ contour length fluctuation (CLF) along a partially dilated tube with its diameter being determined by the constraint release (CR) activated tension equilibration along the chain backbone (Matsumiya et al. Macromolecules 2013, 46, 6067). In relation to this finding, we re-examined the dielectric and viscoelastic terminal relaxation times of components in linear PI blends having various component molecular weights and volume fractions, Mi and υi (i = 1 and 2 for the short and long components). In entangling blends with M2 ≫ M1 and large υ2 (>critical volume fraction υ2e for the onset of long−long entanglement), the relaxation time τ2,b of the long chain decreases with decreasing υ2 but stayed considerably larger than τ2,soln of the same long chain in a solution having the same υ2. This result suggested that the CR-activated tension equilibration retards the reptation/CLF motion of the long chain in such blends. A simple “solution model” considering this retardation due to the CR relaxation of short−long entanglements was formulated. Utilizing data for the CR relaxation time τdil‑2,CR of dilute long chains (with υ2 < υ2e), the model described the τ2,b data for υ2 > υ2e very well. Nevertheless, this model could not apply to the cases where M2 and M1 are rather narrowly separated and the short−long entanglements considerably survive in the time scale of the long chain relaxation. For this case, a “blend model” was formulated to consider self-consistently, though in an approximate way, the CR relaxation of all species of entanglements (short−short, short−long, long−short, and long−long entanglements) thereby mimicking coupled relaxation of the long and short chains. The component relaxation times deduced from this model (again on the basis of the τdil‑2,CR data) were surprisingly close to the data, not only for the PI/PI blend having narrowly separated M2 and M1 but also for those with M2 ≫ M1 (the latter being described satisfactorily also with the solution model), suggesting that reptation/CLF of the components in the terminal relaxation regime occurs along partially dilated tube with the diameter being determined by the CR-activated tension equilibration. Furthermore, the “blend model” worked satisfactory also for literature data for polystyrene blends having various M2/M1 ratios. These results demonstrate the importance of CR-activated tension equilibration in the blends, which is consistent with the finding for monodisperse bulk.

1. INTRODUCTION Entanglement dynamics is one of the central subjects in polymer physics and has been investigated extensively.1−4 The entanglement undoubtedly results from mutual uncrossability of real polymer chains. In the widely utilized tube model, this uncrossability is represented as a tube surrounding a focused chain referred to as a “probe”, and the probe is constrained to move essentially along the axis of this tube via several mechanisms, for example, reptation and contour length fluctuation (CLF) for linear chains and arm retraction (AR) for branched chains.1−4 This treatment in the tube model greatly simplifies description of the probe motion. Nevertheless, the motional coupling between the probe and the tube-forming chains (that surround the probe) strongly affects the probe © 2013 American Chemical Society

dynamics, in particular when the probe motion is equally slow and/or slower compared to the motion of the tube forming chains. This motional coupling effectively allows the tube for the probe to move in space. The model incorporates the tube motion as the constraint release (CR) mechanism1−4 to consider that the probe motion is determined by competition between the CR mechanism and the mechanisms occurring along the tube axis, e.g., reptation, CLF, and AR. The CR molecular picture regards the entanglement segment as the motional unit to describe the probe motion. A simplified Received: September 11, 2013 Revised: November 2, 2013 Published: November 20, 2013 9296

dx.doi.org/10.1021/ma4018795 | Macromolecules 2013, 46, 9296−9312

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relationship, μ(t) = φ′(t)/β(t) with β(t) being the maximum possible number of entanglement segments that can be CRequilibrated in time for a given value of μ(t).14,17b,19 In relation to this result, it should be emphasized that the long chain component in PI/PI blends recovers validity of eq 1 in its terminal relaxation regime, because this component exhibits a plateau of its modulus at long times: In this plateau zone, βfull‑DTD is just weakly dependent on t and stays rather small thereby allowing the CR-equilibration over βfull‑DTD entanglement segments to occur in time. The validity of full-DTD relationship for monodisperse linear PI allowed us to evaluate the magnitude of spatial coarsegraining in its terminal regime, βfull‑DTD ≅ 3 for PI with various molecular weights17b (also see Figure S2 in the Supporting Information of ref 15). Thus, ≅3 entanglement segments together behave as a stress sustaining unit in this regime. However, a recent experiment15 indicated that the above full/ partial-DTD picture is still incomplete in a sense that the picture does not specify the terminal relaxation time (instead, the picture just specifies the relationship between μ(t) and φ′(t) in a given time scale): The experiment examined the dielectric relaxation of dilute linear PI (probe; having molecular weight M1 = 21k−179k and volume fraction υ1 = 0.1 or 0.2) in a matrix of much longer PI (M2 = 1100k) to reveal that the relaxation in this matrix is retarded, by a factor of 1.3−1.8 (that increased with decreasing M1), compared to the relaxation of the same probe in monodisperse bulk. This retardation, due to suppression of CR/DTD in the long matrix, is qualitatively consistent with the validity of the full-DTD relationship for monodisperse bulk PI. However, quantitatively, the observed retardation is less significant compared to the retardation by a factor of βfull‑DTD1.5 ≅ 5.2 that is expected if reptation/CLF in the monodisperse bulk occurs f reely along the fully dilated tube. (Because βfull‑DTD is equivalent to the effective entanglement molecular weight Me,eff describing the modulus in the terminal regime, the relaxation time in monodisperse bulk systems scale as Me,eff−1.5 ∼ βfull‑DTD−1.5 and the retardation factor is given by βfull‑DTD1.5 if reptation/CLF in those systems freely occurs as in entangled solutions in low-M solvents: Experimentally, the terminal relaxation time of those solutions, defined as a product of the zero-shear viscosity and steady state compliance, is known to scale as M2+δ/Me,solnδ with δ ≅ 1.5.3,4,21,22 The same scaling can be also deduced from a tube model-based argument that the entanglement retards the relaxation, compared to the intrinsic Rouse relaxation, by a factor of {M/Me,soln}δ with δ ≅ 1.5 if the chain relaxes through reptation/CLF.) The above difference between the observed and expected retardation factors led us to hypothesize that reptation/CLF requires the chain tension to be CR-equilibrated along the chain backbone, not just locally over βfull‑DTD entanglement segments in a direction lateral to the backbone.15 Then, the chain exhibits reptation/CLF along a partially dilated tube that wriggles in the fully dilated tube, the latter being characterized with βfull‑DTD ≅ 3. Analysis based on the CR time data allowed us to estimate the number of entanglement segments characterizing this partially dilated tube, β* ≅ 1.2−1.4 (that decreased gradually with increasing M1).15,23 These β* values were in harmony with the observed retardation factor (β*1.5 agreed with this factor),15 which suggested the importance of the tension equilibration for reptation/CLF. This importance was also deduced from theoretical analysis.13 Thus, the chain should have two dif ferent length scales of DTD, namely lateral and along the chain backbone, the former due to the local CR

version of CR, referred to as the dynamic tube dilation (DTD),2,3,5,6 focuses on mutual CR-equilibration of those entanglement segments to utilize the group of equilibrated segments as a coarse-grained (dilated) motional unit, with its size enlarging on an increase of the time scale of our focus. Within this DTD molecular picture, the probe is composed of enlarged units and always moves along the dilated tube via the reptation/CLF/AR mechanisms. Thus, the combination of DTD and reptation/CLF/AR mechanisms allows us to simplify the description of the probe motion. (It should be noted that the coupling between the probe and tube-forming chains can be described in a dual form, with either CR or DTD molecular picture, although the latter picture is formulated in the coarsegrained time scale thereby smearing some details of the fast CR motion.) The current version(s) of the tube model formulated as above has been compared with experimental data.7−13 In particular, the combination of the CR/DTD mechanism with the reptation/CLF/AR mechanisms has been most critically tested for binary blends of chemically identical components having different molecular weights and/or architectures. For this test, it is very useful to compare physical quantities that differently average the same, stochastic chain motion.14,15 cisPolyisoprene (PI) has the so-called type-A dipoles parallel along the chain backbone, and its dielectric data reflect the orientational memory of the end-to-end vector (for linear chains) and/or end-to-branching point vector (for starbranched chains), i.e., the correlation of these vectors at two separate times (0 and t).3,14,16 In contrast, viscoelastic data detect the anisotropy of orientational distribution of the entanglement segments at a given time and do not directly reflect this two-time correlation.3,14,16 Thus, comparison of the dielectric and viscoelastic data of PI serves as an ideal test for the above purpose. This comparison has been made to reveal that monodisperse linear PI obeys the full-DTD relationship between the normalized viscoelastic and dielectric relaxation functions, μ(t) and Φ(t),14,15,17 μ(t ) = {φ′(t )}1 + d ≅ {Φ(t )}2 for linear monodisperse PI (1)

where d (≅ 1.3 for PI) is an effective dilation exponent experimentally defined in the terminal relaxation zone, and φ′(t) is a dielectrically determined survival fraction of the dilated tube: φ′(t) is indistinguishable from Φ(t) at short/ intermediate t and becomes moderately larger than Φ(t) at long t in the terminal regime, and is numerically close to {Φ(t)}2/(1+d) = {Φ(t)}0.87 in the entire range of t; see Figure S2 in Supporting Information of ref 15. Equation 1 is valid if the system satisfies the full-DTD criterion that βfull‑DTD entanglement segments, with βfull‑DTD = 1/{φ′(t)}d, are quickly CR-equilibrated in the time scale of t to allow the relaxed portion of the chain to behave as a “solvent”. In fact, validity of this criterion was confirmed for monodisperse linear PI.14,17b In contrast, monodisperse star PI does not obey eq 1 because star PI has a broad relaxation mode distribution to exhibit intensive relaxation at short/intermediate times.14,18−20 For this case, βfull‑DTD in the terminal regime is too large to allow the CR-equilibration over βfull‑DTD entanglement segments to occur in time. Similarly, for blends of long and short linear PI, βfull‑DTD becomes too large at intermediate times just after the short chain relaxation and thus eq 1 fails in that time scale.14,17 For these cases, experiments showed validity of a partial-DTD 9297


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blend. Thus, we utilized the G* and ε″ data to evaluate τ[X] 1,b in this study. For the PI 329k/PI 14k blend having various volume fractions υ2 of the long chain, no dielectric measurement was made but the G* data were obtained in the previous study.24 Those G* data and the corresponding τ[G] i,b data were published in ref 24 only for the case of CR dominant relaxation of the long chain achieved at small volume fraction υ2 (=0.01) of this chain. Thus, in this study, we utilized the G* data (reproduced in Supporting Information) to evaluate τ[G] i,b for all υ2 values. The method of evaluation, based on the subtraction of the short component modulus from the blend modulus,15,17 is also described in Supporting Information. For the two PS/PS blends, the G* data were reported in ref 25 but the relaxation times shown therein were evaluated from the analysis of the relaxation spectra. (PS does not have the type-A dipole and no dielectric measurement was attempted.) Thus, we utilized those G* data to re-evaluate τ[G] i,b with the subtraction method explained in Supporting Information. These newly obtained τ[G] i,b values were close to the previous values obtained from the spectra analysis. 2.2. CR Contribution in the Blends Examined. The Struglinski−Graessley parameter26 rSG shown in Table 1 is defined by rSG = 3M2Me02/M13 = 3N2/N13, where Ni = Mi/Me0 with Me0 being the entanglement molecular weight in bulk. This parameter gives a measure of a ratio of the pure reptation time of the long chain, τ2,rep ∝ M23/Me0, to the longest Rouse−CR time of the dilute long chain entangled only with the short chains, τdil‑2,CR ∝ M13M22/Me03. Qualitatively, we can consider that the long chain motion in the blend is dominated by reptation if rSG is small, and by CR if rSG is large. However, rSG just compares the characteristic times of the slowest eigenmodes of the CR and reptation mechanisms and neglects coupling between all eigenmodes of these mechanisms. In real blends, this coupling occurs more or less,15 so that rSG gives just a rough measure for judging whether CR or reptation dominates the long chain relaxation. Nevertheless, a large difference of the rSG values of the blends listed in Table 1 suggests that analysis of the τ[X] i,b data (X = G, ε) made in this study covers a wide variety of situations having different contributions of the CR dynamics to the long chain motion.

equilibration determining the modulus level, whereas the latter being related to the CR-activated tension equilibration that governs the reptation/CLF time along the partially dilated tube.15,23 In relation to this result, we remember that eq 1 characterizing the local CR/DTD is valid for the terminal relaxation of the long chain in the PI/PI blend with M2 ≫ M1.14,17a However, it has not been critically examined whether the tension equilibration due to the large scale CR along the chain backbone affects the relaxation time of the components therein. Thus, we analyzed the relaxation time data for PI/PI blends17,24 as well as polystyrene/polystyrene (PS/PS) blends25 reported in literature. It turned out that the relaxation of the long chain in the blend with M2 ≫ M1 remains slower than that in the corresponding solution. This result suggests that the reptation/CLF of the long chain in the blend is indeed retarded until the tension equilibration is achieved through the CR-relaxation of the short−long entanglements. On the basis of this result, we formulated a simple “solution model” that mimics this retardation and a little more elaborated “blend model” that considers relaxation of all species of entanglements self-consistently (though in an approximate way). These models, utilizing the CR time data of dilute long chains in short chain matrices and the relaxation times of monodisperse bulk, were found to describe satisfactorily the component relaxation times in the blends. Details of these results, demonstrating the importance of the CR-activated tension equilibration in the blends, are presented in this article.

2. EXPERIMENTAL SECTION 2.1. Materials. Table 1 summarizes the component molecular weights and experimental temperatures for the PI/PI and PS/PS

Table 1. Characteristics of Binary Blends of Linear Components Examined in This Study blend code PI 308k/PI 21ka PI 308k/PI 94ka PI 329k/PI 14kb PS 316k/PS 39kb,c PS 316k/PS 89kb,c

10−3M1 (N1) 21.4 94.0 14.4 38.9 88.6

(4.28) (18.8) (2.88) (2.16) (4.92)

10−3M2 (N2) 308 308 329 316 316

(61.6) (61.6) (65.8) (17.6) (17.6)

rSG

T/°C

ref

2.36 0.03 8.26 5.22 0.44

40 40 40 167 167

17a 17b 24 25 25

3. RESULTS AND DISCUSSION 3.1. Overview. Figures 1 and 2 show the second-moment average, dielectric and viscoelastic terminal relaxation [G] times,3,14,16 τ[ε] i,b and τi,b (blue squares), obtained for the long and short chains (i = 2 and 1) in the two PI/PI blends as indicated. This well-defined average relaxation time is unequivocally evaluated in the terminal relaxation regimes of those chains, and is close to the longest relaxation time.3,14,16 [G] Thus, in this study, we utilize the τ[ε] i,b and τi,b data as the longest relaxation time data. These data are double-logarithmically plotted against the long chain volume fraction, υ2. Hereafter, the superscripts [ε] and [G] standing for the dielectric and viscoelastic times are shown only when necessary. In Figures 1 and 2, the horizontal, purple dotted lines show the τ1,m data of the short chain in its monodisperse bulk, and the green solid lines, the CR relaxation time τdil−2,CR of the dilute long chain (PI 308k) being entangled only with the short chains (PI 21k or PI 94k). τdil‑2,CR was evaluated from previously reported empirical equations,15,24,27

a Both viscoelastic and dielectric data are available. bOnly viscoelastic data are available. cIn the previous report,25 the PS samples were coded with the number-average molecular weights.

blends examined in this study. All components had narrow molecular weight distribution (Mw/Mn < 1.1), and Mi (with i = 1 and 2 for the short and long components) indicates the weight-average molecular weight. The number of entanglements per component chain, Ni = Mi/ Me0 with Me0 being the entanglement molecular weight in bulk (10−3Me0 = 5 and 18 for PI and PS, respectively21), is also shown in Table 1. For all those blends, viscoelastic data were reported in the previous papers.17,24,25 Dielectric data were available for the two PI/PI blends17 containing PI 308k as the long chain component. In this study, we compare the viscoelastic and dielectric relaxation [ε] [G] [ε] times τ[G] i,b and τi,b of the components in those blends with τi,m and τi,m in respective bulk systems. These relaxation times, defined as the second-moment average relaxation times,3,14,16 were already published and/or evaluated in this study, as described below. For the PI 308k/PI 21k and PI 308k/PI 94k blends, the viscoelastic modulus (G*) and dielectric loss (ε″) data as well as the τ[X] i,b and τ[X] i,m data (X = G, ε) have been reported in refs 17a and17b, except for the τ[X] 1,b data of the short chain component (PI 21k) in the former

[G] −25 3 τdil M1 M 2 3 (in s) − 2,CR = 1.0 × 10

(2a)

[ε] −25 3 τdil M1 M 2 3 (in s) − 2,CR = 2.0 × 10

(2b)

τdil−2,CR given by these equations are the second-moment average CR relaxation times that can be directly compared with the τ data. Because those data are utilized as the longest relaxation times, τdil−2,CR is also regarded as the longest CR time in this study.27 For the PI 308k/PI 21k and PI308k/PI 94k 9298


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Figure 2. Component relaxation times in PI 308k/PI 94k blends at 40 °C.

Figure 1. Component relaxation times in PI 308k/PI 21k blends at 40 °C. Note that τ1,b is multiplied by a factor of 10.

For υ2 > 2/N2, respective long chains have more than two long−long entanglements, on average, and their dynamics is significantly affected by those entanglements as well as the short−long entanglements. For such mutually entangled long chains, the average number of short−long entanglements for a subchain (a section of backbone) between the long−long entanglements is given by υ1/υ2, as shown in Figure 3. The PI 308k/PI21k and PI 308k/PI94k blends exhibit a clear difference in the CR-equilibration of those υ1/υ2 short−long entanglements, as explained below. The Rouse-type CR-equilibration time of the subchain constrained by the υ1/υ2 short−long entanglements, τsub,CR, can be related to the CR time τdil‑2,CR of the dilute long chain in the short chain matrix as τsub,CR ≅ {(υ1/υ2)/N2}2τ[ε] dil‑2,CR{τ1,b/ τ1,m}, where the factor (υ1/υ2)/N2 is a ratio of the short−long entanglement number of the subchain to that of the dilute long chain as a whole, and the τ1,b/τ1,m ratio represents an effect of the retardation of the short chain motion in the blend. In the PI 308k/PI21k blend with any υ2 value, τ2,b of the long chain is much longer than τ1,b of the coexisting short chain (τ2,b/τ1,b > 300) and never becomes shorter than τdil‑2,CR; see Figure 1. Thus, τsub,CR never exceeds the τ2,b data. Then, in a simple analysis, the enlarged entanglement segment similar, in size, to

blends having large and small rSG values (cf. Table 1), respectively, τdil‑2,CR is in agreement with and much longer than the τ2,b data for the dilute long chain. The horizontal black dashed lines show the intrinsic Rouse relaxation time of the long chain, τ2,Rouse.28 The filled black circles connected by the black lines indicate the τ2,soln data of PI308k solutions in an oligomeric solvent (vinyl-rich oligomeric butadiene (oB) with M ≅ 2k) in the iso-frictional state.17 This oB is a marginal solvent for PI.29 Clearly, τ2,soln is shorter than τ2,b of the long chain even in the PI 308k/PI 21k blend where the short chain relaxes much faster than the long chain. In addition, τ1,b of the short chain in the blend increases on approach of υ2 to unity. Models are formulated later to address these characteristic features of the blends. Here, we focus on the number of short−long and long−long entanglements per long chain, υ1N2 and υ2N2 (with υ1 = 1−υ2; short chain volume fraction). If the short chain relaxes much faster than the long chain, the number of ef fective long−long entanglements per long chain decreases to υ2dN2 (d ≅ 1.3) in long time scales, which is attributed to the CR-activated tension equilibration of the long chain.13 However, the focus here is placed on the nominal entanglement number at short times. 9299


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For this problem, a more elaborated model, later referred to as the “blend model”, is formulated, with an attempt of describing this full coupling. The blend model incorporates the coupling through self-consistent Rouse−CR calculation of the survival fractions of all entanglement species (short−short, short−long, long−short, and long−long entanglements), although the calculation is made in an approximate way. In the remaining part of this paper, details are explained first for the starting analysis (comparison with the solution data) and then for the two models. The model predictions are compared with the data of the PI/PI and PS/PS blends listed in Table 1. 3.2. Analysis for PI 308k/PI 21k Blend Based on Solution Data. 3.2.1. Comparison of Blend and Solution Data. In the PI 308k/PI 21k blend with υ2 > 2/N2 ≅ 0.03, the υ1/υ2 short−long entanglements (cf. Figure 3) have been CRequilibrated in the time scale of terminal relaxation of the long chain component, as explained in the previous section. Thus, naively, τ2,b of this component is expected to agree with τ2,soln of the PI 308k/oB solution in the iso-frictional state. However, for 0.03 ≤ υ2 ≤ 0.2, the τ2,b data are considerably larger than the τ2,soln data, as noted in Figure 1 (compare blue squares and black circles). This difference can be mainly attributed to the prerequisite for reptation/CLF of the long chain,15 the tension equilibration, as explained below. In the solution, the subchain between the long−long entanglements equilibrates its tension through the intrinsic Rouse motion; see the right part of Figure 3. The time required for this equilibration is specified as τsub,Rouse = KRouse(Me0/υ2)2, where Me0/υ2 is the molecular weight of this subchain and KRouse (=4.2 × 10−13 s for dielectric τsub,Rouse)28 is an empirically determined prefactor that describes the intrinsic Rouse time as a function of the subchain molecular weight. In contrast, the tension equilibration of the subchain in the blend is achieved through the CR relaxation of the υ1/υ2 short− long entanglements constraining the subchain; see the left part of Figure 3. The time required for this equilibration is given by τsub,CR ≅ {(υ1/υ2)/N2}2τ[ε] dil‑2,CR{τ1,b/τ1,m}, as explained earlier. Thus, to the first approximation, the terminal relaxation time (reptation/CLF time) of the mutually entangled long chains having υ2 > 2/N2 in the blend, τ2,b, is expected to be longer than τ2,soln of the same chain in the solution by an equilibration factor specified as

Figure 3. Schematic illustration of entanglement equilibration for long chain in PI 308k/PI 21k blend and PI 308k/oB solution with υ2 > 2/ N2.

that in the solution can be taken as the motional unit for the terminal relaxation of the long chain in the blend. In contrast, in the PI 308k/PI 94k blend having a small rSG value (cf. Table 1), the τ2,b/τ1,b ratio never exceeds 30 and τ2,b is smaller than τdil‑2,CR; see Figure 2. For this case, a considerable fraction of short−long entanglements still survives in the time scale of the terminal relaxation of the long chain, and the motional unit in this relaxation should be smaller than the enlarged segment explained above. Considering the above difference, we analyzed the υ2 dependence of τ2,b and τ1,b in the PI 308k/PI 21k and PI 308k/PI 94k blends and further formulated two models. Before going into details of the analysis and models, the molecular idea(s) underlying the analysis and models is summarized here as a guide for the remaining part of this paper. The starting point is the difference between τ2,b and τ2,soln of the long chain in the blend and solution: τ2,b is longer than τ2,soln, as already noted in Figures 1 and 2. In the analysis for the PI 308k/PI 21k blend, the solution is regarded as a reference system to relate this difference to delay of the tension equilibration for the long chain in the blend due to CR relaxation of the short−long entanglements (as depicted in Figure 3). Then, a simple model, later referred to as the “solution model”, is formulated on the basis of this molecular picture. Specifically, the retardation is quantified through Rouse−CR calculation of the fraction of short−long entanglements surviving in the time scale of τ2,b. The increase of τ1,b of the short chain on blending is also described as a result of suppression of CR for the short chain on blending. In this simple model, the solution is utilized as the reference so that the coupling of the dynamics of the short and long chains is not fully taken into account. Actually, the long chain motion is affected by the short chain motion and this effect is fed back to the short chain motion, and vice versa, to fully couple the motion of those chains. Thus, the solution model is applicable to the PI 308k/PI 21 blend (in which the short chain relaxes much faster than the long chain and the coupling is rather weak) but not to the PI 308k/PI 94k blend.

θeq =

τsub,CR τsub,Rouse

≅

[ε] ⎧ [ε] ⎫ υ12τdil ‐ 2,CR ⎪ τ1,b ⎪ ⎬ ⎨ [ε] ⎪ KRouseM 2 2 ⎪ ⎩ τ1,m ⎭

(for υ1/υ2 > 1) (3)

(Here, we have assumed that the number of short−long entanglements per subchain, υ1/υ2, is larger than unity.) τ[ε] 1,b and τ[ε] 1,m appearing in eq 3 are the dielectric relaxation time (end-toend relaxation time) of the short chain in the blend and monodisperse bulk, respectively. Note that the viscoelastic τ[G] 1,b and τ[G] 1,m are affected by the local DTD lateral to the chain backbone and thus not adequate as the times defining just the retardation of reptation/CLF of the long chain. [ε] We evaluated the θeq factor from the data of τ[ε] 1,b , τ1,m, and [ε] τdil‑2,CR (cf. eq 2b). The relaxation time thus expected for the long chain in the blend, θeqτ[X] 2,soln (X = ε, G), are shown in Figure 1 with the red curves. These curves are close to the τ[X] 2,b data of the long chain with large υ2 (>0.1; see Figure 1), suggesting that the long chain in the PI 308k/PI 21k blend relaxes essentially through reptation/CLF along the dilated 9300


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tube similar to that in the solution but this relaxation is retarded in the blend due to the CR-activated tension equilibration. This molecular picture can be cast in a simple “solution model” if we self-consistently consider the times required for the tension equilibration and reptation/CLF of the long chain, as explained below. 3.2.2. Solution Model Considering CR-Equilibration of Short−Long Entanglements. 3.2.2a. Dielectric Relaxation Time of Long Chain in PI 308k/PI 21k Blend. For the mutually entangled long chains in the PI 308k/PI 21k blend with υ2 > 2/ N2 ≅ 0.03, we first focus on the dielectric relaxation time (endto-end relaxation time) τ[ε] 2,b . The end-to-end relaxation is rather insensitive to the local relaxation including the DTD lateral to the chain backbone, which allows us to straightforwardly consider the similarity between this blend and the solution explained in section 3.2.1 and also the CR-equilibration in the [ε] blend to formulate the solution model for τ2,b . (For convenience of the readers, the parameters utilized/analyzed in the model are summarized in Appendix.) Because the viscoelastic τ[G] 2,b is sensitive to this lateral DTD, it is not so straightforward to formulate a model for it. Thus, τ[G] 2,b is later evaluated on the basis of the model for the dielectric τ[ε] 2,b combined with the partial-DTD analysis. In the solutions in the iso-frictional state, the dielectric τ[ε] 2,soln of the long chain can be described by an empirical equation, [ε] 1.5 τ[ε] with τ[ε] 2,soln = τ2,mυ2 2,m being the data in monodisperse bulk, as noted for the black circles in the top panel of Figure 1. Thus, we can express τ[ε] 2,b in the blend as [ε] [ε] [ε] 1.5 τ2,b = τ2,m {υ2 + υ1*(τ2,b )} θeq*

for υ2 > 2/N2

τq(s ‐ l) = τ1(s ‐ l)

θeq* =

{υ1 −

[ε] ⎫ ⎧ ⎪ τ1,b ⎪ ⎨ ⎬ ⎪ [ε] ⎪ ⎩ τ1,m ⎭

υ1N2

⎞ t ⎟ (s ‐ l) ⎟ ⎝ τq ⎠

q=1

(7a)

(7b)

As shown in eq 7b, the slowest CR time defined for the υ1N2 short−long entanglements is shorter than τ[ε] dil−2,CR for the N2 short−long entanglements (of the dilute long chain) by the factor of υ12 because of the Rouse−CR feature. Equation 7b also indicates that the CR modes are activated by the short chain motion thereby being retarded by the relaxation time [ε] ratio of this chain in the blend and monodisperse bulk, τ[ε] 1,b /τ1,m. [ε] As noted from eqs 4−6, τ2,b and υ1* are mutually coupled so that a consistent set of τ[ε] 2,b and υ1* values is obtained through [ε] iterative calculation given that the τ[ε] 1,b /τ1,m ratio is known. If we [ε] [ε] utilized the τ1,b and τ1,m data of the short chain, we can straightforwardly make this calculation. However, in this study, we attempt to formulate a model also for τ[ε] 1,b so as to correlate the behavior in the blends with that in monodisperse bulk examined in a recent study.15 This model, being combined with [ε] eqs 4−7, enables us to consistently calculate τ[ε] 1,b and τ2,b , as shown later in Figure 4. It should be noted that eqs 4−7 apply only to concentrated and mutually entangled long chains having υ2 > 2/N2 (≅ 0.03 for PI 308k). For less concentrated long chains, we may simply assume the Rouse−CR behavior,

(4)

[ε] [ε] τ2,b = τdil − 2,CR

for υ2 < 2/N2

(8)

although the long chain should exhibit a rather broad crossover from the reptation/CLF behavior to the CR behavior on a decrease of υ2 to and below 2/N2. The chain motion specified by eq 8 is analogous to but slower than the intrinsic Rouse motion in dilute solutions (compare the black and green horizontal lines in Figure 1). Namely, the CR-equilibration of the short−long entanglements retards the relaxation also for the dilute long chain. 3.2.2b. Dielectric Relaxation Time of Short Chain in PI 308k/PI 21k Blend. For discussion of τ[ε] 1,b of the short chain in the blends, it is informative to first summarize the dynamics in monodisperse bulk, reptation/CLF along the partially dilated tube.15 The partially dilated tube diameter is expressed as a* = aβ*1/2, where a is the diameter of undilated tube and the dilation parameter β* specifies the number of dilated entanglement segments, N/β*, that remain independently among the whole (N) entanglement segments. Suppression of this partial-DTD increases the relaxation time by a factor of β*1.5. This β* is determined from the prerequisite that the chain can exhibit reptation/CLF along the partially dilated tube only when the CR equilibration along the chain backbone over g entanglement segments (g = N − N/β* + 1) occurs in time.15 For monodisperse bulk, this prerequisite is cast in a simple [ε] relationship τ[ε] m ≥ τp*, where τm is the measured dielectric relaxation time and τp* is the characteristic time of the slowest possible CR mode (p*th mode) available for the in-time equilibration: With the aid of the Rouse−CR model, τp* is [ε] expressed in terms of the longest CR time τ[ε] CR,m as τp* = τCR,m 2 −2 sin (π/2N) sin (p*π/2N), and the mode order p* is related

(5)

⎛

∑ exp⎜⎜−

qπ 2N2υ1

τ(s−l) 1

[ε] [ε] υ1*(τ2,b ) = υ1Ψ(1)(τ2,b )

1 υ1N2

2

[ε] ⎫ ⎧ ⎪ τ1,b ⎪ [ε] ⎨ ⎬ τ1(s ‐ l) = υ12τdil ‐ 2,CR ⎪ [ε] ⎪ ⎩ τ1,m ⎭

For υ2 just above 2/N2, τ[ε] 2,b becomes considerably short (and [ε] close to τ[ε] dil−2,CR ; cf. Figure 1) so that some fraction υ1*(τ2,b ) of the short−long entanglements may survive in the time scale of [ε] τ[ε] 2,b thereby increasing τ2,b . This effect has been considered in [ε] eq 4 through the υ1*(τ2,b ) term added to the net fraction of the long chain, υ2. (The sum, υ2 + υ1*, shows an effective fraction of the entanglements constraining the long chain motion.) Those surviving short−long entanglements are not involved in the CR-activated tension equilibration that retards reptation/ CLF of the long chain. For this reason, the equilibration factor (originally given by eq 3) has been modified to θeq* shown in eq 5. With the aid of the Rouse−CR model, the surviving fraction υ1*(τ[ε] 2,b ) defined for the υ1N2 short−long entanglements (per long chain) can be expressed as17b

with Ψ(1)(t ) =

π 2N2υ1

with

with [ε] υ1*(τ2,[εb])}2 τdil ‐ 2,CR KRouseM 2 2

( ) sin ( ) sin 2

(6)

Here, τ(s−l) indicates the relaxation time of qth CR mode given q by 9301


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to the equilibrated entanglement number g as15 1/p* = {(N−2) g+1}/(N−1)2 (→ g/N for N ≫ 1). This relationship specifies the smallest possible p* value thereby allowing us to determine the maximum β* (and g) value satisfying τ[ε] m ≥ τp*. This maximum β* gives the retardation factor on suppression of partial-DTD, β*1.5. Now, we turn our attention to the PI 308k/PI 21k blend. In this blend, the long chain motion is much slower than the short chain motion so that the long−short entanglement for the short chain can be treated, to the first approximation, as the fixed entanglement in the time scale of the short chain relaxation, τ[ε] 1,b . For this case, the tube for the short chain dilates only through the CR relaxation of N1υ1 short−short entanglements, and its dilation parameter β1,b* is smaller than β1,m* in monodisperse bulk. Then, the short chain relaxation is retarded on blending by a factor of (β1,m*/β1,b*)1.5, and τ[ε] 1,b is related to 1.5 [ε] [ε] τ[ε] 1,m in monodisperse bulk as τ1,b = (β1,m*/β1,b*) τ1,m. The slowest CR time of the N1υ1 short−short entanglements, τ[ε] CR1,b, 2 is shorter than τ[ε] CR1,m in monodisperse bulk by a factor of υ1 1.5 but should be also increased by the factor of (β1,m*/β1,b*) (= [ε] τ[ε] 1,b /τ1,m). Thus, the above reptation/CLF prerequisite for the chain in monodisperse bulk can be modified for the short chain in the blend as [ε] τ1,b

sin 2 ⎛ β * ⎞1.5 ⎛ β * ⎞1.5 1,m 1,m ε ε [ ] 2 [ ] ⎟ τ1,m ≥ ⎜ ⎟ = ⎜⎜ ⎟ ⎜ β * ⎟ υ1 τCR1,m 2 ⎝ β1,b* ⎠ ⎝ 1,b ⎠ sin

(for υ2 > 1/N1)

detects decay of orientational anisotropy of the entanglement segments. This decay occurs not only through reptation/CLF along the partially dilated tube but also through the local DTD lateral to the chain backbone,15 as explained earlier. Thus, it is not so straightforward to formulate a model for the viscoelastic relaxation time. Nevertheless, for the PI 308k/PI 21k blend in the entire range of υ2, experiments17 indicated that the viscoelastic and dielectric relaxation functions of the long chain, μ2(t) and Φ2(t), satisfy the full-DTD relationship, μ2(t) ≅ {Φ2(t)}2 (cf. [ε] eq 1) that leads to τ[G] 2,b ≅ τ2,b /2, at long times in the terminal relaxation zone (although not at shorter times). The same relationship also holds for the monodisperse bulk PI.14,15 We can utilize these empirical facts to express τ[G] 2,b of the long chain [G] in the blend in terms of the viscoelastic τ2,m data of monodisperse bulk as ⎛ τ [ε] ⎞ [G] [G]⎜ 2,b ⎟ τ2,b = τ2,m ⎜ [ε] ⎟ ⎝ τ2,m ⎠

(12) [ε] τ[ε] 2,b /τ2,m

where the dielectric ratio is obtained from the analysis explained earlier (eqs 4−8). (In the blend model explained later, this empirical method of specifying τ[G] 2,b is refined to a method based on self-consistent Rouse−CR calculation of the entanglement survival fraction that is conceptually similar to the method explained below for the short chain with υ2 > 1/N1.) Now we focus on the short chain in the blend. For υ2 < 1/N1, the short chain is not entangled with the long chain and its dynamics should be close to that in monodisperse bulk. The full-DTD relationship should be still valid for such cases, and [G] τ[G] 1,b of the short chain can be related to the τ1,m data of this [G] [G] [ε] [ε] chain in monodisperse bulk as τ1,b = τ1,m{τ1,b /τ1,m} (cf. eq 12). [ε] Because τ[ε] 1,b = τ1,m for υ2 < 1/N1 (cf. eq 11), we find

( ) ( ) π 2N1υ1 p*π 2N1υ1

(9)

with [ε] [ε] 2 τCR1,m = τdil − 2,CR (N1/ N2)

(10a)

(N υ − 2)2 g + 1 1 = 11 p* (N1υ1 − 1)2

(10b)

N1 +1 g = N1 − β1,b*

(10c)

[G] [G] τ1,b = τ1,m

(for υ2 < 1/N1)

(for υ2 < 1/N1)

(13)

In contrast, for larger υ2 > 1/N1, the full-DTD relationship may become invalid for the short chain because of the enhanced constraint from the long chains. For such cases, we can make the partial-DTD analysis reported previously.14,17b This analysis focuses on competition of the full-DTD and Rouse−CR processes to describe the local, lateral DTD that occurs to the maximum level allowed by the CR mechanism: Specifically, the lateral dilation parameter β(t) is estimated as14,17b β(t) = min[βfull‑DTD(t), βCR(t)]. Here, βfull‑DTD(t) is the number of entanglement segments equilibrated during the lateral full-DTD process, and is expressed in terms of the dielectrically evaluated tube survival fraction φ′(t) as βfull‑DTD(t) = 1/{φ′(t)}d (with d = 1.3 for PI). βCR(t) is the maximum number of entanglement segments that can be CR-equilibrated in a given time scale of t. For the partial-DTD process, the viscoelastic relaxation function μ(t) is expressed as μ(t) = φ′(t)/β(t), and the viscoelastic τ[G] is evaluated for this φ′(t)/ β(t) ratio. (For β(t) = βfull‑DTD(t) and β(t) = 1, this ratio reduces to {φ′(t)}1+d and φ′(t) (full-DTD and no-DTD limits), respectively.14,17b) For the short chain with υ2 > 1/N1, the solution model regards the long−short entanglements for the short chain to be [G] fixed in the time scale of τ[ε] 1,b (≥τ1,b ), as explained in the previous section. For this case, βCR(t) for the Rouse−CR process of the short chain can be expressed in terms of the known quantities of this chain, the viscoelastic CR time in

(In eq 9, we have limited ourselves to the short chain having more than one long−short entanglements, i.e., for the short chain with υ2 > 1/N1.) The CR time of the short chain in its monodisperse bulk, τ[ε] CR1,m appearing in eq 9, is evaluated from empirical eq 2b with M2 therein being replaced by M1. This 0 5 [ε] τ[ε] CR1,m (∝ M2M1) can be also related to the τdil−2,CR data (∝ M22M31) of the dilute long chain, as shown in eq 10a. [ε] We utilized the τ[ε] dil−2,CR and τ1,m data in eqs 9 and 10 to specify the lowest possible p* value and the maximum possible 1.5 β1,b* (and g) value, and further evaluated τ[ε] 1,b = (β1,m*/β1,b*) τ[ε] (with β * = 1.81; determined from eqs 9 and 10 with υ 1,m 1,m 1= 1 for monodisperse bulk PI 21k). τ[ε] thus obtained was utilized 1,b in eqs 5 and 7b to calculate τ[ε] 2,b of the long chain in the blends. The results are later shown in Figure 4. Equations 9 and 10 are valid only when υ2 > 1/N1 and the short chain has more than one entanglement with the long chains. For smaller υ2 ( 1/N1)

case (i) β1,b(t) = βfull‑DTD(t) in the entire range of τ[ε] 1,b /2 < t < τ[ε] 1,b . case (ii) β1,b(t) = βfull‑DTD(t) for τ[ε] 1,b /2 < t < t*, β1,b(t) = βCR(t) . for t* < t < τ[ε] 1,b [ε] case (iii) β1,b(t) = βCR(t) in the entire range of τ[ε] 1,b /2 < t < τ1,b . (t* for case (ii) is a characteristic time where βfull‑DTD(t) and βCR (t) coincided with each other.) Cases (i) and (ii) correspond to lateral full-DTD and partial-DTD, respectively, and the case (iii) mostly corresponds to no-DTD. The short chain exhibited the crossover from case (i) to case (ii) and further to case (iii) with increasing υ2. In all cases, μ1,b(t) = [ε] φ1,b′(t)/β1,b(t) in the range of τ[ε] 1,b /2 ≤ t ≤ τ1,b monotonically decayed with t and was well approximated as an single exponential function, μ1,b(t) = K exp{−(t − τ[ε] 1,b /2)/τ*} with K = μ1,b(τ[ε] 1,b /2). Thus, the decay time τ* was simply evaluated as

(14)

with Θ1(t ) =

1 υ1N1

⎛ ⎞ t ⎟ ⎜ − exp ∑ ⎜ [G] ⎟ ⎝ τCR1, p ⎠ p=1 υ1N1

(15a)

and [G] τCR1, p

[ε] ⎫ ⎧ sin 2 ⎪ τ1,b ⎪ 2 [G] ⎨ ⎬ = ⎪ [ε] ⎪υ1 τCR1,m ⎩ τ1,m ⎭ sin 2

( ) ( ) π 2υ1N1 pπ 2υ1N1

τ* =

(15b)

[ε] τ1,b

{

2 ln

The function Θ1(t) (= 1 at t = 0) specified by eq 15a is the survival fraction of the υ1N1 short−short entanglements for the short chain in the blend during the viscoelastic Rouse−CR process, and the time τ[G] CR1,p included in eq 15a is the relaxation time of pth CR mode that is related to the longest CR time [ε] 2 [G] {τ[ε] 1,b /τ1,m}υ1τCR1,m for those υ1N1 entanglements through eq [ε] 15b. Note that the factor {τ[ε] 1,b /τ1,m} appearing in eq 15b is equivalent to the factor (β1,m*/β1,b*)1.5 in eq 9. We can compare βCR(t) specified by eqs 14 and 15 with βfull‑DTD(t) to evaluate τ[G] 1,b for μ(t) = φ′(t)/β(t), with β(t) = min[βfull‑DTD(t), βCR(t)]. This comparison requires us to know βfull‑DTD(t) = 1/{φ′(t)}d (d = 1.3). Full comparison can be made if the dielectric relaxation function Φ(t) (that gives the tube survival fraction φ′(t)) is known in the entire range of t, as was the case in the previous empirical test17b of the full/partialDTD picture. However, the analysis in the solution model allows us to find just the terminal dielectric relaxation time τ[ε] 1,b of Φ(t) (cf. eqs 9 and 10), not Φ(t) in the entire range of t. Thus, τ[G] 1,b needs to be estimated from the known values of βCR(t) and τ[ε] 1,b with a method that is compatible with the solution model. We can find this method by focusing on an experimental observation for monodisperse linear PI that φ′(t) is numerically close to {Φ(t)}2/(1+d) = {Φ(t)}0.87 in the entire range of t and [ε] the value of βfull‑DTD(t) = 1/{φ′(t)}1.3 at t = τ[G] 1,m ≅ τ1,m/2 is close to 3; see Supporting Information in ref 15. Because the relaxation mode distribution of Φ(t) and φ′(t) hardly changes on blending,15,17a this observation allows us to approximate φ1,b′(t) of the short chain in blend at long t as φ1,b′(t) ∝ exp(− 0.87t/τ[ε] 1,b ), and the full dilation parameter, as βfull‑DTD(t) ≅ Iβ/ 1.3 1.13/2 {exp(−0.87t/τ[ε] = Iβ exp(1.13t/τ[ε] = 1,b )} 1,b ) where Iβ = 3/e 1.71 is a numerical prefactor ensuring βfull‑DTD(t) = 3 at t = τ[ε] 1,b / 2. An exception is noted for the case of no-DTD where φ1,b′(t) ∝ exp(− t/τ[ε] 1,b ). Keeping this exception in mind, we can compare βfull‑DTD(t) approximated as above and βCR(t) = {υ1Θ1(t) + υ2}−1 to determine β1,b(t) and evaluate τ[G] 1,b for μ1,b(t) = φ1,b′(t)/β1,b(t). In the actual evaluation of τ[G] 1,b , we considered a fact that the [ε] [ε] viscoelastic τ[G] 1,b is between τ1,b /2 and τ1,b (corresponding to lateral full-DTD and no-DTD, respectively) and compared [ε] βfull‑DTD(t) and βCR(t) in the range of τ[ε] 1,b /2 ≤ t ≤ τ1,b . We found that both βfull‑DTD(t) and βCR(t) increase with t monotonically and β1,b(t) = min[βfull‑DTD(t), βCR(t)] is classified in one of the three cases:

[ε] μ(τ1,b / 2) [ε] ) μ(τ1,b

}

(16a)

The viscoelastic relaxation time of the short chain in the blend, τ[G] 1,b , essentially coincides with this τ*. An exception is found for the case (iii) (for no-DTD): For this case, φ1,b′(t) ∝ exp(−t/ [G] [ε] τ[ε] 1,b ) and τ1,b = τ1,b so that the use of the approximate φ1,b′(t) [G] overestimates τ[G] 1,b . Thus, we evaluated τ1,b as [G] [ε] τ1,b = min[τ *, τ1,b ]

(16b)

τ[G] 1,b

For the case (i), this coincided with the relaxation time for lateral full-DTD, τ[ε] 1,b /2. 3.2.3. Comparison of Solution Model with Data for PI 308k/PI 21k Blend. Figure 4 compares the relaxation times calculated from the “solution model” with the data for the PI 308k/PI 21k blend. As noted in the top panel, the dielectric τ[ε] 2,b and τ[ε] 1,b calculated from the model (red circles) are very close to the data (blue squares). In particular, the model calculation for the long chain is close to the data even for υ2 just above 2/ N2 (≅ 0.03) where the simplest analysis based on the solution [ε] data (θeqτ[ε] 2,soln in Figure 1) underestimates τ2,b in the blend. This improvement is mostly due to the incorporation of the surviving short−long entanglements (eq 4) that do not exist in the solution. The fraction υ1* of those surviving entanglements at the time τ[ε] 2,b was negligibly small for large υ2 but became nontrivial for small υ2 to effectively increase τ[ε] 2,b . In the bottom panel, the orange circles show the viscoelastic τ[G] 2,b of the long chain calculated from the solution model (eqs 4−11) combined with the full-DTD relationship (eq 12), and the filled green circles indicate τ[G] 1,b of the short chain calculated from the model combined with the partial-DTD analysis (eqs 14−16). These τ[G] i,b are surprisingly close to the data, despite the rather crude approximation in the partial-DTD analysis (use of single exponential functions for φ1,b′(t) and βfull‑DTD(t)). In fact, the increase of υ2 resulted in the crossover of τ[G] 1,b of the short chain from lateral full-DTD for υ2 ≤ 0.1 (case (i) explained in the previous section) to partial-DTD for υ2 = 0.2 (case (ii)), and further to no-DTD for υ2 = 0.5 (case (iii)); see green circles. This crossover is in harmony with the experimental observation15 that the CR/DTD process is fully quenched for dilute probe chains in a matrix of much longer chains. It should be emphasized that all parameters utilized in the above model calculation were determined experimentally and the calculation required no parameter fitting/adjustment. Thus, 9303


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entanglements for each component chain, the short−long entanglement for the long chain and short−short entanglement for the short chain. This treatment is valid when the short chain relaxes much faster than the long chain. However, in general, the two components do not necessarily have widely separated relaxation times. For such cases, the dynamics of the long and short chains are strongly coupled, and we need to consider all species of entanglements self-consistently and simultaneously. In the remaining part of this paper, we attempt to formulate the blend model that achieves this self-consistent treatment. 3.3. Blend model Considering All Species of Entanglements. As noted in Figure 2, the CR time of the dilute long chain in the PI 308k/PI 94k blend, τdil−2,CR (horizontal green line), is longer than the τ2,b and τ2,m data of this chain in the blend and monodisperse bulk. The solution model is not applicable to such cases: eq 4 indicates that the solution model 1.5 is applicable only when υ21.5θeq* < {υ2 + υ1*(τ[ε] 2,b )} θeq* < 1, but this relationship did not hold for the PI 308k/PI 94k blend. Considering this limitation of the solution model, we formulated, though in an approximate way, the blend model that considers all species of the entanglements for the long and short components self-consistently to express the reptation/ CLF time along the partially dilated tube. This formulation, applicable to both PI 308k/PI 94k and PI 308k/PI 21k blends, is explained below. (For convenience of the readers, the parameters utilized/analyzed in the model are summarized in Appendix.) The relaxation times deduced from the model are compared with the data later in Figures 5−7 and 9. As explained earlier in section 3.2.2, the dielectric relaxation time detecting the global, end-to-end relaxation is rather insensitive to the local relaxation including lateral DTD whereas the viscoelastic relaxation time is sensitive to lateral DTD. Thus, as done for the solution model, the blend model is formulated for the dielectric relaxation time, and the result is utilized to evaluate the viscoelastic relaxation time with a method similar to that explained in section 3.2.2c. 3.3.1. Dielectric Relaxation Times of Components in Blend. A key in the blend model is the survival fraction of the entanglements for the component i (=1, 2 for the short and long chains), ϕi,s*, where the second subscript s specifies the system; s = b (blend) or m (monodisperse bulk). This ϕi,s* decays with time through the CR relaxation of the entanglements. In the monodisperse bulk, ϕi,m* decays through the CR relaxation of a single species of entanglement, for example, through the CR relaxation of the short−short entanglement for bulk short chain. However, in the blends, all species of entanglements are to be considered selfconsistently. For this case, the previous method of specifying the partially dilated tube in terms of the CR eigenmode index p* (cf. eqs 9 and 10) requires us to utilize a CR model for a chain having two different entanglement lifetimes (e.g., short− long and long−long entanglement lifetimes) distributed along its backbone. However, this distribution should be random so that the analysis of the CR eigenmode index requires tedious averaging for this distribution, which does not allow us to formulate a simple model. Thus, in this study, we make an approximation of summing the survival fractions that decay through the CR relaxation of respective entanglement species to cast ϕi,b* in a form

Figure 4. Comparison of component relaxation times in PI 308k/PI 21k blends calculated from solution model combined with the partialDTD analysis (filled circles) with the data at 40 °C (unfilled blue squares). Note that τ1,b is multiplied by a factor of 10.

the close agreement between the model calculation and data suggests the basic validity of the molecular picture underlying the solution model: (i) reptation/CLF of the long chain along the partially dilated tube being retarded by the tension equilibration achieved through the CR relaxation of the short−long entanglements and (ii) the reptation/CLF motion of the short chain along the partially dilated tube retarded by suppression of CR for this chain on blending. It should be again emphasized that the τ2,b data of the long chain in the blend are longer than the τ2,soln data (Figure 1) because of the retardation of the CR-activated tension equilibration. In addition, Figure 4 suggests that τ2,b is almost proportional to υ2 in the range of υ2 > 2/N2 (≅ 0.03). This proportionality seems to be a consequence of the retardation of tension equilibration. Nevertheless, stronger υ2 dependence of τ2,b is observed for the PI 329k/PI 14k blend having a larger rSG value compared to the PI 308k/PI 21k blend examined here, as explained later for Figure 7. Thus, the proportionality seen in Figure 4 is to be interpreted as a crossover occurring on the increase of rSG. Here, we should note that the simple analysis in the solution model (eqs 4−11) mainly focused on single species of

ϕi ,b*(t ) = υ1Ψ1(i)(t ) + υ2 Ψ(2i)(t ) 9304

(17)


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(i) where Ψ(i) 1 (t) and Ψ2 (t) describe the Rouse−CR decay for the component i activated by the short and long chains, respectively. These Rouse−CR decay functions can be expressed in terms of the number of entanglements of the component i formed with the component j, υjNi, and the local CR time for those entanglements (determined by the motion of the component j), [τj°]b, as17b

Ψ(ji)(t ) =

1 υjNi

υjNi

⎞ t ⎟ (i) ⎟ ⎝ τj , p ⎠

⎛

∑ exp⎜⎜− p=1

Here, we note that a product Niϕi,b* represents the number of the independently surviving entanglement segments per component i in the blend. This product is equivalent to a ratio Ni/gi,b, where gi,b is the number of the entanglement segments that have been CR-equilibrated along the backbone of this component, as explained earlier for eq 10c. Thus, we find gi,b = 1/ϕi,b*. The equilibration number gi,b is in turn related to the tube dilation parameter βi,b* through eq 10c (or equivalent equation). Thus, βi,b* and ϕi,b* are related to each other as Ni 1 = Ni − +1 ϕi ,b* βi ,b*

(18)

with ⎛ pπ ⎞ ⎟⎟ τj(,ip) = [τj°]b sin−2⎜⎜ ⎝ 2υjNi ⎠

From eqs 17−21, the survival fraction of the entanglements for the component i in the blend, ϕi,b*, is obtained given that all of β1,b*, β2,b*, β1,m*, and β2,m* are known. These β*’s are in turn related to ϕ* through eq 21. Thus, self-consistent values of β* and ϕ* are obtained from iterative calculation based on eqs 17−21. (This calculation requires no parameter fitting/ adjustment.) For monodisperse bulk, this iterative calculation involves only one component and is rather simple. Namely, for the short chain in its monodisperse bulk, we utilized the known value of [τ1°]m =τdil−2,CR sin2(π/2N2) to obtain β1,m* and ϕ1,m* from iterative calculation based on eqs 17−19 and eq 21 with υ1 = 1 and υ2 = 0. (eq 20 is not involved in this iteration.) Similarly, for the long chain in bulk, we utilized the known value of [τ2°]m =τdil−2,CR(N2/N1)3sin2(π/2N2) to determine β2,m* and ϕ2,m* from eqs 17−19 and eq 21 with υ1 = 0 and υ2 = 1. The parameters for the blends, β1,b* and β2,b*, are obtained from iterative calculation based on eqs 17−21 utilizing the β1,m* and β2,m* values in monodisperse bulk thus determined above, together with the known values of [τ1°]m and [τ2°]m. This iteration involves eq 20 (that couples β1,b* and β2,b* through eqs 17−19 and 21) and should be made consistently for both long and short chains. Nevertheless, the iteration was satisfactorily achieved with a good convergence to determine the β1,b* and β2,b* values (together with the ϕ1,b* and ϕ2,b* values). In this iterative calculation, we had to introduce additional conditions so as to keep the consistency of the calculation with the molecular picture underlying the blend model. These conditions are summarized below. 3.3.2. Additional Conditions for Consistency between Iterative Calculation and Blend Model. Condition (i). For all blends examined, the tube dilation parameter β1,b* of the short chain in the iteration sometimes became larger than β1,m* of this chain in monodisperse bulk. Because the short chain relaxation is never accelerated on blending and thus β1,b* cannot exceed β1,m*, we replaced β1,b* by β1,m* whenever β1,b* became larger than β1,m*, and continued the iteration until a good convergence was obtained. Namely, the known β1,m* value was utilized as a physically required upper bound for β1,b*. Condition (ii). The short chain motion is much faster than the long chain motion in the blend having a large N2/N1 ratio. In the iterative calculation for such blends with small but finite υ2 (not very larger than 2/N2), the survival fraction ϕ2,b* of the entanglements for the long chain decreased to 1/N2, which resulted in a significant increase of β2,b* and a decrease of τ[ε] 2,b . For this case, the fast short chains quickly CR-equilibrate the whole backbone of the long chain so that the long chain does not exhibit reptation/CLF assumed in the above model. In the other case for the long chain having small but finite υ2 in the

(1 ≤ p ≤ υjNi) (19)

Here, τ(i) j,p denotes the relaxation time of pth CR mode for the υjNi entanglements (and the highest CR mode has τ(i) j,υjNi = [τj°]b). For example, τ(2) is the relaxation time of pth CR mode 1,p for the long chain (i = 2) over its υ1N2 entanglements formed with the short chains (j = 1), and τ(2) 1,1 (p = 1) reduces, for υ1 → 1, to the CR time τdil−2,CR for the whole backbone of the dilute long chain in the short chain matrix. The CR process in this example is activated by the short chain motion and thus [τ1°]b is related to the relaxation time τ[ε] 1,b of this chain in the blend, as explained later for eq 20. If the CR-activating motion of the component j is identical to that in its monodisperse bulk, [τj°]b is equal to [τj°]m in monodisperse bulk and can be straightforwardly obtained from the known value of τdil−2,CR: For this case, [τ1°]b = [τ1°]m = τdil−2,CRsin2(π/2N2) (cf. eq 19 with p = 1, Ni = N2, and υj = υ1→ 1, which gives τ(2) 1,1 = τdil‑2,CR), and [τ2°]b = [τ2°]m = [τ1°]m (N2/ N1)3. [τ1°]m and [τ2°]m in turn correspond to the dielectric relaxation times (reptation/CLF times) τ[ε] j,m of the short and long chains that are necessary for activating the local CR in respective bulk systems. However, in general, the long and short chains exhibit coupled motion in the blend. Namely, the long chain motion is accelerated by the short chain whereas the short chain motion is retarded by the long chain. This acceleration/retardation effect can be characterized with the tube dilation parameter β* defined along the chain backbone, as explained earlier for eq 9. Utilizing βj,b* and βj,m* of the CR-activating component j in the blend and monodisperse bulk, we can relate [τ1°]b and [τ2°]b of the short and long chains in the blend to the known CR time τdil−2,CR of the dilute long chain as ⎛ β * ⎞1.5 ⎛ β * ⎞1.5 ⎞ ⎛ 1,m 1,m ⎜ ⎟ ⎟ τdil − 2,CR sin 2⎜ π ⎟ [τ1°]b = ⎜ [τ1°]m = ⎜⎜ ⎟ ⎟ ⎝ 2N2 ⎠ ⎝ β1,b* ⎠ ⎝ β1,b* ⎠ (20a)

and ⎛ β * ⎞1.5 ⎛ β * ⎞1.5⎛ N ⎞3 2,m 2,m ⎜ ⎟ ⎟ ⎜ 2 ⎟ τdil − 2,CR [τ2°]b = ⎜ [τ2°]m = ⎜⎜ ⎟ ⎟ ⎝ β2,b* ⎠ ⎝ β2,b* ⎠ ⎝ N1 ⎠ ⎛ π ⎞ ×sin 2⎜ ⎟ ⎝ 2N2 ⎠

(21)

(20b)

In eqs 20a and 20b, we have utilized the relationship for the [ε] dielectric times in the blend and monodisperse bulk, τ[ε] j,b /τj,m = 1.5 (βj,m*/βj,b*) (cf. eq 9). 9305


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A Comment. Here, a brief comment is to be made for condition (ii) explained above. The model does not regard the blends as the solutions of the long chain. Thus, the retardation of reptation/CLF due to the CR-equilibration, the key feature in the solution model explained earlier, is not explicitly formulated in the blend model. Nevertheless, condition (ii) prohibits the long chain from exhibiting physically unreasonable fast relaxation. Namely, this condition effectively retards reptation/CLF and is interpreted to be equivalent to the prerequisite in the solution model (CR-activated tension equilibration being required for reptation/CLF along the partially dilated tube). 3.3.3. Viscoelastic Relaxation Times of Components in Blend. As explained earlier, the viscoelastic relaxation times of the components in the blend, τ[G] i,b , are sensitively affected by the local relaxation including lateral DTD and cannot be straightforwardly modeled. Thus, we again make the partialDTD analysis14,17b (as in section 3.2.2c) to evaluate τ[G] i,b from the dielectric τ[ε] i,b deduced from the blend model (eqs 17−24). Specifically, for the component i, we compare the number of equilibrated entanglement segments for lateral full-DTD, β(i) full − DTD(t), and the maximum number of the segments that can be CR-equilibrated in a given time scale, β(i) CR(t), to evaluate τ[G] i,b . (For these β’s as well as the other parameters described below, the superscript (i) stands for the quantity of the (1) (1) component i; βfull − DTD (t) and βCR (t) are equivalent to βfull−DTD(t) and βCR(t) discussed in section 3.2.2c.) In the blend model, the CR-induced decay of the survival fraction of the entanglement segments was approximated as a sum of contributions from the short and long chains (cf. eq 17). Here, we make a conceptually similar approximation to express β(i) CR(t) as

blend with a small N2/N1 ratio, this approach to the CR limit [ε] did not occur, but τ[ε] 2,b sometimes decreased below τdil −2,b of the [ε] dilute long chain. (τdil −2,b was calculated from eqs 17−21 with υ2 = 0 and β1,b* = β1,m*, i.e., in the absence of the effect of the dilute long chain on the short chain behavior. This calculation does not involve the coupling of β1,b* and β2,b* and was [ε] conducted easily). The decrease of τ[ε] 2,b below τdil −2,b is [ε] physically unreasonable, because τ2,b should always increase with increasing υ2. Thus, for both cases, we had to introduce a lower bound of τ[ε] 2,b . We set this lower bound in terms of the known value of τ[ε] dil −2,b in an approximate but simple way, as explained below. Each entanglement of the long chain is considered to be sustained by υ1 short chains and υ2 long chains, so that its effective lifetime, [τ2°]eff, can be approximated as an average of the lifetimes determined by single short and/or long chain, [ε] 1/2 [ε] 1/2 2 [τ2°]eff ∼ {υ1(τ1,b ) + υ2(τ2,b ) }

(22)

The dielectric relaxation times of the short and long chains in [ε] the blend, τ[ε] 1,b and τ2,b appearing in eq 22, specify the lifetime determined by each single chain and correspond to the local CR times [τ1°]b and [τ2°]b explained for eqs 19 and 20. Equation 22 has adopted the squared average of τ1/2 so as to reproduce some limiting cases in a way consistent with eqs 19 2 [ε] and 20: [τ2°]eff ∼ υ22τ[ε] 2,b and [τ2°]eff ∼ υ1 τ1,b respectively for 2 [ε] [ε] the cases of τ[ε] ≫ (υ /υ ) τ and τ ≪ (υ1/υ2)2τ[ε] 2,b 1 2 1,b 2,b 1,b (for the cases that the CR equilibration is governed by the long and [ε] short chains, respectively), and [τ2°]eff ∼ τ[ε] 2,m for the case of τ2,b = τ[ε] , i.e., for monodisperse bulk. 1,b Now, we focus on the lower bound of the relaxation time of [ε] the long chain, [τ[ε] 2,b ]min. Because τ2,b never increases with [ε] decreasing υ2, [τ2,b ]min is essentially given by τ[ε] dil −2,b of the dilute long chain (having υ2 → 0) but should be larger than dil τ[ε] dil −2,b by a factor, [τ2°]eff/[τ2°]eff, that represents an increase of [ε] the entanglement lifetime with υ2. Namely, [τ2,b ]min = [ε] dil τdil −2,b([τ2°]eff/[τ2°]eff). Here, [τ2°]eff is given by eq 22 and 1/2 1/2 2 satisfies an inequality [τ2°]eff ≥ {υ1 (τ[ε] + υ2(τ[ε] 1,m) dil −2,b) } [ε] [ε] [ε] because τ[ε] ≥ τ and τ ≥ τ . The lifetime of the 1,b 1,m 2,b dil −2,b entanglement for the dilute long chain, [τ2°]dil eff, can be also [ε] obtained from eq 22 with υ2 → 0 as [τ2°]dil eff ∼ τ1,m (note that [ε] [ε] τ1,b →τ1,m for υ2 → 0). From these considerations, we can specify the range of τ[ε] 2,b of the long chain in terms of the known [ε] quantities, τ[ε] dil −2,b and τ1,m, as

1 (i) βCR (t )

where is the survival fraction defined for υ j N i entanglements of the component i formed with the component j. (Θ(1) 1 (t) is equivalent to Θ1(t) defined/utilized in section 3.2.2c.) The decay of Θ(i) j (t) is activated by the motion of the component j, and can be described by the Rouse−CR function similar to that shown in eq 15a: Θ(ji)(t )

⎛ ⎞ t ⎟ ⎜ ∑ exp⎜− (i) ⎟ ⎝ τCR, j , p ⎠ p=1 υjNi

(26)

Here, is the viscoelastic relaxation time of the pth CR mode for the υjNi entanglements of the component i:

[ε] τ1,m

(23) (i) τCR, j,p

[ε] (This [τ[ε] 2,b ]min approaches τdil −2,b with decreasing υ2, and τ[ε] coincides with the CR time τ[ε] dil −2,b dil −2,CR of the dilute long chain if N2/N1 ≫ 1.) The corresponding upper bound of β2,b* is specified as

⎛ π ⎞ sin 2⎜ 2υ N ⎟ ⎝ j i⎠ (i) = τCR, j ,1 ⎛ pπ ⎞ sin 2⎜ 2υ N ⎟ ⎝ j i⎠

(27a)

[ε] ⎫⎧ ⎧ ⎛ Nj ⎞3⎛ υjNi ⎞2 ⎫ ⎪ τj ,b ⎪⎪ ⎪ [G] ⎬ = ⎨ [ε] ⎬⎨ τ ⎟⎪ ⎜ ⎟⎜ ⎪ ⎪⎪ dil ‐ 2,CR N ⎝ 1 ⎠ ⎝ N2 ⎠ ⎭ ⎩ τj ,m ⎭⎩

(27b)

with

1/1.5

[β2,b*]max

1 = υjNi

τ(i) CR,j,p

[ε] 1/2 [ε] 1/2 2 {υ1(τ1,m ) + υ2(τdil } ‐ 2,b)

⎛ τ [ε] ⎞ 2,m ⎟ = β2,m*⎜⎜ [ε] ⎟ ⎝ [τ2,b ]min ⎠

(25)

Θ j(i) (t)

[ε] [ε] τ2,b ≥ [τ2,b ]min [ε] [ε] with [τ2,b ]min = τdil ‐ 2,b

= υ1Θ1(i)(t ) + υ2 Θ(2i)(t )

(i) τCR, j ,1

(24)

τ[G] dil‑2,CR

In the iterative calculation based on eqs 17−21, we replaced β2,b* by [β*2,b]max (eq 24) whenever β2,b* became larger than [β2,b * ]max and continued the iteration until a good convergence was obtained.

In eq 27b, is the viscoelastic CR time for the dilute long chain in the short chain matrix, and its value is known from the empirical eq 2a. This τ[G] dil‑2,CR represents the time necessary to complete the viscoelastic CR relaxation of N2 entanglement 9306


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segments activated by the short chain motion, and the factor (Nj/N1)3(υjNi /N2)2 in eq 27b converts this τ[G] dil‑2,CR into the time for the CR relaxation of the υjNi entanglement segments [ε] activated by the motion of the component j. The factor τ[ε] j,b /τj,m accounts for a change of the local CR relaxation rate due to a change in the relaxation time of the component j on blending [ε] [ε] [ε] (τ[ε] 1,b /τ1,m ≥ 1 and τ2,b /τ2,m ≤ 1). Equations 25−27 might appear to be very similar to eqs 17−21 utilized in the blend model. However, this is not the case. Equations 17−21 include the unknown parameters related to the reptation/CLF time along the partially dilated tube, ϕi,b* and βi,b*, and these parameters were determined from iterative calculation based on eqs 17−21. In contrast, all parameters in [ε] is eqs 25−27 are known: τ[G] dil‑2,CR is obtained from eq 2a, τj,b [ε] deduced from the blend model, and the τj,m data of monodisperse bulk are known. Thus, β(i) CR(t) is straightforwardly specified by eqs 25−27. The partial-DTD analysis compares this β(i) CR(t) with the lateral full-DTD parameter, β(i) (t). As explained in section full‑DTD 3.2.2c, the latter can be approximated as β(i) full‑DTD(t) ≅ 1.71 exp(1.13t/τ[ε] ′ (t) ∝ i,b ), and the tube survival fraction, as φi,b [ε] exp(−0.87t/τ[ε] i,b ), with τi,b being the dielectric relaxation time deduced from the blend model. An exception is found for the ′ (t) ∝ exp(−t/τ[ε] case of no DTD where φi,b i,b ). Thus, we were (i) (i) (t) to evaluate the able to compare βCR(t) and βfull‑DTD viscoelastic relaxation function of the component i in the b l e n d , μ i , b ( t ) = φ i ,′b ( t ) / β i , b ( t ) w i t h β i , b ( t ) = (i) min[β(i) full‑DTD(t),βCR(t)]. This μi,b(t) was well approximated as an single exponential function (as was the case also for the analysis combined with the solution model), and the viscoelastic relaxation time of the component i in the blend, [G] [ε] τ[G] i,b , was simply evaluated as τi,b = min[τi,b*,τi,b ] with τ* i,b = [ε] [ε] [ε] {τi,b /2}/ln{μi,b(τi,b /2)/μi,b(τi,b )} (cf. eq 16). 3.3.4. Comparison of Blend Model with Data. Figures 5 and 6, respectively, compare the component relaxation times deduced from the blend model with the data for the PI 308k/PI 21k and PI308/PI 94k blends (blue squares). All parameters utilized in this model calculation were determined experimentally and no parameter fitting/adjustment was necessary (which was the case also for the solution model). As noted in the top panels, the blend model (red circles) well describes the dielectric τ[ε] i,b data in both blends, despite the approximation in the model (eq 17). In particular, the model worked for the long chain with small υ2 that exhibited either lateral full-DTD behavior (Figure 5) or partial-DTD behavior (Figure 6). Thus, the blend model has a wider applicability compared to the solution model. In the bottom panels of Figures 5 and 6, the green circles indicate the viscoelastic τ[G] i,b calculated from the blend model combined with the partial-DTD analysis explained in the previous section. The calculated τ[G] i,b are close to the data, despite the above approximation in the model as well as the rather crude approximation in the partial DTD analysis (use of single exponential functions for β(i) full‑DTD(t) and φi,b′(t)). The increase of τ[G] i,b of the short chain with υ2 reflects the crossover from lateral full-DTD to partial-DTD (and further to noDTD), as similar to the behavior explained for Figure 4. We also compared the model prediction with the τ[G] i,b data of the PI 329k/PI 14k blend24 having very large rSG (cf. Table 1). The results are shown in Figure 7. For this blend, no dielectric [G] data are available. However, the τ[ε] dil−2,CR and τdil−2,CR values of the dilute long chain necessary for the model calculation and the partial-DTD analysis were obtained from eq 2 (cf. green

Figure 5. Comparison of component relaxation times deduced from the blend model combined with the partial-DTD analysis (filled circles) with the data for 308k/PI 21k blends at 40 °C (blue squares). Note that τ1,b is multiplied by a factor of 10.

horizontal line in Figure 7), and τ[G] i,b was calculated without any parameter fitting/adjustment. The calculated τ[G] i,b is close to the data, again suggesting validity of the blend model. In Figure 7, we also note that υ2 dependence of the τ[G] 2,b data is a little enhanced on a decrease of υ2 from 0.4 to 0.05 and an δ empirical relationship, τ[G] 2,b ∝ υ2 with δ ≅ 1.5, is observed in this range of υ2. This strong υ2 dependence may be the asymptotic dependence in the limit of large r SG (and thus the proportionality between τ[G] 2,b and υ2 noted in Figure 4 is to be interpreted as crossover to the asymptotic limit). This asymptotic υ1.5 2 dependence is similar to the dependence seen for the solution in the oligomeric solvent (cf. Figure 1). Nevertheless, τ[G] 2,b of the long chain in the PI 329k/PI 14k blend is still larger than τ[G] 2,soln in the corresponding solution, again suggesting the importance of the CR-activated tension equilibration required for reptation/CLF in the blend. It is informative to compare the diameter a2,b* of the partially dilated tube for the long chain (along which the chain exhibits reptation/CLF) with the diameter a2,full‑DTD of the fully dilated tube. Because the dielectric relaxation time of the long chain, −1.5 τ[ε] ∝ a2−3, the ratio of a2,b* to a2,full‑DTD is 2 , scales as Me,eff 9307


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straightforwardly obtained from τ[ε] 2,b deduced from the blend model as a 2,b* a 2,full ‐ DTD

⎛ τ [ε] ⎞1/3 ⎛ τ [ε] ⎞1/3 2,full ‐ DTD ⎟ ⎜ ⎜ 2,soln ⎟ =⎜ ⎟ = ⎜ [ε] ⎟ [ε] ⎝ τ2,b ⎠ ⎝ τ2,b ⎠

(28)

In eq 28, we have utilized the dielectric relaxation time of the [ε] 1.5 long PI chain in the solution, τ[ε] (cf. Figure 1), as 2,soln = τ2,mυ2 the time for reptation/CLF along the fully dilated tube. For the mutually entangled long chain (having υ2 > 2/N2) in the PI/PI blends examined above, the a2,b*/a2,full‑DTD ratio thus deduced from the model is shown in Figure 8. The thick horizontal line

Figure 8. Normalized diameter of partially dilated tube (reptation/ CLF path) for the long chain in PI/PI blends as indicated.

shows the ratio expected for the case of full-DTD along the chain backbone, and the dashed line shows the ratio for the case of no DTD (a2,b*/a2,full‑DTD = a2/a2,soln = υ20.5 for PI). Clearly, the ratio for the long chain in the actual PI/PI blends is located between these two extreme cases, and the deviation from the full-DTD behavior becomes more significant with decreasing rSG and with decreasing υ2. This result confirms that the diameter of the partially dilated tube and the corresponding reptation/CLF path length are determined by competition between the reptation/CLF and CR mechanisms. 3.4. Comparison of Blend Model with Data for PS/PS Blends. The viscoelastic τ[G] i,b data at 167 °C have been reported for the PS/PS blends listed in Table 1. PS has no typeA dipole and thus no dielectric τ[ε] i,b data reflecting the end-toend relaxation are available. However, an empirical equation is available for the viscoelastic CR time of the dilute long chains in −12 the PS/PS blends (τ[G] M13M22Me−3 ≅ 2.1 × dil‑2,CR/s ≅ 1.2 × 10 3 −25 3 2 10 M1 M2 at 167 °C). Thus, we estimated the CR time for [G] the end-to-end relaxation as τ[ε] dil‑2,CR = 2τdil‑2,CR and utilized this estimate in the blend model combined with the partial-DTD analysis to calculate τ[G] i,b for the PS/PS blends. The results are compared with the data in Figure 9. No data are available for solutions of the long PS chain in oligomeric styrene that is dynamically similar to the oligomeric butadiene (oB) utilized for PI. Nevertheless, the solution data in good and marginal low-M solvents for PS, dibutyl phthalate (DBP) at T ≥ 6 °C and dioctyl phthalate (DOP) at T ≥ 35 °C,

Figure 6. Comparison of component relaxation times deduced from the blend model combined with the partial-DTD analysis (filled circles) with the data for 308k/PI 94k blends at 40 °C (blue squares).

Figure 7. Comparison of component relaxation times deduced from the blend model combined with the partial-DTD analysis (green circles) with the viscoelastic data for PI 329k/PI 14k blends at 40 °C (blue squares). Note that τ1,b is multiplied by a factor of 100. 9308


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have been reduced at an iso-frictional state with respect to bulk PS at 167 °C.30,31 The data in these solvents agreed well with each other.31 These solution data at the iso-frictional state are shown in Figure 9 with the black circles. Those data can be cast

tension equilibration for reptation/CLF along the partially dilated tube. In addition, Figure 9 also demonstrates the increase of τ[G] 1,b of the short chain with υ2 that reflects the crossover from lateral full-DTD to partial DTD (and further to no DTD), as similar to the behavior noted for the PI/PI blends. Figure 10 examines the diameter a2,b* of the partially dilated tube for the long chain in the PS/PS blends deduced from the

Figure 10. Normalized diameter of partially dilated tube (reptation/ CLF path) for the long chain in PS/PS blends as indicated.

blend model. A ratio of a2,b* to the diameter a2,full‑DTD of the fully dilated tube in the solution, a2,b*/a2,full‑DTD = {τ2,soln/ τ2,b}1/3 (cf. eq 28), is plotted against υ2. This ratio is located between the two extreme cases of full-DTD and no-DTD, and the deviation from the full-DTD behavior becomes more significant for smaller rSG and υ2. This result is in harmony with that observed for the PI/PI blends (Figure 8), again confirming the importance of the CR-activated tension equilibration for reptation/CLF along the partially tube. 3.5. Comment for Difference between PS and PI. The above results demonstrate that the blend model is commonly valid for the PI/PI and PS/PS blends. Nevertheless, we also note a delicate difference between these blends: The CR time (τ[G] dil‑2,CR) data of the long dilute chains in those blends have been utilized to evaluate the CR time in monodisperse bulk, [G] [G] τ[G] CR,m, and a ratio of τCR,m to the measured relaxation time τm in bulk has been compared for PI and PS, as explained in ref 24 and also in Supporting Information of ref 15 (see Figure S11 [G] therein). This comparison revealed that the τ[G] CR,m/τm ratio is larger for PS than for PI having the same entanglement number, M/Me. Thus, the CR time is not uniquely determined by the entanglement number (of the long and short chains in the case of the blends) but depends on the chemical structure of the chain, possibly because the number of chains sustaining one entanglement is different for PI and PS.15,24 This difference does not explicitly affect the prediction of the blend model because it is implicitly included in the τ2‑dil,CR and τm data utilized in the model, which led to the validity of the model for both PI and PS. However, the effect of the chemical structure on the CR time needs to be considered when a more elaborated molecular model is formulated. This effect deserves a further study.

Figure 9. Comparison of component relaxation times deduced from the blend model combined with the partial-DTD analysis (green circles) with the viscoelastic data for PS/PS blends at 167 °C (blue squares). Note that τ1,b in the PS 316k/PS 39k blend (top panel) is multiplied by a factor of 10. [G] 2.2 (black solid curve). in an empirical equation τ[G] 2,soln = τ2,mυ2 [G] This υ2 dependence of τ2,soln is stronger than that seen in Figure 1 for the PI solutions in oB. This difference may be partly related to a difference in the solubility of DBP/DOP for PS and oB for PI (and also to a difference in the viscosity; oB is much more viscous compared to DBP and DOP and could have, in a sense, some polymeric character). As noted in Figure 9, the τ[G] 2,b data of the long chain in the [G] blends are larger than the τ[G] 2,soln data. Furthermore, τi,b of the long and short chains in the blend calculated from the blend model combined with the partial-DTD analysis (green circles) are close to the data for the PS 316/PS 39k and PS 316k/PS 89k blends having large and small rSG (cf. Table 1). These results again suggest the importance of the CR-activated

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4. CONCLUDING REMARKS We have examined the component relaxation times in binary blends in relation to the molecular picture that the mutually entangled long chains in the blend exhibit reptation/CLF along the partially dilated tube only after the CR-activated tension equilibration is completed. For this purpose, we formulated the solution model treating the blend as an equivalent solution but with retardation due to this tension equilibration. This model successfully described the dielectric and viscoelastic relaxation time data for the PI 308k/PI 21k blend, lending support to that molecular picture. The retardation due to the CR-equilibration of the tension, not existing in the real solution in a low-M solvent, appears to be essential when we f ully coarse-grain the time scale in the analysis of the chain relaxation thereby regarding the short chains as a solvent. The solution model is not applicable to the PI 308k/PI 94k blend in which the short chain does not relax much faster than the long chain. For this case, the full coarse-graining explained above becomes invalid. For the PI 308k/PI 94k blend as well as the PI 308k/PI 21k blend, we formulated the blend model that self-consistently considers all species of entanglements for the long and short chains in a partially coarse-grained time scale. The minimum time required for the CR process to occur in the blend having finite υ2, estimated from the relaxation time of dilute long chains (with υ2 → 0) and that of the short chain, was incorporated in the model through the physically required upper bound of the tube dilation parameter for the long chain, [β2,b*]max. (This minimum time corresponds to the time required for the CR-activated tension equilibration considered in the solution model.) The blend model described the data for both PI 308k/PI 21k and PI 308k/PI 94k blends, and is superior to the solution model. The blend model also described the data for the other PI/PI blend and PS/PS blends, again lending support to the molecular picture of CR-activated tension equilibration incorporated in the model. Despite the success summarized above, it should be emphasized that the models formulated in this study are based on the τi,m data of the long and short chains in monodisperse bulk and the τdil‑2,CR data (empirical equation) for the dilute long chain entangled with much shorter chains. Namely, the models just utilize those data to calculate τi,b in the blends, without explicitly formulating the dielectric and viscoelastic relaxation functions, Φ(t) and μ(t), in the entire range of t. Thus, those models are not a full molecular model that should be able to predict, in a self-consistent way, these functions and all relaxation times including τi,m and τ2‑dil,CR. This full molecular model may be formulated in a dual form, either in the fully/partially coarse-grained time scale where the tube dilation picture is utilized to describe the CR process, or in the non-coarse-grained time scale where the Rouse-like feature of the CR process and the unit process of reptation/CLF at short time scale (that would occur along the undilated tube) are to be explicitly taken into account. In either case, the concept of CR-activated tension equilibration would be essential, as noted in the previous studies13,15 and also demonstrated in this study. Formulation of the full molecular model is a challenging subject of future work. Finally, it should be pointed out that the blend model formulated in this study can be, in principle, extended to polydisperse melts having arbitrary molecular weight distributions. However, all components in such melts are motionally coupled, and this extension requires us to solve an infinite

number of equations, each being similar to one of eqs 17−24, so that the current form of the blend model becomes intractably complicated. Thus, some mean-field treatment would be necessary for the extension. This treatment is another interesting/important subject of future study.

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APPENDIX. SUMMARY OF PARAMETERS UTILIZED/ANALYZED IN THE SOLUTION AND BLEND MODELS

A1. Known Parameters

• component index: i = 1 for the short chain, i = 2 for the long chain. • υi: volume fraction of the component i in the blend. • υjNi (=υjMi/Me0): number of entanglements of the component i formed with the component j. CR relaxation of these entanglements is activated by the motion of the component j. [G] • τ[ε] i,m and τi,m : dielectric and viscoelastic relaxation time data of the component i in monodisperse bulk. [ε] [G] and τdil‑2,CR : dielectric and viscoelastic CR • τdil‑2,CR relaxation times of the dilute long chain in the short chain matrix (evaluated from empirical eq 2). [ε] [G] and τCR1,m : dielectric and viscoelastic CR • τCR1,m relaxation times of the short chain in monodisperse bulk (evaluated from empirical eq 2). τ[ε] CR1,m is related to τ[ε] dil‑2,CR through eq 10a. A2. Output of the Models

• τ[ε] i,b : dielectric relaxation time of the component i in the blend calculated from the models. • τ[G] i,b : viscoelastic relaxation time of the component i in the blend calculated from the models combined with the partial-DTD analysis. A3. Parameters Analyzed in the Solution Model

• θeq*: retardation factor for the CR-activated tension equilibration of the long chain in the blend. Solution of the long chain is utilized as the reference for this retardation (cf. eqs 4 and 5). • υ1*(τ[ε] 2,b ): fraction of the short−long entanglements of the long chain surviving in the time scale of τ[ε] 2,b . This fraction is obtained from Rouse−CR calculation (cf. eq 6). • τ(s‑l) q : dielectric relaxation time of qth CR mode for N2υ1 short−long entanglements of the long chain. The time [ε] for the lowest CR mode, τ(s‑l) 1 , is related to the τdil‑2,CR data (cf. eq 7b). • β1,x*: dilation parameter giving the diameter aβ1,x*1/2 of the partially dilated tube along which the short chain in the blend (x = b) or monodisperse bulk (x = m) exhibits reptation/CLF. • p*: index specifying the CR modes that are available for partial tube dilation allowing reptation/CLF of the short chain (p* is related to β1,b* through eqs 10b and 10c). A4. Parameters in the Partial-DTD Analysis Associated with the Solution Model

• βfull‑DTD(t): number of entanglement segments of the short chain equilibrated during the lateral full-DTD process. β full‑DTD (t) is approximated as a single exponential function. • βCR(t): maximum number of entanglement segments of the short chain that can be CR-equilibrated in a given 9310


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time scale of t. βCR(t) is calculated from viscoelastic Rouse−CR function Θ1(t) defined for υ1N1 short−short entanglements of the short chain in the blend.

AUTHOR INFORMATION

Corresponding Author

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

A5. Parameters Analyzed in the Blend Model

Notes

The authors declare no competing financial interest.

• ϕi,s*: survival fraction of Ni entanglements of the component i in the blend (s = b) and monodisperse bulk (s = m). ϕi,b* in the blend is approximated as a sum of contributions from the long and short chains (cf. eq 17). • Ψ(i) j (t): Rouse−CR function defined for υjNi entanglements of the component i formed with the component j. (For example, Ψ(2) 1 (t) is the Rouse−CR function for υ1N2 short−long entanglements of long chain.) Decay of Ψ(i) j (t) is activated by the motion of component j. (i) • τ(i) j,p : dielectric relaxation time of pth CR mode of Ψj (t). • [τ j °] b : local dielectric CR time defined for υ j N i entanglements of the component i formed with the component j. ([τj°]b coincides τ(i) j,p with p = υjNi.) [τj°]b is determined by the motion of component j in the blend, and is expressed in terms of the local dielectric CR time of this component in monodisperse bulk, [τj°]m (∼ τ[ε] j,m). • βi,x*: dilation parameter giving the diameter aβi,x*1/2 of partially dilated tube along which component i in the blend (x = b) or monodisperse bulk (x = m) exhibits reptation/CLF. β1,x* of the short chain is analyzed also in the solution model. • τ[ε] dil‑2,b: dielectric relaxation time of dilute long chain in the matrix of short chain calculated from the blend [ε] model. (τ[ε] dil‑2,b coincides with the known CR time τdil‑2,CR [ε] if N2/N1 ≫ 1. Otherwise, τ[ε] is smaller than τ dil‑2,b dil‑2,CR.) τ[ε] is utilized to specify the physically required lower dil‑2,b [ε] bound ([τ[ε] 2,b ]min) for τ2,b of the long chain in the blend. • [τ2°]eff: effective local dielectric CR time for the long chain in the blend approximated as an average of the contributions from the short and long chains (cf. eq 22). [ε] For the dilute long chain, [τ2°]eff reduces to [τ2°]dil eff ∼ τ1,m [ε] . [τ2°]eff is introduced together with τdil−2,b to specify the [ε] lower bound ([τ[ε] 2,b ]min) for τ2,b of the long chain in the blend.

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

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REFERENCES

(1) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, U.K., 1986. (2) McLeish, T. C. B. Adv. Phys. 2002, 51, 1379. (3) Watanabe, H. Prog. Polym. Sci. 1999, 24, 1253. (4) Rubinstein, M.; Colby, R. H., Polymer Physics: Oxford University Press: New York, 2003. (5) Marrucci, G. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 159. (6) (a) Milner, S. T.; McLeish, T. C. B. Macromolecules 1997, 39, 2159. (b) Milner, S. T.; McLeish, T. C. B. Macromolecules 1998, 31, 7479. (7) Graham, R. S.; Likhtman, A. E.; McLeish, T. C. B. J. Rheol. 2003, 47, 1171. (8) Kapnistos, M.; Vlassopoulos, D.; Roovers, J.; Leal, L. G. Macromolecules 2005, 38, 7852. (9) Das, C.; Inkson, N. J.; Read, D. J.; Kelmanson, M. A.; McLeish, T. C. B. J. Rheol. 2006, 50, 207. (10) Likhtman, A. E.; Graham, R. S. J. Non-Newtonian Fluid Mech. 2003, 114, 1. (11) Wang, Z.; Chen, X.; Larson, R. G. J. Rheol. 2010, 54, 223. (12) van Ruymbeke, E.; Kapnistos, M.; Vlassopoulos, D.; Liu, C. Y.; Bailly, C. Macromolecules 2010, 43, 525. (13) van Ruymbeke, E.; Masubuchi, Y.; Watanabe, H. Macromolecules 2012, 45, 2085. (14) Watanabe, H. Polymer J. 2009, 41, 929. (15) Matsumiya, Y.; Kumazawa, K.; Nagao, M.; Urakawa, O.; Watanabe, H. Macromolecules 2013, 46, 6067. See also Supporting Information therein. (16) Watanabe, H. Macromol. Rapid Commun. 2001, 22, 127. (17) (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. (18) Watanabe, H.; Matsumiya, Y.; Inoue, T. Macromolecules 2002, 35, 2339. (19) Watanabe, H.; Sawada, T.; Matsumiya, Y. Macromolecules 2006, 39, 2553. (20) Qiao, X.; Sawada, T.; Matsumiya, Y.; Watanabe, H. Macromolecules 2006, 39, 7333. (21) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (22) (a) Graessley, W. W. Adv. Polym. Sci. 1974, 16, 1. (b) Graessley, W. W. Polymeric Liquids & Networks: Dynamics and Rheology: Garland Science: New York, 2008. (23) For reptation/CLF to occur along the partially dilated tube, some entanglement segments need to be CR-equilibrated along this tube (to achieve the tension equilibration of the chain). The number g of those equilibrated segments is related to the number β* characterizing the diameter of the partially dilated tube as g = N − N/β* + 1 with N = M/Me0,15 and the reptation/CLF time along the partially dilated tube scales as β*−1.5. For monodisperse linear PI, g increases with increasing N and is larger than βfull‑DTD characterizing the fully dilated tube diameter.15

A6. Parameters in the Partial-DTD Analysis Associated with the Blend Model

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Article

• β(i) full‑DTD(t): number of entanglement segments of the component i equilibrated during the lateral full-DTD (i) process. β full‑DTD (t) is approximated as a single exponential function. (In the partial-DTD analysis associated with the solution model, β(1) full−DTD(t) of the short chain is designated as βfull‑DTD(t).) • β(i) CR(t): maximum number of entanglement segments of the component i that can be CR-equilibrated in a given time scale of t. β(i) CR(t) is calculated from viscoelastic (i) Rouse−CR functions Θ(i) 1 (t) and Θ2 (t) defined for υ1Ni and υ2Ni entanglements of the component i in the blend. (In the partial-DTD analysis associated with the solution model, Θ(1) 1 (t) is designated as Θ1(t).)

ASSOCIATED CONTENT

S Supporting Information *

Viscoelastic data of PI/PI blends with a very large Struglinski− Graessley parameter. This material is available free of charge via the Internet at http://pubs.acs.org. 9311


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(24) Sawada, T.; Qiao, X.; Watanabe, H. Nihon Reoroji Gakkaishi (J. Soc. Rheol. Jpn.) 2007, 35, 11. (25) Watanabe, H.; Sakamoto, T.; Kotaka, T. Macromolecules 1985, 18, 1008. (26) Struglinski, M. J.; Graessley, W. W. Macromolecules 1985, 18, 2630. (27) The empirical equations reported/utilized in ref 15, τ[G] dil‑2,CR/s = −25 M13M22, were for the 1.5 × 10−25 M13M22 and τ[ε] dil‑2,CR/s = 3.0 × 10 viscoelastic and dielectric relaxation times of the slowest CR eigenmode. In contrast, this study focuses on the second-moment average CR relaxation time that is most straightforwardly evaluated from experiments and can be directly compared with the blend data. For this case, the numerical prefactor in the above empirical equation decreases by a factor of 15/π2 (which is well-known for the Rouse−CR mechanism3) to give eq 2 in the text. A small difference between 15/π2 and unity hardly affects the results presented in this paper. (28) For entangled monodisperse linear PI in bulk at 40 °C, an empirical equation of the relaxation time τ was reported previously:15 τ = KM3.5 with K (in s) = 4.2 × 10−19 and 2.0 × 10−19 for the dielectric and viscoelastic τ. For the long chain with the molecular weight M2, the intrinsic Rouse relaxation time was evaluated on the basis of this empirical equation as τ2,Rouse = K(2Me0)3.5(M2/2Me0)2 = KRouseM22, where Me0 (=5k) is the bulk entanglement molecular weight and KRouse = 1.0 × 106K. (In this evaluation, we located the onset of entanglement effect on τ at M2 = 2Me0, as known for the τ data of linear PI.15,17,19,24) (29) Watanabe, H.; Sato, T.; Osaki, K. Macromolecules 1996, 29, 104. (30) Watanabe, H.; Kotaka, T. Macromolecules 1986, 19, 2520. (31) (a) Watanabe, H.; Yoshida, H.; Kotaka, T. Macromolecules 1988, 21, 2175. (b) Yoshida, H.; Watanabe, H.; Kotaka, T. Macromolecules 1991, 24, 572.

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Component Relaxation Times in Entangled Binary Blends of Linear

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