1. Overview

The constraint release (CR) relaxation can be most clearly observed for dilute high-M probe chains entangled only with much shorter matrix chains. As an example, Figure 4 shows G′ and G″ data (symbols) of blends of linear dilute PI 626k probe (M₂ = 626 × 10³; volume fraction υ₂ = 0.005) in entangling linear PI matrices with various molecular weights M₁ as indicated.⁴⁴ For comparison, the imaginary part of the complex viscosity, η″ = G′/ω, with ω being the angular frequency, is also shown (top panel). The time–temperature superposition excellently worked for those data⁴⁴ (and for all other data presented in this review). For the blends, the indices “1” and “2” are hereafter used to represent the short and long chain components, respectively.

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As noted in Figure 4, the data of the blends (symbols) with such small υ₂ are almost indistinguishable from the matrix data (dashed curves) at high ω where the matrix has not relaxed. In contrast, at low ω where the matrix has fully relaxed, the relaxation of the dilute PI 626k probe is clearly observed, in particular for G′ and η″ being much more sensitive to weak but slow relaxation compared to G″. (Thus, G′ clearly exhibits a double-step decrease at high and low ω, and η″ shows two peaks at those ω.) We also note that the mode distribution of the probe relaxation is insensitive to M₁ of the matrix given that M₁ is much smaller than M₂ of the probe. In this extreme situation, the probe relaxation is activated by the motion of much shorter matrix chains, namely, pure CR relaxation of the probe is detected experimentally. The corresponding CR behavior of dilute probes has been observed also for the dielectric relaxation of PI probes^26,34 and for the viscoelastic relaxation^45–48 and diffusion^49,50 of polystyrene (PS) probes. Viscoelastic data of binary blends have been reported extensively also for the other polymer species, for example, polybutadiene (PB).¹ However, to the best of our knowledge, no systematic data are available for long and dilute PB probes entangled only with much shorter matrix PB chains (i.e., for the dilute PB probes in the CR regime).

For υ₂ (=1 − υ₁) ≪ 1, the behavior of the matrix chains in the blend is negligibly affected by the dilute probe, as noted in Figure 4. Then, the matrix contribution to the complex modulus G_b * (ω) of the blend is safely evaluated as υ₁G_1,m * (ω), with G_1,m * (ω) being the complex modulus data of pure matrix. Correspondingly, the complex modulus of the dilute probe in the blend is obtained from the G_b * (ω) and G_1,m * (ω) data as^{34,44–46,51}

Figure 5 compares the storage modulus G_2,b′(ω) thus evaluated for dilute linear PI probes⁴⁴ in the CR regime (namely, for the PI probes entangled only with much shorter PI matrix chains). Also shown for comparison is the G_2,b′(ω) data of PS probes in the CR regime reported in the literature.^45,46 For clarity of the figure, the data are shown only for representative probes. Those PI and PS probes have various M₂ and υ₂, and their G_2,b′(ω) data were obtained at different temperatures (40 and 167 °C for the PI and PS probes, respectively). Thus, in the comparison in Figure 5, the G_2,b′(ω) data are normalized by the Rouse factor, {M₂/ρυ₂RT} with ρ, R, and T being the mass density of the blend, gas constant, and absolute temperature, respectively, and plotted against the reduced frequency . Here, is the second-moment average viscoelastic relaxation time of the probe evaluated from the G_2,b′(ω) and G_2,b″(ω) data as^34,44–46

is identical to a ratio of weighed integrals of the relaxation modulus G_2,b(t) of the probe¹ (as shown in the second line of Eq. 7). Note that the subtraction in Eq. 6 is just a minor correction for the G_b′(ω) data of the blend at low ω examined in Figure 5 (because G_b′(ω) ≫ G_1,m′(ω) at those ω; see middle panel of Figure 4) and that the normalized modulus {M₂/ρυ₂RT}G_2,b′(ω) and the relaxation time were experimentally confirmed to be independent of υ₂ (see, for example, Watanabe et al.³⁸ and its Supporting Information, Sawada et al.⁴⁴ and Watanabe and Kotaka⁴⁵): namely, the probe was confirmed to be dilute and entangled only with the matrix chains.

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In Figure 5, the G_2,b′(ω) data of various PI and PS probes having M₂ ≫ M₁ collapse onto a master curve, indicating that these chemically different probes exhibit universal CR relaxation mode distribution; see also figure 12 of Watanabe et al.²⁶ and figure 4 of Sawada et al.⁴⁴ Furthermore, at low ω, those data are well described by the CR–Rouse model⁴² shown with the solid black curve (Fourier transformation of G_CR(t) given below),

Here, is the longest viscoelastic CR relaxation time, and the second-moment average relaxation time (used in the horizontal axis for the black curve in Figure 5) is given by = (π²/15) in the continuous treatment¹ for N ≫ 1. (The numerical pre-factor of π²/15 is obtained from the second line of Eq. 7, with G_2,b(t) therein being replaced by G_CR(t) given by Eq. 8.)

The universality of the mode distribution seen in Figure 5 vanishes when M₂ and M₁ are not sufficiently separated and the CR mechanism does not dominate the probe relaxation; see figure 3 of Sawada et al.,⁴⁴ figure 4 of Qiao et al.,⁵¹ and figure S5 in Supporting Information of Matsumiya et al.³⁷ Thus, this universality is an important criterion for judging if the probe relaxation is dominated by the CR mechanism.

This criterion can be cast as a critical value of the Struglinski–Graessley parameter SG = M₂. This SG is defined as a ratio of the reptation time of the probe, τ_rep,2 ∝ /M_e, to the Rouse-type CR relaxation time of the probe, τ_CR,2 ∝ τ_rep,1 (M₂/M_e)² ∝ , and numerical pre-factors in τ_rep,2 and τ_CR,2 are omitted in SG. (The matrix reptation time τ_rep,1 involved in τ_CR,2 differs from the actual relaxation time of the matrix, as discussed later in relation to Figures 9 and 10.) The universal mode distribution (namely, the CR dominance in the probe relaxation) is experimentally found only in ranges of SG ≥ 0.2 and SG ≥ 0.5 for PI⁴⁴ and PS^1,46–48 probes, respectively. The critical SG value for the CR dominance is smaller for PI than for PS, suggesting that the CR effect on the relaxation emerges more prominently for PI than for PS when CR is competing with other relaxation mechanisms such as reptation. This difference between PI and PS is further discussed later for Figures 9, 11, and 12.

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The CR relaxation has been observed also for dilute star PI probes entangled with much shorter matrix star PI chains.³⁶ Figure 6 shows data of the normalized storage modulus of those star probes, {M_arm,2/ρυ₂RT}G_2,b′(ω) with M_arm,2 being the arm molecular weight of the probe, plotted against the reduce frequency . The black solid curve indicates the reduced storage modulus obtained from the CR–Rouse relaxation modulus of tethered chains,

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(The average relaxation time associated with Eq. 9 is given by = (π²/12) for N_arm,2 ≫ 1, where the pre-factor of π²/12 is obtained from the second line of Eq. 7 with G_2,b(t) therein being replaced by G_CR(t) given by Eq. 9.) The reduced modulus data of the star probes exhibit almost universal dependence on ω at low ω < 2/ irrespective of M_1,arm and M_2,arm of the matrix and probe arms, and are close to the CR–Rouse modulus, although deviation from this universal dependence is noted at high ω because of fast relaxation of the probe attributable to shallow CLF.³⁶ It needs to be added that the deviation from the universal ω dependence is not clearly observed for the linear probes (cf. Figure 5), because those linear probes have much higher M₂ compared to the star probe arms tested in Figure 6, and thus the non-universality due to CLF is not clearly resolved at the frequencies examined in Figure 5.

2. Data of CR Relaxation Time

For the dilute probes exhibiting the universal behavior of G_2,b′(ω) at low ω (cf. Figures 5 and 6), the data can be used as the second-moment average viscoelastic CR relaxation time . The data available in the literature are summarized in Figure 7 for various linear probes in various linear matrices^34,44–46 and in Figure 8 for two star PI probes in various star PI matrices.³⁶ The plots shown with the same color indicate the data obtained for a given probe in different matrices (and the matrices are mostly common for different probes), as shown in the legend. Those data, cast in empirical equations shown later, allow us to estimate the CR relaxation time of monodisperse linear and star chains in bulk, thereby offering an experimental basis for discussion of the CR contribution to the relaxation of those chains.

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In Figure 7, the data of various linear probes in various linear matrices collapse on a universal line when plotted double-logarithmically against M₁³M₂², as noted in the top and bottom panels for PS and PI, respectively. The proportionality to is in accord with the Rouse-like feature of the actual CR process observed for the PI and PS probes (cf. Figure 5). Similarly, for two star PI probes in various star PI matrices, the normalized data with being the CR–Rouse factor for the probe arm relaxation time collapse on a line when plotted semi-logarithmically against N_arm,1, where N_arm,1 and N_arm,2 indicate the entanglement number per arm of the probe and matrix star chains, respectively: N_arm,j = M_arm,j/M_e with M_e = 5.0 × 10³ for PI. (The normalization by the CR–Rouse factor was necessary for collapse of the plots for the two star probes.) These results can be cast in the following empirical equations (shown in Figures 7 and 8 with black lines):

for the linear PS probe in linear PS melt at 167 °C⁴⁶

for the linear PI probe in linear PI melt at 40 °C^44,51

for the star PI probe in star PI melt at 40 °C³⁶

These empirical equations are used in a test of the CR contribution to the terminal relaxation in monodisperse bulk shown below and in an analysis of the DTD process explained later. (The data of linear probes are well described by those equations, but an equally good description is given by an empirical equation with slightly weaker M₁ dependence.⁴⁴ Nevertheless, the results of the test and analysis hardly changed even if the latter equation was used. Thus, for definiteness, we adopt Eqs. 10–12 as the empirical equations that serve as the basis of our test/analysis.)

Setting M₁ = M₂ = M in Eqs. 10 and 11, we can experimentally evaluate the average viscoelastic CR relaxation time of monodisperse linear PS and PI samples of the molecular weights M. Similarly, from Eq. 12 with N_arm,1 = N_arm,2 = N_arm, we obtain of monodisperse star PI samples. For those PS and PI samples, the second-moment average relaxation time (terminal relaxation time) has been obtained as a product of the zero-shear viscosity and recoverable compliance, and thus the ratio can be evaluated in a purely empirical way. Figure 9 compares this ratio for the linear PS and PI as well as for the star PI. The segmental friction involved in and is canceled in the ratio, so that the comparison in Figure 9 allows us to unequivocally examine differences of the CR contribution to the terminal relaxation of respective polymers.

Comparing the monodisperse linear and star PI having the same entanglement number per chain span, in other words having N = 2N_arm (blue and green symbols in Figure 9), we note that the ratio is considerably smaller and thus the CR contribution to the terminal relaxation is significantly larger for the star PI. This fact is also deduced in our later analysis of the DTD process. (The ratio smaller for the star chain is also expected from the tube model assuming the full dilation of the tube for both linear and star chains, but this model itself fails for the monodisperse star chain, as shown later in that analysis.)

We also note that the ratio is considerably smaller and the CR contribution is considerably larger for linear PI than for linear PS having the same entanglement number N; cf. blue and red symbols. In our ordinary understanding, the terminal viscoelastic relaxation of entangled monodisperse linear polymers is uniquely determined by the plateau modulus G_N and the terminal relaxation time . However, the considerable difference of the ratio noted for linear PI and PS (Figure 9) suggests that this understanding needs to be refined for the CR relaxation of those polymers. This difference is further tested below in relation to the M₁ dependence of the CR time of probes in blends. Readers who like to skip this test can directly proceed to the “D. Experimental Test of Dynamic Tube Dilation Mechanism” section where the DTD mechanism is examined experimentally.

3. Factors Determining CR Relaxation Time

In the simplest molecular view, the viscoelastic CR relaxation time of a dilute probe, , is expected to be proportional to the terminal relaxation time of the pure matrix, . However, this is not the case in experiments. For the star PI matrices examined in Figure 6 (M_arm,1 = 9.5k–41k), the relaxation time is well described by an empirical equation, = 2.3 × 10⁻⁵exp{0.75N_arm,1} (in s) at 40 °C, with N_arm,1 being the entanglement number per matrix star arm.⁵¹ This N_arm,1 dependence of is stronger than that of (Eq. 12). Similarly, of the linear PS and PI matrices is proportional to , and this M₁ dependence is stronger than that of (cf. Eqs. 10 and 11). These differences between and are important in our discussion of the CR mechanism and are further examined below for the raw G′ and η″ data in order to avoid any small uncertainty in the subtraction and zero-ω extrapolation made for in Eqs. 6 and 7.

For the PI/PI blends that contain the same, dilute PI 626k probe (exhibiting the CR relaxation) but different linear PI matrices, Figure 10 plots the G′ and η″ data⁴⁴ against the reduced angular frequency , with the matrix relaxation time being evaluated as the product of the zero-shear viscosity and recoverable compliance of the matrix. (The η″ data are normalized by the matrix relaxation time .) The dashed curves show the data of the matrices plotted in the same format.

At high ω where the matrix (occupying 99.5% of the blends) dominantly contributes to the blend relaxation, the data of the three blends (symbols) and the three matrices (dashed curves) agree with each other when plotted against , as clearly noted in Figure 10. This agreement is consistent with our understanding that the terminal viscoelastic relaxation of monodisperse linear polymer is determined by G_N (common for the three matrices) and : for the blends and matrices, the difference of N₁ has been compensated by plotting the data against (∼ω), which results in the agreement of the high-ω data. In contrast, no agreement is noted for the low-ω data of the blends detecting the CR relaxation of the dilute PI 626k probe. Specifically, in the reduced frequency scale of , the probe relaxation is systematically accelerated with increasing M₁, as clearly noted for both G′ and data. This result indicates, without any further analysis, that of the probe is more weakly dependent on M₁ compared to of the matrix. The factor of appearing in Eqs. 10 and 11 gives just a quantitative description of this experimental fact.

This dependence of could be due to multiplicity of chains sustaining one entanglement (argued by Klein⁵³) combined with the CLF contribution to the CR process,¹ or a very broad crossover in the local CR relaxation on an increase of M₁ (>M_e) as argued by Read and coworkers⁵⁴ (although the corresponding crossover of has not been experimentally resolved for monodisperse matrices). The molecular origin(s) of the dependence of deserves further studies.

In relation to the above test of the M₁ dependence of , it is also informative to examine the difference of for chemically different PS and PI probes by focusing on the raw G′ and η″ data so as to again avoid any small uncertainty in the subtraction and zero-ω extrapolation (Eqs. 6 and 7). For this purpose, Figure 11 shows the normalized G′/G_N and data for PI and PS blends^44,46 that contain dilute probes in linear matrices of very similar entanglement numbers, N₁ = 6.9 and 6.8 for PS 124k and PI 34k in the top panel and N₁ = 4.0 and 4.2 for PS 72k and PI 21k in the bottom panel: the normalization by G_N is necessary for direct comparison of the data of chemically different blends. For PS and PI at 167 and 40 °C, respectively, G_N/MPa = 0.21 and 0.48 and M_e = 1.8 × 10⁴ and 5.0 × 10³, the latter being used for evaluation of N₁.

At high ω where the matrix chains dominantly contribute to the relaxation, the data of the blends (symbols) and matrices (dashed curve) collapse on a universal curve when plotted against the reduced frequency , as noted in Figure 11. This collapse is again in harmony with our understanding that the terminal viscoelastic relaxation of monodisperse linear polymers (matrix in this case) is determined by G_N and . However, at low ω where the CR relaxation of the dilute probe dominates the blend relaxation, the PS and PI blends exhibit a clear difference in their data.

In Figure 11, the differences in the volume fraction υ₂ and the entanglement number N₂ of the probes, υ₂ = 0.01 and N₂ = 156 for PS 2.8M and υ₂ = 0.005 and N₂ = 125 for PI 626k, partly contribute to the difference of the low-ω CR relaxation of these probes. However, this difference due to the differences in N₂ and υ₂ can be removed when the G′ and data are normalized by the Rouse factor for the probe modulus, F_R = {M₂/ρυ₂RT}, and plotted against a reduced frequency , where is the Rouse factor for the CR time of the probe. However, the difference of the low-ω CR relaxation of the PI and PS probes clearly remains even after this normalization, as shown in Figure 12. This remaining difference does not vanish even if the normalized F_RG′ and data are plotted against another type of reduced frequency, with α = 3 or 3.5 (because the PS and PI matrices have almost identical N₁).

The difference between the PS and PI probes seen in Figure 12 clearly indicates that these probes have different CR relaxation times even when N₁ and N₂ are common for them. In other words, is not uniquely determined by N₂ and the matrix relaxation time but is affected by an extra factor that changes with the chemical structure of the chain. This extra factor, not clearly recognized for monodisperse polymers, may correspond to the number z of local constraints per entanglement considered in the CR–Rouse model of Graessley:⁴² z reflects the number of matrix chains forming an entanglement for the probe chain. If the entanglement is exclusively due to the binary (pair-wise) constraint between the chains irrespective of their chemical structure, the above difference between the PS and PI probes may not be straightforwardly deduced within the CR–Rouse model. A correlation between the extra factor and the chemical structure of the chain is closely related to the nature of entanglement (binary or multiple-chain constraint) and is further discussed later in the “F. Comments on Recent Theoretical Model/Analysis” section in relation to the tube dilation exponent.

Here, it should be emphasized that the extra factor discussed above is absorbed in the numerical front factors in Eqs. 10 and 11, and thus the CR behavior of chemically different PS and PI probes is still described commonly by the CR–Rouse model (with its being specified by Eqs. 10 and 11). In fact, the CR relaxation of the PS 2.8M and PI 626k probes examined in Figures 11 and 12 exhibits the same, Rouse-type mode distribution, as noted from the agreement of the data and black solid curves, where the curves for G′ represent a sum of G_CR′ of the probe (obtained from G_CR(t) shown in Eq. 8) and the terminal tail of the υ₁G_1,m′ data (∝ ω²) of the matrix, and the curves for η″, the corresponding sum reduced by ω. Note that the difference between the PS 2.8M and PI 626k probes in Figures 11 and 12 vanishes when their G_2,b′ data are normalized by the Rouse factor and plotted against the frequency normalized by their CR time data ; see collapse of the plots for those probes around the Rouse–CR curve in Figure 5 (the filled gray square and circle for the PS 2.8M probe in PS 72k and PS 124k matrices, and the unfilled red square and circle for the PI 626k probe in PI 21k and PI 34k matrices). This collapse again indicates the validity of the CR–Rouse model. Thus, in the following sections, we use this model to examine some details of the dynamic tube dilation (DTD) mechanism.

1. Test of Molecular Picture of Full-DTD

Several molecular models^43,57–59 assume that the relaxed portion of the chain behaves as a solvent not contributing to the entanglement, and thus the tube diameter in a melt is fully dilated to the diameter in a solution having a polymer volume fraction υ = ϕ′(t). With this full-DTD assumption, the number of equilibrated entanglement segments, β(t), is related to ϕ′(t) as^24,33,34,43

Equation 17 is equivalent to the scaling of the plateau modulus of the solution, G_N,soln = G_N,bulkυ^1+d.

The value of the dilation exponent d is a subject of recent theoretical analyses^60,61 (that prefer d = 1 at short t), as explained later in detail. However, in our experimental test of the molecular picture of full-DTD, d is to be determined directly from the modulus data of blends containing the long chains that are entangled among themselves and with much shorter chains. The G′ data of PI 308k/PI 21k blends³⁴ with υ₂ ≥ 0.1 serve this purpose; the long chains therein (PI 308k) are entangled among themselves as well as with the short chains (PI 21k). In Figure 14, the G′ data of those blends are normalized by a factor of with d = 1.3 (top panel) and d = 1.0 (bottom panel) and plotted against the normalized frequency . Here, is the second-moment average viscoelastic relaxation time of the long chain in the blend evaluated by Eqs. 6 and 7, except that G_1,m * (ω) in Eq. 6 is replaced by G_1,m * (λω) where the factor λ represents minor retardation of the matrix relaxation due to the entanglement with the concentrated long chains.³⁴ (This entanglement suppresses CR for the short matrix chains in the blends to retard their relaxation compared to that in the pure matrix.)

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For d = 1.3, the G′ data of the PI 308k/PI 21k blends are well superposed on the G′ data of bulk PI 308k (black circles) in the terminal relaxation regime at < 2, as noted in the top panel of Figure 14. In contrast, the bottom panel shows that d = 1.0 gives much poorer superposition, as most clearly noted from comparison of the insets in the top and bottom panels. The corresponding difference, good and poor superposition for d = 1.3 and 1.0, is noted also for the data of the blends as well as for the data of PI 308k solutions in an oligomeric butadiene.³⁴ Thus, our experimental test of the full-DTD picture adopts d = 1.3, as already shown in Eq. 17.

Substituting Eq. 17 into Eqs. 15 and 16, we can use the dielectric Φ(t) data of monodisperse linear and star PI to evaluate their ϕ′(t) for the case of full-DTD. The viscoelastic relaxation function obtained from this ϕ′(t), μ_f-DTD(t) = {ϕ′(t)}^1+d with d = 1.3 (cf. Eqs. 14 and 17), is shown in Figures 15 and 16 with green curves for comparison with the μ(t) data of monodisperse linear^34,37 and star PI,⁵¹ respectively. (For this comparison, the G*(ω) data of linear PI reported in Watanabe et al.^34,37 were converted into μ(t).)

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For the PI 308k/PI 21k blends examined in Figure 14, we first need to decompose the Φ(t) data of the blend as a whole into contributions Φ₁(t) and Φ₂(t) of the short and long linear chains (components 1 and 2):³⁴ Φ(t) = (1 − υ₂)Φ₁(t) + υ₂Φ₂(t). The mode distribution of Φ₁(t) was found to agree well with that of the short chain in monodisperse bulk,³⁴ which enabled easy decomposition. Then, from those Φ_j(t), ϕ_j′(t) of the component j was evaluated through a relationship analogous to Eq. 15 combined with Eq. 17, Φ_j(t) = ϕ_j′(t) − (1/4N_j)[{ϕ′(t)}^−d/2 − 1]² with j = 1,2 and ϕ′(t) = (1 − υ₂)ϕ₁′(t) + υ₂ϕ₂′(t).³⁴ (This relationship considers the dilated tube diameter to be common for the long and short chains, as noted for the factor in the second term, {ϕ′(t)}^−d/2 = a_f-DTD′(t)/a for full-DTD.) The corresponding viscoelastic relaxation function, μ_f-DTD(t) = {ϕ′(t)}^1+d with d = 1.3, is compared with the μ(t) data in Figures 17.³⁴

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For the monodisperse linear PI chains (Figure 15), μ_f-DTD(t) (green curves) is in good agreement with the μ(t) data (circles) in the entire range of t. Thus, the full-DTD assumption is valid for consistently describing the Φ(t) and μ(t) data of those chains. This validity has been confirmed also for G′(ω) and G″(ω) data in the frequency domain (see figure S1 in Supporting Information of Matsumiya et al.³⁷).

In contrast, for the monodisperse star PI (Figure 16) as well as for the blends of linear PI (Figure 17), μ_f-DTD(t) (green curves) is considerably smaller than the μ(t) data at intermediate t. Namely, the molecular picture of full-DTD significantly overestimates the viscoelastic relaxation at those t, as noted also in the frequency domain.^33,34 (Similar results have been found also for a Cayley-tree type branched PI.⁵⁶) In particular, for the blends with a small volume fraction of the long chain (υ₂ = 0.1 and 0.2), this overestimation is most significant at t = 10⁻³–10⁻¹ s where the short matrix chain (majority in the blends) has fully relaxed but the long chain has not; see Figure 17. Nevertheless, at either longer or shorter time scales, μ_f-DTD(t) agrees with the μ(t) data of the blends. These results suggest the origin of the failure of the full-DTD picture, as discussed in the following section.

Here, it is informative to examine the prediction of the Milner–McLeish (MM) model⁵⁸ for entangled monodisperse star chains. The MM model is a sophisticated tube model that incorporates the stochastic, first-passage nature of the arm retraction but still adopts the full-DTD picture. The normalized viscoelastic and dielectric relaxation functions of this model can be summarized as^33,58

and

with

In Eqs. 18 and 19, L_eq = N_arma is the equilibrium contour length of the star arm consisting of N_arm entanglement segments, τ_MM(z) is a time required for the arm retraction over a contour length z (that includes both shallow and deep retraction, the latter being associated with an entropic penalty), and d is the dilation exponent. Figure 18 compares the MM calculation (with d = 1.3) and the μ(t) and Φ(t) data of six-arm star PI (80k)₆ sample. (The data and calculation in the frequency domain published in Watanabe et al.³³ were converted into the time domain.) For an adequate choice of the model parameters, the MM model excellently describes the μ(t) data; see red curve in the top panel. However, the model with the same parameters gives the dielectric Φ(t) considerably larger than the data at long t, as shown with the red curve in the bottom panel. An adjustment of the parameters allows the MM model to describe the Φ(t) data (cf. green curve in the bottom panel), but the viscoelastic μ(t) calculated with those adjusted parameters (green curve in the top panel) is significantly smaller than the data. Namely, even the sophisticated tube model fails to consistently describe the viscoelastic and dielectric data of star PI, given that the model adopts the full-DTD picture. (The failure is noted also for the model calculation with d = 1.) This result for the MM model is consistent with the results of experimental test of the full-DTD picture presented in Figure 16: the MM fitting of the Φ(t) data gives μ(t) being smaller than observed (cf. green curve in top panel of Figure 18), which is similar to the experimental results seen in Figure 16.

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2. Test of Molecular Picture of Partial-DTD

The local CR process is accumulated during the relaxation to expand a length scale of lateral displacement allowed for the entanglement segments. DTD is the molecular picture that makes coarse-graining of this accumulation in both length and time scales to define an effectively dilated tube in a given time scale.^1,2,24,43 Thus, the DTD picture should be valid if the length and time scales are consistently coarse-grained. For a test of this consistency, the number of equilibrated entanglement segments assumed in the full-DTD picture, β_f-DTD(t) = {ϕ′(t)}^−d (Eq. 17), should be compared with the maximum number of equilibrated segments allowed by the CR mechanism, β_CR(t) = 1/ψ_CR(t), with ψ_CR(t) being a survival function specified below.

For a monodisperse linear chain composed of N entanglement segments, ψ_CR(t) can be expressed as^36,51

for monodisperse linear chain

with

and

Here, τ_CR denotes the CR time defined for the displacement of entanglement segments and is equivalent to the known dielectric CR time for the end-to-end fluctuation, = ( has been confirmed to coincide with for dilute PI probes in the CR regime³⁴). Equation 20 considers the CR–Rouse process of a given chain (probe) activated by the global motion of the surrounding chains (the first summation in Eq. 20) and by the contour length fluctuation (CLF) of the probe (equivalent to the surrounding chains in the monodisperse system; the second summation). N_CLF is the number of entanglement segments relaxing through CLF that occurs with a known characteristic time τ_CLF (intrinsic Rouse time of the whole chain backbone). The factor indicates a relaxation time ratio of p-th CR mode to the slowest CR mode, and the other factor , the ratio for p-th and slowest CLF modes. (For high-M linear PI, the CLF term in Eq. 20 is not important at long t. Thus, Eq. 20 with N_CLF = 0 was used in Watanabe et al.³⁵ However, for completeness, Eq. 20 with N_CLF = was used in this article to evaluate ψ_CR(t) for monodisperse linear PI examined in Figure 15.)

For the star chain having N_arm entanglement segments per arm, ψ_CR(t) is also given by Eqs. 20 and 21, where N is replaced by N_arm and the relaxation time ratios, r_p, by the Rouse-type ratios for a tethered chain,^36,51

The tube model generally assumes that the branching point of the star chain is fixed in space until the star arm fully relaxes, which leads to the use of r_p for the tethered chain in Eq. 22.

Here, a comment needs to be made for the dilute linear probes examined in Figure 5. In principle, the CLF contribution specified above also needs to be considered for these probes. However, all those probes have large N₂ (≥24), so that their relaxation at ≤ 100 (examined in Figure 5) is dominated by the low-order CR–Rouse modes with the index up to p ≅ 10 and is negligibly contributed from the fast CLF modes. For this reason, those linear probes exhibit the universal terminal relaxation that coincides with the CR–Rouse relaxation (solid curve in Figure 5). (At low ≤ 100, μ(t) = ϕ′(t)/β_CR(t) = ϕ′(t)ψ_CR(t) evaluated on the basis of Eqs. 14 and 20 is numerically indistinguishable from a reduced modulus {M₂/N₂ρυ₂RT}G_CR(t), with G_CR(t) being given by Eq. 8.) The situation is a little different for the star probes examined in Figure 6. Those star probes have just moderately large N_arm (= 12 and 16), and their terminal relaxation is non-negligibly contributed from the CLF modes. Because of this CLF contribution, the star probes exhibit less universal terminal relaxation compared to the linear probes (cf. Figures 5 and 6), as explained earlier.

Now, we compare β_f-DTD(t) and β_CR(t) (= 1/ψ_CR(t)) for the monodisperse linear and star chains examined in Figures 15 and 16. (μ_f-DTD(t) shown therein is equivalent to β_f-DTD(t) = {μ_f-DTD(t)}^−d/(1+d).) For those chains, τ_CLF and τ_CR (= = ; cf. Eqs. 11 and 12 with M₁ = M₂ = M) are known, so that the comparison can be straightforwardly made. The results are shown in Figures 19 and 20, where the black arrows indicate the second-moment average viscoelastic relaxation time of the monodisperse chain, (= product of the zero-shear viscosity and recoverable compliance).

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For the linear chains (Figure 19), β_f-DTD(t) (blue circles) agrees with β_CR(t) (red curves) within uncertainties of evaluation of these β (∼10% for each) in the entire range of t < , namely, during the whole process of viscoelastic relaxation. (For PI 308k at long t ∼ , β_f-DTD(t) tends to become even smaller than β_CR(t).) Thus, the CR–Rouse dynamics allows the equilibrated entanglement number β(t) to increase up to β_f-DTD(t) considered in the full-DTD picture. For this reason, the normalized viscoelastic relaxation function μ_f-DTD(t) deduced from this picture agrees with the μ(t) data, as noted in Figure 15.

In contrast, for the monodisperse star chains, β_f-DTD(t) is significantly larger than β_CR(t) (by a factor beyond the uncertainty explained above), and thus the CR–Rouse dynamics does not allow the actual β(t) to increase up to β_f-DTD(t) at intermediate to long time scales (at t > 10⁻² s); see Figure 20. Consequently, μ_f-DTD(t) deviates from the μ(t) data at those t, as noted in Figure 16. In fact, in Figure 20, the failure of the full-DTD picture can be noted for β_f-DTD(t) itself: β_f-DTD(t) reaches its maximum possible value, β_max = N_arm (horizontal dashed line) corresponding to the full equilibration of the whole arm, at t considerably shorter than , which does not allow the full-DTD picture to work in the terminal relaxation regime. (In contrast, the CR-equilibrated number β_CR(t) remains smaller than β_max in the entire range of t < .)

Here, it is informative to consider how the full-DTD picture fails for the star chains and works for the linear chains. For an arm of a given (probe) star chain, the fully dilated tube diameter (= a{β_f-DTD(t)}^1/2 = a{ϕ′(t)}^−d/2) is available as a length scale of equilibration in a time scale of t thanks to the motion of surrounding star chains; however, this length scale is too large for the CR–Rouse motion of the probe arm to explore within that time scale. Such an unusably large a_f-DTD′(t) is a consequence of the broad relaxation mode distribution of the star chains that significantly decreases ϕ′(t) at short t. In contrast, the monodisperse linear chains exhibit a narrow terminal relaxation mode distribution that allows β_f-DTD(t) to stay small and comparable to β_CR(t) in the entire range of t (cf. Figure 19): β_f-DTD(t) ≅ 3 ≪ N even at t = . This narrow mode distribution, enabling the CR–Rouse motion to cover the length scale of a_f-DTD′(t) in time, is the reason for the validity of the full-DTD picture for the linear chains noted in Figure 15.

For the PI 308k/PI 21k binary blends, we can similarly examine validity of the full-DTD picture. Specifically, β_f-DTD(t) (= {μ_f-DTD(t)}^−d/(1+d) with μ_f-DTD(t) being shown in Figure 17) needs to be compared with β_CR,1(t) and β_CR,2(t) defined for the short and long chains, PI 21k and PI 308k. The comparison is made by separately considering the CR–Rouse times for the long and short chains on the basis of empirical Eq. 11 and by taking into account a dielectrically detected moderate retardation of the short chain relaxation due to the entanglement with the long chains.³⁵ The results of this comparison are presented in Figure 21. For simplicity, β_CR,1(t) and β_CR,2(t) used in the comparison were evaluated just for the CR–Rouse relaxation, namely, with the aid of Eq. 20 but without incorporating the CLF contribution.³⁵ However, for the dominant part of the long chain relaxation at long t, the CLF contribution is minor and the lack of this contribution in β_CR,1(t) and β_CR,2(t) is not important.

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For the blends with a small volume fraction of the long chain, υ₂ = 0.1 and 0.2, Figure 21 shows that β_f-DTD(t) (blue circle) remains close to β_CR,1(t) (green curve) and β_CR,2(t) (red curve) at short t up to the terminal relaxation time of the short chain, (green arrow). (Note also that β_CR,1(t) in that range of t remains smaller than its maximum possible value, N₁, shown with the horizontal green dotted line.) However, at intermediate t where the short chain has fully relaxed and only the long chain sustains the modulus, β_f-DTD(t) becomes considerably larger than β_CR,2(t) so that the long chain cannot explore, in time, the length scale of fully dilated tube diameter a_f-DTD′(t) = a{β_f-DTD(t)}^1/2. Nevertheless, at longer t where β_CR,2(t) approaches its maximum possible value, N₂ (horizontal black dotted line), β_f-DTD(t) becomes smaller than β_CR,2(t) and the long chain can be equilibrated over the diameter a_f-DTD′(t) in time. For those blends, the failure of the full-DTD picture is noted exactly in the intermediate time scale where β_f-DTD(t) > β_CR,2(t); compare Figures 17 and 21 for υ₂ = 0.1 and 0.2. It should be noted that the rapid and intensive relaxation of the short chain leads to a large decrease of ϕ′(t) and a large increase of β_f-DTD(t) at t ≥ 2 (cf. Figure 21) thereby resulting in the failure of the full-DTD picture. This role of the fast relaxation of the short chain in the blends with small υ₂ is similar to that of intensive fast relaxation of the star chains having a broad mode distribution. Consequently, the blend with larger υ₂ = 0.5 exhibits less intensive fast relaxation of the short chain so that its β_f-DTD(t) is only slightly larger than β_CR,2(t) at intermediate time scale (cf. Figure 21), which results in just slight failure of the full-DTD picture for this blend (Figure 17). Finally, for υ₂ = 1 (bulk PI 308k), β_f-DTD(t) ≤ β_CR,2(t) and thus the full-DTD picture is valid in the entire range of t up to the terminal relaxation time of the long chain, (red arrow); cf. Figures 17 and 21.

The results of the test of the full-DTD picture presented in Figures 19–21 clearly indicate that the full-DTD picture fails in the range of t where β_f-DTD(t) > β_CR(t). However, the test does not rule out the tube dilation up to β_CR(t). In fact, direct comparison of the viscoelastic and dielectric data of monodisperse PI unequivocally indicates that the entanglement is not fixed in space (cf. Figure 3). Namely, in the terminology of the tube model, the CR/DTD process undoubtedly occurs for the monodisperse polymers. Thus, a molecular picture of partial-DTD is naturally introduced as^24,35,51

The corresponding survival fraction of the partially dilated tube, ϕ′(t), is evaluated from the dielectric Φ(t) data and the τ_CR data (= = ; cf. Eqs. 11 and 12 with M₁ = M₂ = M): the τ_CR data give β_CR(t) = 1/ψ_CR(t) through Eqs. 20–22, and Eqs. 15 and 16 combined with Eq. 23 allow us to determine ϕ′(t) and β_p-DTD(t) from the β_CR(t) value and the Φ(t) data.^35,51 The viscoelastic μ_p-DTD(t) for the partial-DTD picture is simply given by Eq. 14 with β(t) = β_p-DTD(t). In Figures 15 and 16, μ_p-DTD(t) thus obtained is shown with the red curves. For monodisperse linear PI, β_f-DTD(t) ≤ β_CR(t) (cf. Figure 19) so that μ_p-DTD(t) coincides with μ_f-DTD(t) and excellently describes the μ(t) data (cf. Figure 15). For monodisperse star PI, μ_p-DTD(t) is considerably larger than μ_f-DTD(t) and agrees well with the μ(t) data (cf. Figure 16).

For the PI 308k/PI 21k blends, the above partial-DTD picture is extended to separately evaluate β_p-DTD(t) for the short and long chain components (in a way corresponding to that explained for Figure 21), and the resulting μ_f-DTD(t) is shown in Figure 17 with the red curve.³⁵ This μ_f-DTD(t) is in excellent agreement with the μ(t) data.

All above results indicate that the partial-DTD picture (including the full-DTD picture for the case of β_f-DTD < β_CR) is valid for monodisperse PI as well as blends of linear PI. The success of the partial-DTD picture has been confirmed also for Cayley-tree type branched PI.⁵⁶ The success of the partial-DTD picture indicates that the tube dilates (i.e., the entanglement loosens) to the maximum length scale allowed by the motion of the tube-forming chains and by the CR motion of the probe chain in a given time scale. In other words, the DTD picture is valid given that the length and time scales are consistently coarse-grained, as noted in Figures 19–21.

It is the comparison of dielectric and viscoelastic data that revealed the success of the partial-DTD picture for describing the entanglement dynamics. This success in turn demonstrates the importance of the comparison of those data. In fact, the comparison is useful also for dipole-inverted PI: coherence of the chain motion in entangled bulk and lack of this coherence in unentangled solutions have been successfully deduced from comparison of the viscoelastic and dielectric data.²⁸

1. Viscoelastic and Dielectric Data of Linear PI Probe in DTD-Free Environment

Figure 22 shows viscoelastic and dielectric losses, G″ and ɛ″, measured for a PI 43k/PI 1.1M blend containing dilute linear PI 43k probe (υ₁ = 0.1).³⁷ The blue triangles and black circles represent the data of PI 43k and PI 1.1M in respective monodisperse bulk. The probe relaxation in the blend is clearly detected as the peak of the G″ and ɛ″ data at high ω; see red circles. This relaxation is observed also for the G′ and {ɛ₀ − ɛ′} data, but much less clearly because these data are less sensitive to fast and weak relaxation of the probe compared to the G″ and ɛ″ data.³⁷ For this reason, the following discussion focuses on the G″ and ɛ″ data.

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As noted in Figure 22, the probe relaxation is slower in the blend (in the high-M matrix) than in its monodisperse bulk, which suggests that the CR mechanism for the probe is suppressed in the blend. For quantitative discussion of this suppression, the blend data need to be decomposed into the contributions from the probe and matrix (short and long chains). This decomposition can be made on the basis of a general blending rule for the viscoelastic and dielectric loss data of the blend, G_b″(ω) and ɛ_b″(ω):³⁷

Here, G₂,_b″(ω) and ɛ₂,_b″(ω) represent the viscoelastic and dielectric losses of the long (matrix) chain in the blend reduced by its volume fraction υ₂, and G₁,_b″(ω) and ɛ₁,_b″(ω) are the losses of the short probe reduced by υ₁ (= 1 − υ₂). G₂,_b″(ω) and ɛ₂,_b″(ω) do not coincide with G₂,_m″(ω) and ɛ₂,_m″(ω) of the pure matrix (black circles in Figure 22) because of the partial relaxation of the long matrix chains activated by the short probe motion. However, for the blend with small υ₁ (= 0.1 in Figure 22), we can satisfactorily express G₂,_b″(ω) and ɛ₂,_b″(ω) in terms of the G₂,_m″(ω) and ɛ₂,_m″(ω) data to evaluate G₁,_b″(ω) and ɛ₁,_b″(ω) of the probe of our interest, as explained below.³⁷

The top panel of Figure 22 shows that the probe (PI 43k) in the blend has fully relaxed at high ω ≅ 10 s⁻¹, and the corresponding probe motion activates partial viscoelastic relaxation of the matrix. Consequently, the terminal relaxation of the matrix (and the blend as a whole), seen at much lower ω, is faster and less intensive compared to that of the pure matrix. This feature enables us to express G₂,_b″(ω) of the matrix in the terminal relaxation zone in terms of the G₂,_m″(ω) data of the pure matrix (black circles in the top panel of Figure 22) as

The factors I₂ and represent the fractional intensity reduction and acceleration of the terminal relaxation of the matrix in the blend. Validity of Eq. 26 is noted in the top panel where the green curve shows the shifted data of pure matrix, with the factors I₂ = 0.89 and = 0.89.³⁷ This green curve excellently describes the G_b″(ω) data of the blend at low ω.

At high ω where the probe of our interest (PI 43k) is still relaxing, (ω) given by Eq. 26 should differ from the actual loss modulus G₂,_b″(ω) of the matrix in the blend, because the matrix entanglement with the probe chains has not fully relaxed at those ω. At such high ω, the matrix partially relaxes viscoelastically together with the probe. Then, the loss modulus difference of the matrix representing this partial relaxation, ΔG₂,_b″(ω) ≡ G₂,_b″(ω) − (ω), should be close to G₁,_b″(ω) of the probe in the blend, except that the relaxation intensity is smaller for ΔG₂,_b″(ω) by a factor of 1 − I₂ (because the fraction I₂ of the intensity relaxes at low ω as represented by (ω)). Thus, substituting G₂,_b″(ω) = ΔG₂,_b″(ω) + (ω) ≅ (1 − I₂)G₁,_b″(ω) + I₂G₂,_m″(ω) in Eq. 24, we can express of the probe in terms of the G_b″(ω) and G₂,_m″(ω) data of the blend and pure matrix as

(Note that a modulus difference corresponding to the above ΔG₂,_b″(ω) does not appear in Eq. 6 for short matrix chains in the blends examined in Figures 4–6 because those short chains are the major component therein and fully relax within a time scale of the long–short entanglement relaxation.)

The situation is much simpler for dielectric ɛ₁,_b″(ω). In general, the dielectric relaxation mode distribution (shape of ɛ″ curve) of linear PI is insensitive to CR/DTD.^23,24,34,35 In the bottom panel of Figure 22, this insensitivity is noted as the similarity of the ω dependence of ɛ″ data of the PI 43k/PI 1.1M blend, bulk PI 43k, and bulk PI 1.1M (matrix) at ω > 300 s⁻¹. Thus, the reduced dielectric loss of the matrix appearing in Eq. 25, ɛ₂,_b″(ω), is satisfactorily expressed in terms of the ɛ₂,_m″(ω) data of the pure matrix as ɛ₂,_b″(ω) = ɛ₂,_m″(ω), where is the acceleration factor determined for the viscoelastic data. Then, the reduced dielectric loss of the probe PI, ɛ₁,_b″(ω), is experimentally evaluated as³⁷ (cf. Eq. 25)

Figure 23 compares G₁,_b″(ω) (unfilled symbols) and ɛ₁,_b″(ω) (filled symbols) of short probe chains entangled with long matrix (PI 1.1M) thus evaluated from Eqs. 27 and 28.³⁷ = 0.94 for the PI 179 probe, and = 0.89 for the three other probes. For direct comparison, ɛ₁,_b″(ω) is multiplied by a factor of 10^A in order to match its peak height with that of G₁,_b″(ω), and a contribution from the intrinsic, local Rouse relaxation within the entanglement segment has been subtracted from G₁,_b″(ω). (ɛ₁,_b″(ω) does not have this contribution.) The volume fraction is υ₁ = 0.2 for the PI 179k probe, and υ₁ = 0.1 for the other three probes, PI 21k, PI 43k, and PI 99k. Judging from the effective molecular weight associated to the probe–probe (short–short) entanglement, M_e,pp = M_{e,bulk PI}/υ₁^1.3 = 41k (for υ₁ = 0.2) and M_e,pp = 100k (for υ₁ = 0.1), the PI 179k probe chains are mostly entangled with the PI 1.1M matrix but also among themselves moderately, whereas the three shorter probes are entangled only with the matrix.

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Here, a comment needs to be added for the probe losses shown in Figure 23. The matrix losses G₂,_m″(ω) and ɛ₂,_m″(ω) subtracted in Eqs. 27 and 28 are the extrapolation of the viscoelastic and dielectric losses of the matrix at low ω (where the probe has fully relaxed) to high ω where the probe is still relaxing. Before completion of the probe relaxation, the matrix would behave more or less similar to that in monodisperse bulk, and its actual losses should be smaller than G₂,_m″(ω) and ɛ₂,_m″(ω) appearing in Eqs. 27 and 28. Thus, G₁,_b″(ω) and ɛ₁,_b″(ω) shown in Figure 23 should be regarded as the smallest possible viscoelastic and dielectric losses of the probe in the blend. Nevertheless, the largest possible losses obtained from Eqs. 27 and 28 with = 1 were only slightly larger than those shown in Figure 23, as demonstrated in Matsumiya et al.³⁷ This close coincidence of the smallest and largest possible loss values, reflecting the value close to unity³⁷ ( ≥ 0.89 for the matrix entangled with the four probes examined in Figure 23), enables us to reliably use G₁,_b″(ω) and ɛ₁,_b″(ω) shown in Figure 23 as the real losses of the probes.

For the PI 21k and PI 43k probes, G₁,_b″(ω) agrees surprisingly well with 10^Aɛ₁,_b″(ω) to satisfy Eq. 5b, as clearly noted in Figure 23. This experimental fact indicates that the matrix–probe (long–short) entanglement constraining the probe motion is fixed in space in the time scale of the probe relaxation. Namely, these two probes relax in the DTD-free environment. In fact, the ω dependence of their G₁,_b″(ω) and ɛ₁,_b″(ω) is very close to that expected for a chain being trapped in a fixed tube and relaxing through reptation and CLF; G_rept+CLF″ ∼ ɛ_rept+CLF″ ∼ ω^−1/4 at high ω.^1,2 For the PI 99k probe, the G₁,_b″(ω) and 10^Aɛ₁,_b″(ω) data are slightly different around their peaks, which indicates that the DTD mechanism is not perfectly quenched for this probe entangled with the PI 1.1M matrix (because M₂ of the matrix is not sufficiently larger than M₁ of the probe). Finally, for the PI 179k probe having a larger M₁, a moderate deviation is clearly noted between G₁,_b″(ω) and 10^Aɛ₁,_b″(ω), although this deviation is much less prominent compared to the deviation seen for monodisperse linear PI in bulk; see the top panel of Figure 3 where G″ and ɛ″ exhibit significant differences not only in their relaxation mode distribution but also in the peak frequency. This moderate deviation seen for the PI 179k probe emerges because the matrix–probe entanglement relaxes to some extent in the time scale of probe relaxation to activate DTD for the probe (and because the moderate probe–probe entanglement also relaxes to activate DTD).

2. Relaxation Time in DTD-Free Environment

In relation to the validity of the DTD picture discussed for Figures 15–17, it is informative to examine the magnitude of acceleration of the relaxation due to DTD. For monodisperse linear PI chains, the full-DTD relationship (Eqs. 14 and 17) is valid between the viscoelastic μ(t) and the dielectrically evaluated tube survival fraction ϕ′(t), as revealed in Figure 15. This validity reflects the narrow relaxation mode distribution of the linear chains that allows the number β(t) of the equilibrated entanglement segments to stay small; in Figure 19, β(t) associated to the terminal viscoelastic relaxation is evaluated as β_μ ≅ 3 (log β_μ ≅ 0.5) at t = for all linear PI samples examined.

If the relaxed portion of the chain behaves as a solvent in all aspects of the relaxation dynamics of monodisperse linear PI chains (i.e., not only for the relationship between μ(t) and ϕ′(t)), then the terminal relaxation time of the monodisperse linear PI chain, τ ∼ M^3.5/, is naively expected to increase by a factor of ≅ 5 when the CR/DTD mechanism working in the monodisperse bulk system is quenched. (The effective entanglement molecular weight determining the relaxation time should be proportional to β_μ if the relaxed portion behaves as the solvent in all aspects.) In the following, this expectation is tested with the aid of the data shown in Figure 23.

For the PI probes examined in Figure 23, we can evaluate the viscoelastic and dielectric relaxation times, and , as reciprocal of the peak frequencies of the G₁,_b″(ω) and ɛ₁,_b″(ω) data. (The corresponding “storage parts” of those probes, G₁,_b′(ω) and ɛ₁,_b′(ω), cannot be accurately evaluated because G_b′(ω) and Δɛ_b′(ω) data of the blends are rather insensitive to the fast and weak relaxation of the probes having small υ₁ (= 0.1 or 0.2). For this reason, the second-moment average relaxation time (cf. Eq. 7) cannot be used in the discussion below.) From the G₁,_m″(ω) and ɛ₁,_m″(ω) data of the probes in the monodisperse bulk state, the relaxation times and are similarly evaluated as reciprocal of the peak frequencies. The relaxation mode distribution of ɛ₁,_b″(ω) in the blends agrees with that of the ɛ₁,_m″(ω) data of monodisperse bulk PI,³⁷ so that the ratio is equivalent to a ratio of the longest dielectric relaxation times in the fixed and unfixed entanglement environments. This is not the case for the viscoelastic data: the mode distribution is narrower for G₁,_b″(ω) than for G₁,_m″(ω) (cf. Figures 22 and 23), and thus the ratio should be somewhat different from (larger than) the ratio of the longest viscoelastic relaxation times. However, this small difference does not affect the following discussion that mainly focuses on the dielectric ratio.

For the four PI probes examined in Figure 23, Figure 24 shows plots of the and ratios evaluated above³⁷ against the entanglement number per probe chain, M₁/M_e (= 4.2–35.8); see green circles. For comparison, black squares show the ratios obtained for shorter, lightly entangled probes in their bulk and in the PI 1.1M matrix (after minor correction of the monomeric friction in bulk).³⁹ The red curves are the results of CR–Rouse analysis explained later. The high-M matrix should have no effect on the relaxation time of unentangled probe if the monomeric friction is kept constant, whereas the matrix effect (suppression of CR/DTD) should become prominent for the entangled probe. This behavior is clearly noted in Figure 24 on the increase of M₁/M_e from 1 to ∼4. A further increase of M₁/M_e results in gradual decreases of the and ratios, and thus the matrix effect becomes less prominent with increasing M₁/M_e; see four green circles.

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In the entire range of M₁/M_e, the dielectric ratio (top panel of Figure 24) is significantly smaller than the naively expected ratio explained above, ≅ ≅ 5 (log ≅ 0.7). This experimental fact indicates that the relaxed portion of the chains does not behave as a solvent with respect to the end-to-end relaxation time of the probe, despite the fact that the viscoelastic μ(t) data and the dielectrically evaluated ϕ′(t) of monodisperse linear PI obey the full-DTD relationship (cf. Figure 15). This relationship is just based on the molecular picture that the relaxed portion of the chains, having the fraction 1 − ϕ′(t), widens the entanglement mesh (modeled as the tube) to the level in a solution having a concentration ϕ′(t). Namely, the full-DTD relationship does not specify anything related to the chain motion in the dilated tube over a length scale greater than dilated tube diameter. In contrast, the naive expectation is based on an assumption that the chain moves along the dilated tube with its intrinsic friction. Thus, the deviation between the data and the expectation is not contradictory to the validity of the full-DTD relationship between μ(t) and ϕ′(t) but suggests a hypothesis explained below.

From the above deviation, the linear PI chain in the monodisperse bulk appears to be longitudinally moving not along the fully dilated tube but along a thinner tube that wriggles in the fully dilated tube, as illustrated in Figure 25. The dilated tube, being introduced as a model to represent the loosening of the entanglement constraint, seems to have a dual structure.³⁷ A wide tube having the diameter a′ describes the constraint for the chain motion in a direction lateral to the chain backbone, thereby describing the extra decay of the viscoelastic modulus on the entanglement loosening (by the full-DTD factor of {a/a′}² = {ϕ′}^d for monodisperse linear chains). In contrast, a thinner tube having the diameter a* (with a < a* < a′) specifies the longitudinal motional path of the chain that determines the dielectric relaxation time (= end-to-end fluctuation time), and the wriggling motion of this path tube in the wider tube allows the lateral equilibration over the length scale of a′ and the corresponding modulus decay to occur. Within this hypothesis, the viscoelastic relaxation time is affected by this modulus decay on the entanglement loosening, and thus the suppression of the loosening (i.e., blending in the high-M matrix) increases the viscoelastic of the probe more significantly than the dielectric , which is consistent with the observation in Figure 24. In the following, the dual tube hypothesis is tested through a simple CR–Rouse analysis.

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3. CR–Rouse Analysis of Relaxation Time.³⁷ — We here focus on the dielectric relaxation time (end-to-end fluctuation time) of the linear PI chain in its monodisperse bulk. This is assigned as the time required for reptation (after CLF) along the motional path tube having the diameter a* (cf. Figure 25). For this reptation to occur, all dilated segments of the size a* should balance their tension through simultaneous CR-equilibration because of the coherent nature of the reptative motion. This simultaneous CR-equilibration occurs through accumulation of local CR-equilibration in all dilated segments.

The number of those dilated segments per chain is given by N* = N/ β*, where β* = (a*/a)² is the number of entanglement segments per dilated segment. Among N entanglement segments of the chain, N* segments (one for each dilated segment) remain as independent segments specifying the spatial position of the dilated segments, so that the number of entanglement segments to be involved in the simultaneous CR-equilibration is given by

Here, the extra factor of “1” has been introduced to satisfy two extreme conditions, g = N for β* = N (for CR-equilibration of all entanglement segments) and g = 1 for β* = 1 (for the case of no dilation). The extra factor is necessary to satisfy those conditions but hardly affects the result of the following analysis because the analysis gives g ≫ 1 for large N.

For the reptation (after CLF) along the longitudinal motional path tube of the diameter a* to occur in time, the characteristic time for the simultaneous CR-equilibration of g entanglement segments, τ_sim-CR, should not exceed the longest dielectric relaxation time corresponding to this reptation. This τ_sim-CR is dependent on g and thus on β* (= {a*/a}²); cf. Eq. 29. Because the tube would be dilated to the maximum possible diameter, β* for a linear PI chain in the monodisperse bulk can be determined from a condition,

The simultaneous CR-equilibration of g entanglement segments in the monodisperse bulk corresponds to p*-th CR–Rouse mode therein, with the mode index p* being specified as³⁷

For N ≫ 1, p* is given by N/g as simply expected from the number of entanglement segments involved between nodes of sinusoidal CR–Rouse eigenfunction. From consideration of two extreme conditions, p* = 1 for g = N and p* = N − 1 (the highest CR–Rouse mode index) for g = 1, the main expression in Eq. 31 is obtained as a minor correction of the case of N ≫ 1.³⁷ Using p* thus specified as a function of g and N, τ_sim-CR is expressed in the CR–Rouse form as

Here, (=) is the longest dielectric CR time for the monodisperse bulk PI specified experimentally by Eq. 11 with M₁ = M₂ = M. From Eqs. 30 and 32 together with the data of (=4.2 × 10⁻¹⁹M^3.5 s; determined from the ɛ″ peak frequency for entangled monodisperse linear PI at 40 °C³⁷), we can evaluate the CR mode index p* and further convert this p* into the number of entanglement segments involved in the simultaneous CR-equilibration, g (cf. Eq. 31).

The number β* of the entanglement segments per dilated segment of the size a* at t = is directly obtained from the g value thus determined (cf. Eq. 29). Figure 26 shows changes of β* with the entanglement number M/M_e (= N) of monodisperse linear PI in bulk. β* gradually decreases with increasing M/M_e but does not reach its asymptote (β* = 1) even at M/M_e = 100. This result suggests that the dielectric relaxation time is still affected by the CR/DTD mechanism even in such a well-entangled state (which gives a proof against CR/DTD-independence of argued in the literature²²). More importantly, β* defined for the longitudinal motional path of the chain is well below the number β_μ ≅ 3 of the laterally equilibrated entanglement segments. This difference between β* and β_μ lends support to the hypothesis of the dual structure of the tube illustrated in Figure 25: a* = a β*^1/2 < a′ = a in the time scale of terminal relaxation so that the tube specifying the longitudinal motional path of the chain wriggles laterally in the wider tube (with the diameter a′), the latter describing the extra decay of viscoelastic μ(t) due to DTD.

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Now we examine the dielectric ratio of linear PI probe in the high-M matrix (PI 1.1M) and in the monodisperse bulk. Because the local friction of the entanglement segment, ζ_e, is the same in these two environments, this ratio is simply given by

(Note that ∝ ζ_eM^3.5/ and β* is equivalent to the ratio of the effective entanglement molecular weight for the chain motion, M_e,eff, in the monodisperse bulk and in the high-M matrix.) The ratio thus evaluated from β* (Figure 26) is shown in the top panel of Figure 24 with the red curve. This curve excellently describes the data for the moderately- to well-entangled PI probes (green circles).

Furthermore, the viscoelastic relaxation time ratio is simply evaluated as = 2β*^1.5, because = in the DTD-free high-M matrix (as noted from the agreement of the G₁,_b″ and ɛ₁,_b″ data for short probes; cf. Figure 23) whereas in monodisperse bulk as confirmed experimentally.^{24,30,34,35,37} The viscoelastic ratio thus deduced, shown with the red curve in the bottom panel of Figure 24, is surprisingly close to the data (green circles).

The above success of the simple CR–Rouse analysis for the and ratios, noted also for star PI⁴⁰ (for which the motional path tube is wriggling in a partially dilated tube describing the μ(t) data in Figure 16), lends support to the hypothesis of the dual tube structure. It should be noted that the CR–Rouse analysis for the linear PI chain just considers the consistent coarse-graining of the length and time scales together with the coherent nature of the reptative motion. From that success, we expect that the dielectric ratio approaches unity, whereas the viscoelastic ratio (= 2) approaches 2 for the linear PI chain in the high-M limit (although this limit is not covered by experiments). This behavior of the viscoelastic ratio corresponds to the Viovy–Rubinstein–Colby⁶² (VRC) scenario considering reptation along the undilated tube that laterally wriggles in the dilated tube. We also note a consequence of this VRC scenario for highly entangled blends of linear chains: we expect that the terminal viscoelastic relaxation of concentrated long chains therein is not accelerated by the short chains but the decrease of its intensity (decrease of the long–long entanglement plateau modulus at low ω) scales as with d ≅ 1.3, whereas the terminal viscoelastic relaxation of the short chain is retarded by the long chains (by a factor of 2 if the short chain is dilute). Indeed, this expectation is in accord with the G* data of highly entangled polybutadiene blends reported by Struglinski and Graessley (see figure 9 of Struglinski and Graessley⁶³).

1. Tube Dilation Exponent and Feature of Interchain Constraint

The value of the tube dilation exponent d has been a long-standing subject of theoretical discussion.^60,61 Van Ruymbeke and coworkers⁶⁰ focused on blends of long and short linear chains to conclude, from a theoretical argument for the binary nature of entanglement (resulting from the pair-wise constraint between the chains), that d should be 1 just after the relaxation of entanglement of the long chain with the short chain. They also considered that the CR-activated tension re-equilibration²⁶ follows this long–short entanglement relaxation to effectively increase d up to 4/3 (≅ 1.3).⁶⁰ Their model, deduced from this consideration, well describes the viscoelastic data of the PI 308k/PI 21k blends (cf. Figure 14 of this article) by using the dielectric data as the reference.⁶⁰ Later, they refined their model based on the time marching algorithm, and this refined model describes both dielectric and viscoelastic data of the PI 308k/PI 21k blends simultaneously (i.e., without using the reference data) as well as viscoelastic data of other blends.⁶⁴

Larson and coworkers⁶¹ examined the terminal viscoelastic relaxation time of “solutions” of high-M monodisperse star polybutadienes (PB) in much shorter, unentangled PB. They focused on very strong dependence of this relaxation time on an effective number of entanglements per star arm, N_arm = with υ₂ being the volume fraction of the star, and concluded that the relaxation time is more universally dependent on N_arm calculated with d = 1 than on N_arm with d = 4/3. This conclusion suggests that the terminal relaxation of star PB solutions is better described with d = 1 rather than with d = 4/3, although a considerable scatter (by a factor of ∼10) is noted for the relaxation times for different sets of M_arm and υ₂ giving the same value of N_arm for d = 1; see figures 9–12 of Larson et al.⁶¹ (where several different values of M_arm were examined for each star PB sample).

We note an important difference in the d values explained above: for the entangled blends of linear PI chains, van Ruymbeke and coworkers^60,64 deduced an increase of the effective d from 1 to 4/3 with increasing t, namely, the effective d value of 4/3 in the terminal relaxation regime. In contrast, for the entangled star PB solutions, Larson and coworkers⁶¹ deduced d = 1 in the terminal regime. This difference might reflect dual (dynamic and static) aspects of the dilation exponent, as discussed by Larson and coworkers.⁶¹ At the same time, one might also suspect that the d value changes with either the chemical structure or the topological architecture (or both) of polymers. The data for highly entangled blends of linear PB (chemically identical to the star PB examined by Larson and coworkers⁶¹) unequivocally suggest d ≅ 1.3 in the terminal regime; see the data for 41L/435L blends in figures 9 and 12 of Struglinski and Graessley.⁶³ Thus, the d value might change with the topological architecture of the chain, or more specifically, with the basic mechanism of relaxation (reptation or arm retraction) determined by this architecture, as judged from the difference of the d values for the linear PI blends and the star PB solutions. A further study is desired for this problem.

Now, we turn our attention to a correlation between the d value deduced by van Ruymbeke and coworkers,^60,64 d = 1 and 4/3 at short and long t, and the d value (= 1.3) incorporated in the partial-DTD picture explained in this article. These two sets of the d values are not necessarily contradicting to each other, because the partial-DTD modulus μ_p-DTD(t) in this picture is affected by the d value only in a range of t where β_f-DTD(t) = {ϕ′(t)}^−d ≤ β_CR(t) (cf. Eq. 23). This point can be further examined for PI blends with small υ₂, for example, the PI 308k/PI 21k blend with υ₂ = 0.1 examined in Figures 17 and 21. In a wide range of t where the short PI 21k chains in the blend have fully relaxed but the long PI 308k chains still exhibit a plateau of μ(t) due to the entanglement among themselves, β_f-DTD(t) exceeds β_CR,2(t) of the long chain (cf. Figure 21) so that the partial-DTD modulus μ_p-DTD(t), well mimicking the μ(t) data, is significantly larger than the full-DTD modulus μ_f-DTD(t) (cf. Figure 17). In that range of t, μ_p-DTD(t) is contributed only from the long chain and expressed as³⁵ μ_p-DTD(t) = ϕ′(t)/β_CR,2(t). This expression does not include the dilation exponent, but we can still define an apparent exponent d_app for μ_p-DTD(t) and β_CR,2(t) as

This d_app changes with t, and μ_p-DTD(t) and β_CR,2(t) of the PI 308k/PI 21k blend with υ₂ = 0.1 (shown in Figures 17 and 21) give d_app = 0.98 and 1.3 at t = 10⁻² and 10⁻¹ s, respectively. Namely, the partial-DTD picture for the PI blend gives d_app that increases from ≊1 to 1.3 with time (after the relaxation of the short chain component), which is essentially the same as the evolution of d considered in the model by van Ruymbeke and coworkers.^60,64 Nevertheless, it should be emphasized that d_app is just an apparent dilation exponent, and the partial-DTD picture does not give the power-law relationship, Eq. 34, in the range of t where β_f-DTD(t) > β_CR,2(t).

Concerning this point, we would like to add that the experimentally observed difference of the CR relaxation behavior of chemically different PI and PS probes (Figures 11 and 12) well fits in the partial-DTD picture through the chemistry-dependent number z of local constraints per entanglement considered in Graessley's CR–Rouse model.⁴² It is not clear if this difference of the CR behavior due only to the chemical difference is straightforwardly deduced within the CR–Rouse model given that the entanglement exclusively results from the binary (pair-wise) constraint corresponding to d = 1.

For this problem, it would be informative to briefly visit the results of molecular dynamics (MD) simulations. Everaers and coworkers⁶⁶ made the primitive path analysis based on the MD simulation to visualize interchain hooking (or nodes) attributable to the binary entanglement and demonstrated good agreements between the plateau modulus deduced from this analysis and the literature data. Such nodes, defined in a static (and statistical) sense, are observed also in simulations with different algorithms, for example, the “contour reduction topological analysis” by Tzoumanekas and Theodorou⁶⁷ and the “direct topological analysis” by Kröger and coworkers.⁶⁸ At the same time, the simulation by Likhtman and Ponmurugan⁶⁹ revealed that (coarse-grained) chains form mutual contacts in a dynamic sense at many places along their backbone, and these contacts are tight and long-lived. Likhtman⁷⁰ further showed that those contacts involve both binary and ternary entanglements.

Summarizing these simulation results, one may arrive at a hypothetical molecular view that the interchain constraint has static and dynamic aspects, and both binary and multiple-chain constraints can be dynamically observed as the entanglement: probabilities of forming respective constraints may change according to the chemical structure and topological architecture of the chain. Within the context of the tube model incorporating the CR/DTD mechanism, this molecular view could lead to the (effective) dilation exponent d > 1 in the terminal relaxation regime where both binary and multiple-chain constraints loosen to dynamically dilate the tube. This possible scenario of d is an interesting subject of research, but a further discussion of it goes well beyond the scope of this article emphasizing an experimental viewpoint for analysis of the viscoelastic and dielectric data. A rigid theoretical study is desired for the behavior of d (and for the underlying molecular view).

2. Duality in Description of Reptation along Tube

In relation to the reptation along the motional path tube (cf. Figure 25), a comment needs to be made for the local friction of the entanglement segment, ζ_e. In the CR–Rouse analysis (Eqs. 29–33), ζ_e is treated as the intrinsic friction being identical in the high-M matrix (CR/DTD-free environment) and in the monodisperse bulk. This treatment is consistent with the definition of the friction for the chain motion along the motional path tube having the diameter a*. However, as an equivalent treatment, we may also introduce an effective friction ζ_e,eff for the chain motion along the partially/fully dilated tube having the diameter a′ (>a*). This dilated tube specifies the range of lateral equilibration of the entanglement segments corresponding to the DTD relaxation of viscoelastic μ(t), and the motion along the dilated tube requires a waiting time t_w for an extra tension equilibration along it. This waiting time can be cast as an increase of ζ_e,eff (>ζ_e). Thus, there is a duality in description of the chain motion, for example, reptation along the motional path tube with the diameter a* (cf. Figure 25) occurring with the intrinsic ζ_e, or, reptation along the dilated tube with the diameter a′ occurring with ζ_e,eff. These two types of reptative motion should give the same μ(t) at t > t_w and the same terminal relaxation time.

In fact, an analysis of the waiting time t_w in blends of linear PI well describes the relaxation time of the long and short chains therein³⁸ (to an extent similar to that seen in Figure 24). More quantitatively, van Ruymbeke and coworkers⁶⁴ analyzed t_w in their model to excellently describe not only the relaxation time but also the frequency dependence of the G* and ɛ″ data of those linear PI blends. They further extended their model to entangled blends of star and linear chains (while keeping the analysis of t_w) to describe the G* data of the blends.⁶⁵ Read and coworkers⁵⁴ made detailed analysis of ζ_e,eff (equivalent to t_w) to specify several different regimes to be added in the Viovy–Rubinstein–Colby (VRC) diagram⁶² and compared their results with experimental data and simulation results. Their analysis needed a hypothetical reference state where the intrinsic ζ_e governs the reptation along the tube; this reference state cannot be the monodisperse bulk wherein the CR/DTD mechanism is already operating. Except this point, their analysis allows us to test differences in the modes of relaxation in the VRC diagram in a purely experimental way.

The above duality in description of chain motion along the dilated tube is a natural consequence of coarse-graining.¹ That is, if we average the chain conformation over every t_w, the chain would look like a fuzzy thread with the lateral width of a′, and its motion would look like 1D motion along the dilated tube. Furthermore, the bead-spring (Rouse) model underlying the tube model gives the same terminal relaxation behavior irrespective of the choice of the bead size as long as the chain is composed of many beads, which is regarded as the duality in the simplest form. Such a duality in description of the chain motion is important in polymer physics and deserves further attention.