Nonlinear Stress Relaxation of Miscible Polyisoprene/Poly(<i>p</i>-<i>tert</i>-butylstyrene) Blends in Pseudomonodisperse State

For miscible pair of polyisoprene (PI) and poly­(<i>p</i>-<i>tert</i>-butylstyrene) (PtBS), the component molecular weights, composition, and temperature were tuned to prepare PI/PtBS blends in the <i>pseudomonodisperse</i> state where the component PI and PtBS chains had the same terminal relaxation time, τ₁. These pseudomonodisperse blends had the linear viscoelastic moduli indistinguishable from the moduli of entangled monodisperse bulk homopolymers of particular molecular weights, and satisfied the time-strain separability in their nonlinear stress relaxation behavior under large step strains. The damping function <i>h</i>(γ) of those blends was close to <i>h</i>_DE(γ) calculated from the Doi–Edwards model and classified to be the so-called type-A damping function, even though the major component (PI) in the blends had a large entanglement number <i>per</i> chain (<i>N</i> ≥ 50). Highly entangled monodisperse homopolymers having similarly large <i>N</i> are known to exhibit the so-called type-C damping characterized by <i>h</i>(γ) ≪ <i>h</i>_DE(γ), and this damping behavior was indeed confirmed for high-<i>M</i> bulk PI utilized as the blend component. Thus, the nonlinear damping behavior was different for the pseudomonodisperse PI/PtBS blends and high-<i>M</i> bulk PI, despite the similarity in the entanglement number <i>N</i> for PI therein. This difference was discussed within the molecular scenario of Marrucci and Grizzuti in relation to the topological hindrance for PI segments due to PtBS segments having a much larger friction. This hindrance retarded the Rouse equilibration of the PI backbone in the blends, which possibly provided the highly entangled PI with a slow contour length fluctuation mechanism that competed with reptation. Such a competing mechanism smears the elastic instability underlying the type-C damping as suggested from the Marrucci–Grizzuti scenario, which possibly allowed the pseudomonodisperse PI/PtBS blends containing highly entangled PI to exhibit the type-A damping. Furthermore, the type-A damping was observed also for a chemically homogeneous, highly entangled PI/PI blend being free from the topological hindrance for PI segments. In this PI/PI blends, the partial constraint release of the high-<i>M</i> component, activated by the relaxation of the low-<i>M</i> component, appeared to compete with reptation of the high-<i>M</i> component thereby smearing the instability and suppressing the type-C damping. Thus, the smearing of instability could be a rather universal feature occurring irrespective of the detail of the competing mechanisms

Viscoelastic and Dielectric Relaxation of Reptating Type-A Chains Affected by Reversible Head-to-Head Association and Dissociation

For entangled linear polymer having type A dipoles and undergoing head-to-head association and dissociation reaction, viscoelastic and dielectric behavior is theoretically analyzed on the basis of the reptation dynamics combined with the reaction kinetics. Specifically, for the dissociated unimer and associated dimer (indexed with j = 1 and 2, respectively), the normalized complex modulus gj*­(ω) and the normalized complex dielectric permittivity ε̃j*­(ω) are analytically calculated via eigenfunction expansion of the orientational anisotropy and orientational memory defined in terms of the bond vectors u of entanglement segments. The reaction activates mutual conformational transfer between the unimer and dimer. Multiple coupling occurs for the anisotropy decay modes of the unimer and dimer due to this transfer, and the viscoelastic g1* and g2* of the unimer and dimer, respectively, exhibit considerably retarded and accelerated relaxation compared to the pure reptation case. In contrast, the memory decay modes of the unimer and dimer are only pairwisely coupled, so that the reaction-induced acceleration and retardation for the dielectric ε̃1* and ε̃2* are much weaker than those seen for the viscoelastic g1* and g2*. The orientational anisotropy is the tensorial, second-moment average of u associated with no cancellation in the conformational transfer, whereas the orientational memory is the vectorial, first-moment average accompanied by partial cancellation, which results in the difference between gj* and ε̃j*. This difference between gj* and ε̃j* is noted also for the associating/dissociating Rouse chains. Nevertheless, the reaction-induced retardation of the viscoelastic relaxation is stronger for the reptating unimer than for the Rouse unimer, whereas the reaction-induced acceleration is similar, in magnitude, for the reptating dimer and Rouse dimer. These features of gj* of the unimer and dimer are discussed in relation to the motional coherence along the chain backbone being present and absent in the reptation and Rouse dynamics

Dielectric Relaxation of Type‑A Chains Undergoing Head-to-Tail Association/Dissociation: Difference from Head-to-Head Case and Correlation with Viscoelastic Relaxation

Dielectric relaxation of type-A chains reflects global motion of the chains but is also affected by relative alignment of the dipoles along the chain backbone, namely, by the dipole inversion. Head-to-head association of type-A unimers gives a symmetrically dipole-inverted dimer, and the association/dissociation equilibrium of these unimers and dimer results in motional coupling of these chains, thereby affecting the dielectric behavior. In fact, for this head-to-head case, eigenmode analysis has been reported in the literature to reveal that motional coupling results in moderate retardation and acceleration of the dielectric relaxation of the unimer and dimer obeying the reptation dynamics. In contrast, the coupling has no effect on the dielectric relaxation of the Rouse unimer and dimer, namely, the effect of motional coupling on the dielectric relaxation changes with the type of chain dynamics. This effect was not clarified for head-to-tail associating unimers and their dimer having no dipole inversion. Thus, for completeness, this study makes the eigenmode analysis of the dielectric relaxation for this case of head-to-tail reaction. For the unimer and dimer obeying either Rouse or reptation dynamics, the analysis indicates that the retardation and acceleration of the dielectric relaxation of the unimer and dimer are much more significant for the head-to-tail case than for the head-to-head case irrespective of the chain dynamics, and that the dielectric relaxation function for the former case exactly coincides with the viscoelastic relaxation function if the unimer and dimer obey the reptation dynamics. This result suggests an interesting method of resolving some detail of the chain dynamics under the reaction through comparison of dielectric and viscoelastic responses of the associative type-A chains

Viscoelastic and Orientational Relaxation of Linear and Ring Rouse Chains Undergoing Reversible End-Association and Dissociation

For dilute telechelic linear and ring Rouse chains undergoing reversible end-association and dissociation, the time (<i>t</i>) evolution equation was analytically formulated for the bond vector of the subchain (or segment), <b>u</b>^[c](<i>n</i>,<i>t</i>) with <i>n</i> being the subchain index and the superscript c specifying the chain (c = L and R for the linear and ring chains). The end-association of the linear chain (i.e., ring formation) occurs only when the ends of the linear chain come into close proximity. Because of this constraint for the ring formation, the time evolution equation for <b>u</b>^[L](<i>n</i>,<i>t</i>) of the linear chain was formulated with a conceptually new, two-step expansion method: <b>u</b>^[L](<i>n</i>,<i>t</i>) was first expanded with respect to its sinusoidal Rouse eigenfunction, sin­(<i>p</i>π<i>n</i>/<i>N</i>) with <i>p</i> = integer and <i>N</i> being the number of subchains <i>per</i> chain, and then the series of odd sine modes is re-expanded with respect to cosine eigenfunctions of the ring chain, cos­(2απ<i>n</i>/<i>N</i>) with α = integer, so as to account for that constraint. This formulation allowed analytical calculation of the orientational correlation function, <i>S</i>^[c](<i>n</i>,<i>m</i>,<i>t</i>) = <i>b</i>^–2⟨<i>u</i>_<i>x</i>^[c](<i>n</i>,<i>t</i>)<i>u</i>_<i>y</i>^[c](<i>m</i>,<i>t</i>)⟩ (c = L, R) with <i>b</i> being the subchain step length, and the viscoelastic relaxation function, <i>g</i>^[c](<i>t</i>) ∝ ∫₀^<i>N</i><i>S</i>^[c](<i>n</i>,<i>n</i>,<i>t</i>) d<i>n</i>. It turned out that the terminal relaxation of <i>g</i>^[R](<i>t</i>) and <i>g</i>^[L](<i>t</i>) of the ring and linear chains is retarded and accelerated, respectively, due to the motional coupling of those chains occurring through the reaction. This coupling breaks the ring symmetry (equivalence of all subchains of the ring chain in the absence of reaction), thereby leading to oscillation of the orientational anisotropy <i>S</i>^[R](<i>n</i>,<i>n</i>,<i>t</i>) of the ring chain at long <i>t</i> with the subchain index <i>n</i>. The coupling also reduces a difference of the anisotropy <i>S</i>^[L](<i>n</i>,<i>n</i>,<i>t</i>) of the linear chain at the middle (<i>n</i> ∼ <i>N</i>/2) and end (<i>n</i> ∼ 0)

Entanglement Length in Miscible Blends of <i>cis</i>-Polyisoprene and Poly(<i>p</i>-<i>tert</i>-butylstyrene)

The entanglement length <i>a</i>, being equivalent to the plateau modulus <i>G</i>_N (∝<i>M</i>_e^–1 ∝ <i>a</i>^–2), is one of the most basic parameters that determine the slow dynamics of high molecular weight (<i>M</i>) polymers. In miscible blends of chemically different chains, the components would/should have the common <i>a</i> value. However, changes of <i>a</i> with the blend composition have not been fully elucidated. For this problem, this study conducted linear viscoelastic tests for miscible blends of high-<i>M cis</i>-polyisoprene (PI) and poly­(<i>p</i>-<i>tert</i>-butylstyrene) (PtBS) and analyzed the storage and loss moduli (<i>G</i>′ and <i>G</i>″) data in a purely empirical way, considering the very basic feature that unentangled and entangled blends having the same composition exhibit the same local relaxation. (From a molecular point of view, this local relaxation reflects the chain motion <i>within</i> the length scale of <i>a</i>.) On the basis of this feature, a series of barely entangled low-<i>M</i> PI/PtBS blends having various component molecular weights and a given composition were utilized as references for well-entangled high-<i>M</i> PI/PtBS blends with the same composition, and the modulus data of the reference were subtracted from the data of the high-<i>M</i> blends. For an optimally chosen reference, the storage modulus of the high-<i>M</i> blends obtained after the subtraction (<i>G</i>_ent′ = <i>G</i>_{high‑<i>M</i> blend}′ – <i>G</i>_ref′) exhibited a clear plateau at high angular frequencies ω. The corresponding loss modulus <i>G</i>_ent″ decreased in proportion to ω^–1 at high ω, which characterized the short-time onset of the global entanglement relaxation: A mischoice of the reference gave no plateau of <i>G</i>_{high‑<i>M</i> blend}′ – <i>G</i>_ref′ and no ω^–1 dependence of <i>G</i>_{high‑<i>M</i> blend}″ – <i>G</i>_ref″ at high ω, but a survey for various low-<i>M</i> PI/PtBS blends allowed us to find the optimum reference. With the aid of such optimum reference, the entanglement plateau modulus <i>G</i>_N of the high-<i>M</i> PI/PtBS blends was accurately obtained as the high-ω plateau value of <i>G</i>_ent′. <i>G</i>_N thus obtained was well described by a linear mixing rule of the entanglement length <i>a</i> with the weighing factor being equated to the number fraction of Kuhn segments of the components, not by the reciprocal mixing rule utilizing the component volume fraction as the weighing factor. This result, not explained by a mean-field picture of entanglement (constant number of entanglement strands in a volume <i>a</i>³), is discussed in relation to local packing efficiency of bulky PtBS chains and skinny PI chains

Component Relaxation Times in Entangled Binary Blends of Linear Chains: Reptation/CLF along Partially or Fully Dilated Tube

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 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 2e), 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

Uniaxial Extensional Behavior of (SIS)_<i>p</i>‑Type Multiblock Copolymer Systems: Structural Origin of High Extensibility

Rheological and structural behavior was examined for a series of symmetric styrene (S)–isoprene (I)–styrene (S) multiblock copolymers of (SIS)p-type (p = 1, 2, 3, and 5 corresponding to tri-, penta-, hepta-, and undecablock) in n-tetradecane (C14), a selective solvent that dissolves the I block and precipitates (but swells) the S block. The molecular weights of respective blocks were almost identical for these copolymers (MI ≅ 40K for I block; MS ≅ 20K and 10K for inner and outer S blocks, respectively). At 20 °C, the (SIS)p/C14 systems with the copolymer concentration C = 30 wt % formed a bcc lattice of spherical S domains (with Tg,PS ≅ 38 °C) embedded in the I/C14 matrix. Under small shear and elongation in the linear regime, the systems exhibited gel-like elasticity sustained by the I blocks connecting the S domains. This linear elastic behavior, being associated with affine displacement of the S domains as revealed from small-angle X-ray scattering (SAXS) under small elongation, was very similar for all (SIS)p/C14 systems having the same C. In contrast, a remarkable difference was found for those systems under large (but slow) elongation: The maximum stretch ratio at rupture, λmax, significantly increased with the repeating number p of the SIS units, λmax ≅ 1.7, 2.2, 6.6, and ≥90 for p = 1, 2, 3, and 5, respectively. In particular, λmax ≥ 90 for p = 5 was much larger compared to the full-stretch ratio of the trapped entanglement strand (λfull‑ent ≅ 14) and even to the full-stretch ratio of the (SIS)5 copolymer chain as a whole (λfull‑copolymer ≅ 40). For investigation of the structural origin of such remarkably high extensibility of the undecablock system (p = 5), SAXS and rheological tests were made under elongation followed by reversal. The tests revealed affine stretching of the lattice (affine displacements of the S domains) and negligible stress–strain hysteresis on reversal of elongation from λrev < 3. In contrast, on reversal from larger λrev up to 60, nonaffine stretching of the lattice and the significant stress–strain hysteresis were observed. Thus, under large elongation, some of the S blocks were pulled out from their domains and transferred to the other S domains at 20 °C, the experimental temperature not significantly lower than Tg,PS (≅ 38 °C) of the swollen S domains, to allow the system to deform plasto-elastically. This deformation differed from unrecoverable plastic flow, as evidenced from spontaneous, full recovery of the size, shape, and SAXS profile of the (SIS)5/C14 specimen being kept at rest (without load) at 20 °C for a sufficiently long time after the elongation. This recovery strongly suggests that the material preserved some memory of initial connection between the (SIS)5 chains through the S domains, in particular in the direction perpendicular to the elongation, and the corresponding physical network still percolated the whole material even under large elongation. This argument in turn provides us with a clue for understanding the difference of λmax for the series of (SIS)p/C14 systems. The full percolation can survive and the material can stand with the elongation if at least two PS blocks, on average, remain intact (not pulled out) in each (SIS)p copolymer backbone. The probability of having such intact S blocks obviously increases with the repeating number p of the SIS units, which possibly resulted in the observed difference of λmax

Rouse Analysis of Nonlinear Rheology of Unentangled Polymer Melts under Fast Shear: Viscoelastic Response to Superposed Oscillatory Strain

Nonlinear rheological behavior of unentangled polymer melts can be described by the Rouse model given that its parameters, spring strength κ, bead friction coefficient ζ, and mean-square Brownian force intensity B, are allowed to change under fast flow/large strain (and to take anisotropic tensorial forms when necessary). Within this model, analytic expressions in terms of those parameters have been obtained for measurable quantities that include viscosity η, the first normal stress difference coefficient Ψ1, and complex dielectric permittivity ε*. Those expressions in turn enable us to extract κ, ζ, and B from experimental data of unentangled melts. In particular, the rheo-dielectric ε* data under shear, recently obtained for unentangled low-M poly(butylene oxide) melt having type-A dipoles (PBO-16k; M = 16 × 103), suggest that the tensorial ζ and B have negligibly small off-diagonal components in a range of Weissenberg number Wi up to 1.2. On the basis of that study, we here focus on the complex shear moduli G∥* and G⊥* of the Rouse chain defined as responses to a small oscillatory strain superposed on the steady shear flow, with ∥ and ⊥ representing parallel and perpendicular superposition, respectively. In the case of negligible off-diagonal components of ζ and B, the Rouse analysis gave a very simple expression of those moduli, GX* (ω) = bX[G]Geq* (ωaX[G]) with X = ∥ and ⊥, where Geq*(ω) is the linear viscoelastic (LVE) complex modulus at an angular frequency ω. Namely, in that case, the relaxation time of GX* decreases by a factor of aX[G] (<1) and its terminal relaxation intensity is enhanced by a factor of bX[G] (>1), but a relative distribution of the relaxation modes exhibits no change. Furthermore, the Rouse parameters obtained from the η, Ψ1, and ε* data of PBO-16k were found to satisfy a specific empirical relationship, {b∥[G]}2 ≅ 1/a∥[G]. Because Geq′(ω) ≅ Geq″(ω) ∝ ω1/2 at high ω where the LVE Rouse relaxation has not completed, this relationship suggests G∥′′(ω) = Geq″(ω) at ω > 1/τ∥[G] and G∥″(ω) Geq″(ω) at ω < 1/τ∥[G], where τ∥[G] is the terminal viscoelastic relaxation time defined for G∥*(ω). This behavior of G∥″(ω) is superficially equivalent to that expected for a case of disappearance of viscous contributions of low-order Rouse eigenmodes under fast shear discussed in the literature on the basis of the concept of Pincus blob. However, the current Rouse analysis clearly indicated that all eigenmodes keep their viscous contributions with a magnitude determined by ζ and κ under flow, confirming the importance of nonequilibrium changes of ζ and κ in the nonlinear flow behavior of unentangled melts

Experimental Test for Viscoelastic Relaxation of Polyisoprene Undergoing Monofunctional Head-to-Head Association and Dissociation

A viscoelastic test was made for end-carboxylated polyisoprene (PI-COOH) of the molecular weight <i>M</i> = 30.₅ × 10³ that underwent the interchain association and dissociation through hydrogen bonding of the COOH groups at the chain end. As a reference, the test was made also for neat PI unimer (with no COOH group at the chain end) and for PI₂ dimer (with <i>M</i> = 61.0 × 10³), the latter being synthesized through end-coupling of PI^– anions (precursor of the PI-COOH sample). The PI-COOH, neat unimer, and dimer samples were diluted in oligomeric butadiene (oB) to a concentration of 10 wt %. The neat unimer and dimer exhibited nonentangled Rouse behavior at this concentration, as expected from their molecular weights. At low temperatures (<i>T</i> ≤ 0 °C) the PI-COOH sample relaxed slower than the reference unimer but faster than the dimer, whereas the relaxation of PI-COOH approached that of the unimer with increasing <i>T</i> > 0 °C, and this change of the relaxation time of PI-COOH was associated with changes in the angular frequency (ω) dependence of the dynamic modulus. This behavior of PI-COOH was well described by a recently proposed theory considering motional coupling between the end-associating unimer and its dimer at chemical equilibrium. On the basis of this result, an effect of the polymeric character of PI-COOH chain on the viscoelastically detected association/dissociation of the hydrogen bonding of the COOH groups was discussed

Effect of Head-to-Head Association/Dissociation on Viscoelastic and Dielectric Relaxation of Entangled Linear Polyisoprene: An Experimental Test

For linear high-cis polyisoprene having a monofunctionally associative carboxyl group at the chain head (PI30-COOH; M = 30.5 × 103), linear viscoelastic and dielectric behavior was examined in its entangled bulk system. The PI30-COOH unimer chain had type-A dipoles aligned from the tail to head so that its large-scale motion (over the tail-to-head distance) activated not only viscoelastic but also dielectric relaxation. Consequently, the head-associated dimer of PI30-COOH had symmetrically inverted type-A dipoles, and its large-scale motion also activated both viscoelastic and dielectric relaxation. These unimer and dimer were coexisting in the system at equilibrium because of the association/dissociation reaction at the carboxyl group, and this reaction strongly affected the viscoelastic and dielectric behavior. Experimentally, the reaction effect was examined by utilizing two reference polyisoprenes undergoing no reaction, PI30 and (PI30)2: PI30 was a prepolymer of PI30-COOH before introducing the carboxyl group at the head, and (PI30)2 was a head-to-head dimer of PI30 prepared by coupling of the PI30 anion. The viscoelastic and dielectric relaxation was found to be faster for the PI30-COOH system than for a reference PI30/(PI30)2 blend having the unimer/dimer composition identical to that in the PI30-COOH system (determined from Fourier transform infrared measurement), and this difference between the PI30-COOH system and the blend was more significant for the viscoelastic relaxation than for the dielectric relaxation. This experimental fact unequivocally indicates that the reaction induces motional coupling between the unimer and dimer to significantly affect the relaxation behavior of these chains. This result lends support to a recent model analyzing this coupling for entangled (reptating) unimer and dimer. In fact, the model described low-frequency asymptotes of the viscoelastic and dielectric losses of PI30-COOH surprisingly well, given that the viscoelastic and dielectric asymptotes of the reference PI30 bulk system were separately fitted by the model to determine the terminal relaxation times in the model calculation in the absence of reaction. This success of the model strongly suggests that the difference of the reaction effects on the dielectric and viscoelastic relaxation reflects the vectorial and tensorial nature of the respective relaxation processes. The dielectric relaxation reflects the (vectorial) first-moment average of bond vectors u of the chain so that the dipole inversion of the dimer leads to pairwise coupling of the dielectric modes of the unimer and dimer under the reaction, thereby allowing the reaction to affect the dielectric relaxation just moderately. In contrast, the viscoelastic relaxation detects the (tensorial) second-moment average of u so that the reaction results in multiple coupling of the viscoelastic modes of the unimer and dimer to strongly affect the viscoelastic relaxation