Stress relaxation and dynamic viscoelastic properties of end-linked poly(dimethyl siloxane) networks containing unattached poly(dimethyl siloxane)

Stress relaxation in uniaxial extension and dynamic shear moduli G′ and G″ have been studied in networks of vinyl-terminated poly(dimethyl siloxane) (PDMS) of five different molecular weights (M _n from 1800 to 29,200) crosslinked with cis-dichlorobis (diethyl sulfide) platinum (II) and containing 10 and 15 wt % of two samples of high-molecular-weight unattached linear hydroxyl-terminated PDMS (M _w 700,000 and 950,000). The M _w/M _n ratio of both the network prepolymers and the unattached linear species was approximately 2. In stress relaxation the stretch ratio was 1.25 or less and the shear relaxation modulus was calculated from the neo-Hookean stress-strain relation. In the dynamic measurements, the strain amplitude was 15% or less; after conversion to the timedependent shear relaxation modulus G(t) the two sets of measurements were combined and the contribution of the unattached species G₁(t) was calculated by difference. After multiplication by (1 − v)⁻¹G/G_e, where v₂ is the volume fraction of network, G is the plateau modulus of the uncrosslinked polymer, and G_e is the equilibrium modulus of the network containing unattached molecules, G₁(t) was compared with G₁₁(t), the relaxation modulus was essentially the same in both environments. The relaxation was slower in the networks than in the uncrosslinked polymer by 1 to 2 orders of magnitude, and it increased gradually with increasing G_e, which is a measure of total to pological obstacles represented by crosslinks plus trapped entanglements. A similar but less striking difference between relaxation in a network and in the homologous environment of a linear polymer was previously observed in end-linked polybutadiene networks and the butadiene phase of a styrene-butadiene-styrene block copolymer. It appears that, in these systems where the topology of the obstacles is fixed, the reptation is severely restricted or else alternative modes of configurational rearrangement which contribute to relaxation in the uncrosslinked polymer are suppressed.     

Rubber networks containing unattached macromolecules. IV. Stress relaxation in end-linked polybutadiene with unattached styrene-butadiene copolymer

Stress relaxation has been studied in networks of dihydroxy-terminated polybutadiene (mostly cis:trans:vinyl = 34:40:26) crosslinked by triphenyl methane-4,4′,4″-triisocyanate and containing about 9.5% by weight of unattached linear random styrene-butadiene copolymer with various molecular weights (from 1.4 to 3.3 × 10⁵) and with styrene content and butadiene microstructure chosen to match the average solubility parameter of the end-linked network. Stress relaxation measurements were made also on networks containing no unattached species and containing 9.3% hydrocarbon oil, and on the various uncrosslinked linear polymers. The stretch ratio was 1.25 and the Young's relaxation modulus was calculated from the neo-Hookean stress-strain relation. For the uncrosslinked linear polymers, the relaxation modulus E₁₁(t) corresponds to a rather narrow distribution of relaxation times whose magnitudes were approximately proportional to the 3.4 power of viscosity-average or weight-average molecular weight; for one polymer, the time dependence agreed closely with the prediction of the Doi-Edwards theory modified for a small degree of molecular weight distribution. The disengagement times calculated from the Doi-Edwards theory as modified by Graessley appeared to be of the correct order of magnitude. The contribution of the unattached species in the networks E₁(t) was calculated by difference; after multiplication by (1?v)^?1, where v₂ is the volume fraction of network, and correction for the difference in monomeric friction coefficient associated with the difference in fractional free volume in the two environments, E₁(t) was compared with E₁₁(t) for each linear polymer. The relaxation was slower in the network than in the uncrosslinked polymer by about an order of magnitude, but the form of the relaxation modulus was similar in both environments except for two linear polymers for which the relaxation in the network became very much slower at long times. This behavior appeared to be correlated with a broader molecular weight distribution.     

Phase separation induced by shear quenching in polymer blends with a diblock copolymer

The effects of adding A–B diblock copolymer to a polymer blend (A/B) on phase‐separation kinetics and morphology have been investigated in a fixed shallow‐quench condition (ΔT = 1.5 °C) by in situ time‐resolved light scattering and phase‐contrast optical microscopy. A shear‐quench technique was used in this study instead of a conventional temperature‐quench method. Mixtures of nearly monodisperse low relative‐molecular masses of polybutadiene (M_w = 2.8 kg/mol), polystyrene (M_w = 2.6 kg/mol), and a near‐symmetric butadiene–styrene diblock copolymer (M_w = 6.3 kg/mol) as an interfacial modifier were studied. We observed that the addition of the diblock copolymer could either retard or accelerate the phase‐separation kinetics depending on the concentration of the diblock copolymer in the homopolymer blends. In contrast to the conventional temperature quench, we observed complex phase‐separation kinetics in the intermediate and late stages of phase separation by the shear‐quench technique. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 819–830, 2001     

Multiblock Copolymers Based on Highly Crystalline Polyethylene and Soft Poly(ethylene-co-butadiene) Segments: Towards Polyolefin Thermoplastic Elastomers

Block copolymers based on polyethylene (PE) and ethylene butadiene rubber (EBR) were obtained by successive controlled coordinative chain transfer polymerization (CCTP) of a mixture of ethylene and butadiene (80/20) and pure ethylene. EBR-b-PE diblock copolymers were synthesized using the {Me₂Si(C₁₃H₈)₂Nd(BH₄)₂Li(THF)}₂ complex in combination with n-butyl,n-octyl magnesium (BOMAG) used as both the alkylating and chain transfer agent (CTA). Triblock and multiblock copolymers featuring highly semi-crystalline PE hard segments and soft EBR segments were further obtained by the development of a bimetallic CTA, the pentanediyl-1,5-di(magnesium bromide) (PDMB). These new block copolymers undergo crystallization-driven organization into lamellar structures and exhibit a variety of mechanical properties, including excellent extensibility and elastic recovery in the case of triblock and multiblock copolymers.     

Effects of chain length distribution on viscoelastic properties of entangled linear polymers: Blending laws for binary blends

The reptation idea of de Gennes and the tube model theory of Doi and Edwards are extended to explain the terminal viscoelastic properties of binary blends in the highly entangled state of two linear monodisperse polymers with different molecular weights M₁ and M₂. A modified tube model is proposed that considers the significance of the constraint release by local tube renewal in accounting for the relaxation process of the higher molecular weight chain. Its relaxation by both reptation and the constraint release is remodeled as the disengagement by pure reptation of an equivalent primitive chain. From knowledge of the longest relaxation times of the blend components, the stress equation is formulated from which blending laws of viscoelastic properties for the binary blends are derived. To force better agreement between theory and experiment at the pure monodisperse limits of the blends, a crude treatment to include the effect of contour-length fluctuation in the equivalent-chain model is also discussed. Theoretical predictions of the zero-shear viscosity and steady-state shear compliance are shown to be in good agreement with literature data on undiluted polystyrenes and polybutadienes over a wide range of the blend composition and M₂/M₁ ratio. The asymptotic of the laws for blends with M₂/M₁ → 1 and 0 are comparable to those from the relaxation spectrum proposed by others earlier on the basis of the tube model.     

Matched ring diblock and linear triblock copolymers in dilute solution

A unique diblock copolymer ring and its linear triblock copolymer precursor composed of polystyrene and polydimethylsiloxane have been characterized by static and dynamic light scattering in dilute solution. The measurements were carried out with cyclohexane as the solvent over a temperature range of 12–35°C. Cyclohexane has the useful property that it is nearly isorefractive with the PDMS so that the PDMS block segments are invisible to the light-scattering technique and it is a theta solvent for polystyrene at 34.5°C. The block polymers in this work contain 35.1 wt % of styrene as determined by proton NMR. In the linear triblock polymer, the polystyrene is the center block with PDMS blocks on each side. Static light scattering measurements give 4.31 × 10⁴ for the average molecular weight of the whole polymer. Light scattering also shows that the apparent theta temperature for the linear triblock is shifted by 15°C to a value of 20°C at which point the second virial coefficient drops sharply and phase separation begins to induce aggregation. The diblock ring, however, shows a strongly positive second virial coefficient and no aggregation even at 12°C which is the limit of these experiments. The diffusion coefficients of cyclic diblock (D_c) and linear triblock copolymer (D₁) are measured by dynamic light scattering. The ratio of diffusion coefficients of cyclic and linear copolymers at 14.9°C and 30°C are D_c/D_l = 1.13 and 1.107 respectively. These compare well with prediction of 1.18 for this ratio from consideration of the hydrodynamics of matched linear and cyclic polymer chains. Dynamic light scattering quantitatively confirms that the linear copolymer experiences a solvent quality change near 20°C but the cyclic polymer remains in good solvent over the entire experimental temperature range. © 1993 John Wiley & Sons, Inc.     

1,4‐Dilithio‐1,1,4,4‐tetraphenylbutane in diethyl ether: An effective initiator for the living anionic polymerization of dienes

The anionic polymerization of butadiene initiated with 1,4‐dilithio‐1,1,4,4‐tetraphenylbutane (LiTPB) in diethyl ether (DEE) gives polybutadiene (PBD) with high 1,2 content (>70%), narrow polydispersities (1.04 < M_w/M_n < 1.20), and predicted molecular weights. In THF, this polymerization does not work very well. After removal of DEE and addition of THF, the PBD dianion is end capped quantitatively by addition of 1,1‐diphenylethylene (DPE) to give the diphenylalkyl end capped PBD dianion. Subsequent addition of methyl methacrylate at low temperatures results in the formation of well‐defined PMMA‐b‐PBD‐b‐PMMA triblock copolymers. The results are accounted for by taking into account the effects of Li ion solvation on the BD initiation and end capping of the PBD anion by DPE. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 2198–2206, 2009     

Control of Microphase Separation of Polystyrene-Silica Nanocomposites by Using Perhydropolysilazane and Partially Epoxidized Poly(styrene-block-butadiene-block-styrene)

In order to control microphase separation of polystyrene-silica nanocomposites, perhydropolysilazane (PHPS), which is a preceramic of silica, and epoxidized poly(styrene-block-butadiene-block-styrene) triblock copolymer [E-SBS, M_w = 8.0 × 10⁴, styrene: 40 mol%, degree of epoxidization of butadiene: 20 mol%] or poly(styrene-block-butadiene-block-styrene) triblock copolymer [SBS, M_w = 1.40 × 10⁵, styrene: 30 mol%] as templates of microphase separation were blended, following the calcination of composites in steam at 60°C. Well-arranged microphase separation was formed with E-SBS, though the macrophase separation was formed with SBS. The morphology of the microphase separation of the composites with E-SBS and PHPS was widely controlled by varying the PHPS content based on Molau's law. Silica domains were formed in polybutadiene domains. NMR analysis indicated the interaction between silanyl group of PHPS and epoxy group in E-SBS. The composites on the substrate were highly transparent and the surface of the composite with 73.5 vol% of silica was harder than 4H.     

Construction of PIB‐b‐PDEAEMA well‐defined amphiphilic diblock copolymers via sequential living carbocationic and RAFT polymerization

A series of well‐defined amphiphilic diblock copolymers consisting of hydrophobic polyisobutylene (PIB) and hydrophilic poly(2‐(diethylamino)ethyl methacrylate) (PDEAEMA) segments was synthesized via the combination of living carbocationic polymerization and reversible addition fragmentation chain transfer (RAFT) polymerization. Living carbocationic polymerization of isobutylene followed by end‐capping with 1,3‐butadiene was first performed at ?70 °C to give a well‐defined allyl‐Cl‐terminated PIB with a low polydispersity (M_w/M_n =1.29). This end‐functionalized PIB was further converted to a macromolecular chain transfer agent for mediating RAFT block copolymerization of 2‐(diethylamino)ethyl methacrylate at 60 °C in tetrahydrofuran to afford the target well‐defined PIB‐b‐PDEAEMA diblock copolymers with narrow molecular weight distributions (M_w/M_n ≤1.22). The self‐assembly behavior of these amphiphilic diblock copolymers in aqueous media was investigated by fluorescence spectroscopy and transmission electron microscope, and furthermore, their pH‐responsive behavior was studied by UV‐vis and dynamic light scattering. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2014 , 52, 1478–1486     

Synthesis and characterization of cis‐polybutadiene‐block‐syn‐polystyrene copolymers with a cyclopentadienyl titanium trichloride/modified methylaluminoxane catalyst

This article reports a practical method for preparing cis‐polybutadiene‐block‐syn‐polystyrene (cis‐PB‐b‐syn‐PS) copolymers with long crystallizable syndiotactic polystyrene (syn‐PS) segments chemically bonded with high cis‐1,4‐polybutadiene segments through the addition of styrene (ST) to a cis‐specific 1,3‐butadiene (BD) living catalyst composed of cyclopentadienyl titanium trichloride (CpTiCl₃) and modified methylaluminoxane (MMAO). The incorporation of ST into the living polybutadiene (PB) precursor remarkably depended on the polymerization temperature. A low temperature (?20 °C) suppressed the rate of ST incorporation, but a high temperature (50 °C) tended to decompose the livingness of the active species and enhance the rate of the aspecific ST polymerization initiated by MMAO. Consequently, temperatures of 0–25 °C seemed to be best for this copolymerization system. Because of the absence of ST livingness, the final products contained not only the block copolymer but also the homopolymers. Attempts to isolate the block copolymer were carried out with common solvent fractionation techniques, but the results were not sufficient. Cross‐fractionation chromatography was, therefore, used for the isolation of the cis‐PB‐b‐syn‐PS copolymer. The presence of long syn‐PS segments was confirmed by the observation of a strong endothermic peak at 260 °C in the differential scanning calorimetry curve. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2698–2704, 2004     

Entanglements and elasticity in polybutadiene networks

Polybutadiene networks were prepared by peroxide crosslinking of monodisperse 1,4-polybutadienes both in solution and in bulk. The effect of the entangled sol fraction on the elastic modulus of high-molecular-weight polybutadiene was observed in stress relaxation measurements. Sol fraction was shown to make a large contribution to the Mooney–Rivlin 2C₂ term. This effect was also observed on the molecular level in NMR spin-spin relaxation measurements. For networks crosslinked in bulk the stress relaxation measurements suggest the presence of trapped entanglements. The 2C₂ term is insensitive to sol extraction in these networks. NMR spin-lattice relaxation measurements in the rotating frame at 4.68 kHz verify the presence of additional effective crosslinks in these networks.     

Chain conformations of block copolymers

An analytical theory of block copolymer conformations is developed for systematically studying the effects of the chain length, chain architecture, and segment interactions. The main results obtained for AB diblock and ABA triblock chains are as follows: (i) In the absence of AA and BB interactions, both diblock and triblock chains collapse to a dense form if the AB interaction is attractive. In the collapsed coil form, the mean-square end-to-end distance 〈R²〉 is proportional to the square root of the number of segments n^1/2. (ii) The diblock chain has a dumbbell form if AA and BB interactions are attractive and AB interaction repulsive, but the triblock chain collapses. In the dumbbell form, 〈R²〉 is proportional to n.     

Prediction of microstructures for polydisperse block copolymers,using continuous thermodynamics

A generalization of an earlier theory (Leary–Henderson–Williams) developed for microphase separation in monodisperse block copolymers is made for copolymers having moderate degrees of polydispersity and illustrated for the Schultz molecular weight distribution (MWD). First, an explicit study is made of molecular weight (M) effects for monodisperse poly (styrene–butadiene) diblock (SB) and triblock (SBS) copolymers. For a fixed temperature, it is shown how the critical molecular weight (M_c)—above which the copolymer is phase-separated at equilibrium —varies with molecular composition (?_S, volume fraction of S component) for both molecular architectures. Also predicted are the microstructural parameters ΔT(M) and f(M)—interphase thickness and volume fraction, respectively—and the high-M limiting functions ΔT ∝? M^α2, f ∝? M^α3, D ∝? M^α4 (D is domain repeat distance) and T_s ∝? M^α5 (T_s is separation temperature). Then, for polydisperse systems in the range 1 ? p ? 3 ( where \[ P = \bar M_w /\bar M_n \] ) corresponding predictions at constant \[ \bar M_n \] are made after identifying the mixture free-energy-minimum state with a weight average of the free energy minima of each fraction of the MWD. Calculations are made specifically for ?_S = 0.50 and T_s = 298 K. It is shown that, even when \[ \bar M_n < M_c \] , polydispersity can induce microphase separation if p is sufficiently large. Good success is obtained in comparisons of D predictions with data on blends of two polydisperse diblock samples.     

Nylon 6–polyisobutylene sequential copolymers. II. Synthesis,characterization, and morphology of di-, tri-, and radial block copolymers

Nylon 6–PIB diblock, triblock, and tristar radial block copolymers have been synthesized from telechelic hydroxyl-terminated polyisobutylene, PIB(OH)_n (n = 1,2,3), by conversion of this prepolymer with hexamethylene diisocyanate (HMDI), toluene diisocyanate (TDI), N-chlorocarbonyl diisocyanate (NCCI), and oxalyl chloride (OxCl) and using the resulting materials as macroactivators for anionic caprolactam polymerization. Prepolymers with molecular weights from 6000 to 38,000 have been employed. Derivatization with NCCI and subsequent anionic caprolactam polymerization gave highest yields and blocking efficiencies. The block copolymers have been characterized by molecular weight and composition. In addition to the expected T_g and T_m characteristics of long PIB and nylon 6 segments, DSC studies showed an intermediate glass transition at ca. ?20°C. Transmission electron microscopy of di-, tri-, and radial blocks show increasing segregation and orientation of rubbery/crystalline domains. Tensile strengths and elongations of the block copolymers range from 16.5 to 41 MPa and 15 to 30%, respectively, and stress-strain diagrams show the effect of block architecture on these properties.     

Effects of copolymer composition on the formation of ionic species,hydrogen evolution,and free-radical reaction in γ-irradiated styrene-butadiene random and block copolymers

Block and random copolymers of butadiene and styrene as well as polybutadiene and polystyrene homopolymers have been investigated with respect to formation of trapped electrons, contribution of ionic species to crosslinking, and hydrogen gas evolution due to γ radiation. The decay kinetics of the disubstituted benzyl radical has also been studied. The yields of electron trapping G(e^?) are measured. The G(e^?) increase linearly with increased polystyrene content in block polymers, while in random copolymer a deviation from a linear relation is observed. The contribution of ionic reactions to crosslinking is about 25–35% of the total crosslinking yield. Hydrogen production in block copolymers is approximately a linear function of the weight-fraction additivity of the yield of hydrogen formation in polystyrene and polybutadiene homopolymers. Energy transfer from butadiene units to styrene units in random copolymers resulted in a deviation from such an additivity relation. The decay of the disubstituted benzyl free radical in block copolymers is a second-order reaction. In random copolymer, the decay is best interpreted in terms of equation based on a second-order decay mechanism of a fraction of the free radicals decaying in the presence of other nondecaying free radicals.     

A lutetium allyl complex that bears a pyridyl-functionalized cyclopentadienyl ligand: dual catalysis on highly syndiospecific and cis-1,4-selective (co)polymerizations of styrene and butadiene

A novel linked‐half‐sandwich lutetium–bis(allyl) complex [(C₅Me₄? C₅H₄N)Lu(η³‐C₃H₅)₂] ( 1 ) attached by a pyridyl‐functionalized cyclopentadienyl ligand was synthesized and fully characterized. Complex 1 in combination with [Ph₃C][B(C₆F₅)₄] exhibited unprecedented dual catalysis with outstanding activities in highly syndiotactic (rrrr>99 %) styrene polymerization and distinguished cis‐1,4‐selective (99 %) butadiene polymerization, respectively. Strikingly, this catalyst system exhibited remarkable activity (396 kg copolymer (mol_Lu h)^?1) for the copolymerization of butadiene and styrene. Irrespective of whether the monomers were fed in concurrent mode or sequential addition of butadiene followed by styrene, diblock copolymers were obtained exclusively, which was confirmed by a kinetics investigation of monomer conversion of copolymerization with time. In the copolymers, the styrene incorporation rate varied from 4.7 to 85.4 mol %, whereas the polybutadiene (PBD) block was highly cis‐1,4‐regulated (95 %) and the polystyrene segment remained purely syndiotactic (rrrr>99 %). Correspondingly, the copolymers exhibited glass transition temperatures (T_g) around ?107 °C and melting points (T_m) around 268 °C; typical values for diblock microstructures. Such copolymers cannot be accessed by any other methods known to date. X‐ray powder diffraction analysis of these diblock copolymers showed that the crystallizable syndiotactic polystyrene (syn‐PS) block was in the toluene δ clathrate form. The AFM micrographs of diblock copolymer showed a remarkable phase‐separation morphology of the cis‐1,4‐PBD block and syn‐PS block. This represents the first example of a lutetium‐based catalyst showing both high activity and selectivity for the (co)polymerization of styrene and butadiene.     

Charge-mosaic membrane from gamma-irradiated poly(styrene-butadiene-4-vinylpyridine) triblock copolymer

Poly(styrene-butadiene-4-vinylpyridine) triblock copolymers were prepared from styrene (S), butadiene (B), and 4-vinylpyridine (P) by sequential anionic polymerization with n-butyllithium as initiator and benzene as solvent. The triblock copolymer was characterizated by gel-permeation chromatography (GPC), transmission electron microscopy (TEM), and viscoelastic spectrometry. Films of the triblock copolymer cast from solution in mixtures of chloroform and n-butyraldehyde were subjected to gamma-ray irradiation to form cross-linked networks, Cationic and anionic groups were introduced by sulfonation and quater-nization to obtain charge-mosaic membranes. The resulting membrane had substantial cation-exchange and anion-exchange capacities. The membranes were very permeable to electrolyte (J_KCI = 2.10×10^?8 mol/cm² s). © 1993 John Wiley & Sons, Inc.     

三嵌段共聚物和两嵌段共聚物在水溶液中共混自组装成多室胶束结构的耗散粒子动力学模拟(英文)

利用耗散粒子动力学模拟研究了在水溶液中混合不同的线形三嵌段共聚物AxByCz和线形两嵌段共聚物AmBn对多室胶束的形貌多样性的影响.通过改变线形的三嵌段共聚物和两嵌段共聚物的链长来寻找多室胶束的形成条件.由线形三嵌段共聚物和线形两嵌段共聚物的不同混合形成的多室胶束结构是多种多样的,例如"蠕虫状"胶束、"汉堡包"胶束、"球上球"胶束、"核-壳-壳"胶束等等.多室胶束的整体形貌和内部结构的控制都可以从线形三嵌段共聚物和两嵌段共聚物的二元共混得到.为了表征获得的多室胶束结构,我们计算了密度图和成对分布函数图.在此工作中,可以获得和观察到复杂的多室胶束.结果表明,简单地混合线形的三嵌段共聚物和线形的两嵌段共聚物是一个控制多室胶束形貌和结构的有效方法,在工程实验中可以更简单更经济地形成多室胶束结构.因此,在设计新的多室胶束方面,聚合物共混仍然是未来值得更加关注的一个话题.     

Epoxies toughened with triblock copolymers

Three different commercial triblock copolymers from Arkema were evaluated as potential toughening agents for two different lightly crosslinked epoxies. It was found that the plane strain fracture toughness, K_IC, was on the order of 3.0 MPa√m for 10 parts per hundred resin (phr) of NanoStrength™ E20 resin (a styrene–butadiene–methylacrylate, SBM, type triblock copolymer) in epoxies cured with either aminoethylpiperazine or piperidine. In contrast, 10 phr NanoStrength E40 resin (also an SBM type triblock copolymer) was ineffective in toughening such epoxies. The difference in toughening effectiveness was attributed to the smaller amount of polybutadiene present in the E40 resin. The third toughening agent from Arkema was NanoStrength M22 resin, which is a symmetric triblock copolymer consisting of side blocks of PMMA surrounding a center block of poly(butyl acrylate) and is designated as MAM. At 10 phr MAM both lightly crosslinked epoxies exhibited improvements in toughness. Morphologies were characterized using TEM and toughening mechanisms revealed using SEM and TOM. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1470–1481, 2007     

Thermodynamic Limitations of the Reptation Model

Summary: The paper deals with the question whether the tube/reptation model of polymer chain dynamics is compatible with general laws of statistical physics. Based on a relation between the mean squared fluctuation of the number of segments in a given volume element and the isothermal compressibility of the polymer system, it follows straightforwardly that the tube/reptation model predicts fluctuations larger than permitted by thermodynamics on the time scale t ≳ τ_R, where τ_R is the Rouse relaxation time.