Stress overshoot of polymer solutions at high rates of shear; Polystyrene with bimodal molecular weight distribution

Overshoot of shear stress, σ, and the first normal stress difference, N₁, in shear flow was investigated for dilute solutions of polystyrene with very high molecular weight in concentrated solution of low M PS. In the case that the matrix was a nonentangled system, behavior of overshoot was similar to that of dilute solution of high M PS in pure solvent. The magnitudes of shear, γ_σm and γ_Nm, corresponding to the peaks of σ and N₁ lay on the universal functions of γ˙τ_R, respectively, proposed for dilute solutions in pure solvent. Here τ_R is the Rouse relaxation time for high M PS in the blend evaluated from dynamic modulus at high frequencies. In the case that the matrix was an entangled system, an additional σ peak was observed at high rates of shear at times corresponding to γ_σm = 2–3. This peak can be assigned to the motion of low M chains in entanglement network. When the matrix was entangled, stress overshoot was observed even at relatively low rates of shear, say γ˙τ_R < 10⁻². This is probably due to the motion of high M chains in entanglement of all the chains. In this case the γ_σm and γ_Nm values were higher than those expected for entangled chains of monodisperse polymer in pure solvent. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2043–2050, 2000     

Dynamic mechanical properties of moderately concentrated polystyrene solutions

The storage (G′) and loss (G″) shear moduli have been measured in the frequency range from 0.04 to 630 Hz for solutions of narrow distribution polystyrenes with molecular weights (M) 19,800 to 860,000, and a few of poly(vinyl acetate), M = 240,000. The concentration (c) range was 0.014–0.40 g/ml and the viscosities of the solvents (diethyl phthalate and chlorinated diphenyls) ranged from 0.12 to 70 poise. Data at different temperatures (0–40°C) were combined by the method of reduced variables. Two types of behavior departing from the usual frequency dependence describable by the Rouse-Zimm-Tschoegl theories were observed. First, for M ? 20,000, the ratio (G″ ? ωη_s)/G′ in the neighborhood of ωτ₁ = 1 was abnormally large and the steady-state compliance J was abnormally small, especially at the lowest concentrations studied. Here ω is circular frequency, η_s solvent viscosity, and τ₁ terminal relaxation time. Related anomalies have been observed by others in undiluted polymers at still lower molecular weights. Second, at the highest concentrations and molecular weights, a “crossover” region of the logarithmic frequency scale appeared in which G″ ? ωη_s < G′. The width of this region is a linear function of log c; the frequency dependence under these conditions can be represented by a sequence of Rouse relaxation times grafted on to a sequence of Zimm relaxation times. For each molecular weight, the terminal relaxation time τ₁ was approximately a single function of c for different solvents of widely different η_s. At lower concentrations, τ₁ was close to the Rouse prediction of 6ηM/π²cRT, where η is the steady-flow viscosity; but at higher concentrations, τ₁ was proportional to η/c² and corresponded, according to a recent theory of Graessley, to an average molecular weight of 20,000 between entanglement coupling points in the undiluted polymer.     

Flow properties of poly(methyl methacrylate) solutions

Viscosity and normal stress behavior were measured for poly(methyl methacrylate) samples of various average molecular weights in diethyl phthalate solution at 30 and 60°C. All samples conformed approximately to the most probable distriution (M?_w/M?_n = 2). Concentrations ranged from 0.113 to 0.38 g/ml, and M?_w from 53,800 to 1,620,000. Despite considerable evidence in the literature of unusual linear viscoelastic behavior for this polymer, its nonlinear properties appear to be rather conventional. The viscosity–shear rate master curve was similar to that found earlier for concentrated solutions of polystyrene and poly(vinyl acetate) of comparable molecular-weight distribution. The viscosity time constant τ_o parallels τ_R, the characteristic time of the Rouse model, although the residual dependence of τ_o/τ_R on concentration and molecular weight appears to be slightly different from that for polystyrene and poly(vinyl acetate). Similar conclusions apply to the recoverable compliance J_e,^o estimated from the normal stress behavior of each solution, and its relationship to the Rouse model compliance J_R.     

Scaling analysis of cotton cellulose/LiCl·DMAc solution using light scattering and rheological measurements

Semidilute solution of cotton lint (CC1) in 8 wt % LiCl/N,N‐dimethylacetamide was investigated using static light scattering (SLS) and rheological measurements. The reduced osmotic modulus estimated by SLS measurements for CC1 solutions are proportional to c^1.16 in the semidilute region. From the exponent of 1.16, de Gennes' scaling theory derives the relationship between radius of gyration, R_g, and molecular weight, M_w, of CC1 as R_g ∝ M^0.62 This corresponds to the Mark‐Houwink‐Sakurada exponent of 0.86. This exponent is very close to that estimated from scaling analysis of zero shear rate viscosity, that is 0.85. Apparent radius of gyration, R_g,app, estimated by SLS measurements for CC1 solutions are proportional to c^?0.5 in the semidilute region. R_g,app indicates the mesh size of polymer entanglement in the semidilute region. On the assumption of the Gaussian behavior of CC1 molecule in the semidilute region, the exponent of ?0.5 gives the relationship between the molar mass between entanglements, M_e, and c as following relationship: M_e ∝ c^?1. This agrees with the concentration dependence on plateau modulus estimated from the dynamic viscoelastic measurements. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2155–2160, 2006     

Stress overshoot of polymer solutions at high rates of shear

Overshoot of shear stress, σ, and the first normal stress difference, N₁, in shear flow were investigated for polystyrene solutions. The magnitudes of shear corresponding to these stresses, γ_σm and γ_Nm, for entangled as well as nonentangled solutions were universal functions of γ˙τ_eq, respectively, and γ_Nm was approximately equal to 2γ_σm at any rate of shear, γ˙. Here τ_eq = τ_R for nonentangled systems and τ_eq = 2τ_R for entangled systems, where τ_R is the longest Rouse relaxation time evaluated from the dynamic viscoelasticity at high frequencies. Only concentrated solutions exhibited stress overshoot at low reduced rates of shear, γ˙τ_eq < 1. The behavior at very low rates, γ˙τ_eq < 0.2, was consistent with the Doi–Edwards tube model theory for entangled polymers. At high rates, γ˙τ_eq > 1, γ_σm and γ_Nm were approximately proportional to γ˙τ_eq. At very high rates of shear, the peak of σ is located at t = τ_R, possibly indicating that the polymer chain shrinks with a characteristic time τ_R in dilute solutions. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1917–1925, 2000     

Synthesis and properties of eighteen-arm polybutadienes

A series of eighteen-arm regular star polybutadienes with molecular weights between 9.9 × 10⁴ and 1.9 × 10⁶ were prepared and characterized. Evidence is presented for the expanded configuration of the large eighteen-arm stars in a θ solvent. The intrinsic viscosities of the eighteen-arm stars gave g′ = [η]/[η]_l = 0.28₄ in dioxane at 27°C (θ solvent) and 0.22₅ in toluene at 35°C (good solvent). The linear viscoelastic properties of the melts were also determined. The plateau modulus, G_N°, is the same as for linear polybutadiene. The zero-shear viscosities (η₀) and the longest relaxation times (T_max) increase exponentially with the arm molecular weight M_a and are identical to those of four-arm polybutadienes with the same M_a. The zero-shear recoverable compliance (J_e°) increases linearly with molecular weight. v′ in J_e°G_N° = v′N_a, where N_a is the number of entanglements per arm, is 0.95 slightly larger than 0.66 for four-arm polybutadienes. Similarly, g₂ is higher than calculated from the Rouse–Ham theory.     

Charge density effects in salt‐free polyelectrolyte solution rheology

Solution rheology of 2‐vinyl pyridine and N‐methyl‐2‐vinyl pyridinium chloride random copolymers in ethylene glycol was studied over wide ranges of concentration and effective charge. The fraction of quaternized monomers α and the fraction of monomers bearing an effective charge f of these copolymers were measured using counterion titration and dielectric spectroscopy, respectively. Ethylene glycol is a good solvent for neutral poly(2‐vinyl pyridine), with very few ionic impurities. The viscosity η and relaxation time τ of dilute and semidilute unentangled solutions exhibit the scaling with concentration and effective charge expected by the Dobrynin model. Reduced viscosity data are independent of concentration in dilute solution, giving an intrinsic viscosity that depends on effective charge, and the experimental data obey the Fuoss law in the semidilute unentangled regime. Scaling concentration with the overlap concentration (c/c*) reduces these data to common curves, and c* ～ f ^?12/7 as predicted by the Dobrynin model, where f is the fraction of monomers bearing an effective charge. While the overlap concentration depends strongly on effective charge until counterion condensation occurs, the entanglement concentration c_e is surprisingly insensitive to effective charge, indicating that entanglement effects are not understood using the Dobrynin model. The terminal modulus G = η/τ depends only on the number density of chains G = ckT/N for c* < c < c_e, and G ～ c^3/2 for c > c_e independent of the effective charge. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2001–2013, 2006     

Rheology of polystyrene solutions around the coil overlapping concentration: A phenomenological description of stresses in simple shear flow

Linear viscoelasticity behavior is described with the sum of two terms for polystyrene solutions in tricresyl phosphate around the coil overlapping concentration (K. Osaki, T. Inoue, & T. Uematsu, J Polym Sci Part B: Polym Phys 2001, 39, 211). One is a Rouse–Zimm (RZ) term represented by the Zimm theory with arbitrarily chosen values of the hydrodynamic interaction parameter and the longest relaxation time (τ_RZ). The other (the L term) consists of a relaxation mode with a single relaxation time (τ_L > τ_RZ) and a high‐frequency limiting modulus proportional to the square of the concentration. In this study, we describe the viscosity (η) and first normal stress coefficient (Ψ₁) in steady shear with simple formulas. The stress due to the L term is assumed to be given by a Kaye, Bernstein, Kearsley, and Zapas (K‐BKZ) equation with the damping function h(γ) = (1 + 0.2γ²)^?1/2, where γ is the magnitude of shear. Contributions to η and Ψ₁ from the RZ term are derived from the RZ model, in which the relaxation time in steady flow is given by τ_st = τ + (τ_RZ ? τ)/(1 + 0.35τ_RZ γ˙) instead of τ_RZ. Here, γ˙ is the rate of shear, and τ is the τ_RZ value at the infinite dilution limit. η and Ψ₁ at various concentrations for two polystyrene samples (with molecular weights of 2890 and 8420 kg mol^?1) are well described with parameters derived from dynamic viscoelasticity. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1038–1045, 2002     

Intra- and inter-chain fluctuations in entangled polymer melts in bulk and confined to pore channels

It is known that topological restraints by “chain entanglements” severely affect chain dynamics in polymer melts. In this field-cycling NMR relaxometry and fringe-field NMR diffusometry study, melts of linear polymers in bulk and confined to pores in a solid matrix are compared. The diameter of the pore channels was 10 nm. It is shown that the dynamics of chains in bulk dramatically deviate from those observed under pore constraints. In the latter case, one of the most indicative signatures of the reptation model is verified 28 years after its prediction by de Gennes: The frequency and molecular mass dependencies of the spin-lattice relaxation time obey the power law T_! ∝ M⁰ v^3/4 on a time scale shorter than the longest Rouse relaxation time τ_R. The mean squared segment displacement in the pores was also found to be compatible to the reptation law < r²>∝ M^−1/2t^1/2 predicted for τ_R < t < τ_d, where τ_d is the so-called disengagement time. Contrary to these findings, bulk melts of entangled polymers show frequency and molecular mass dependencies significantly different from what one expects on the basis of the reptation model. The data can however be described with the aid of the renormalized Rouse theory.     

Infinite-dilution oscillatory flow birefringence properties of polystyrene and poly(α-methylstyrene) solutions

Oscillatory flow birefringence (OFB) measurements have been carried out for an extensive series of solutions containing narrow-distribution, atactic, linear polystyrenes PS (10, 000 or 390, 000 M_w) or poly(α-methylstyrene) PMS (400,000 M_w) in a high-viscosity solvent, Aroclor 1248. The concentration ranges examined are such that the concentration dependence is obtained in both the “dilute” and “semidilute” regimes; the data are sufficiently precise to permit extrapolation to obtain for the first time the infinite-dilution properties. Various plotting formats are explored to determine an appropriate extrapolation procedure. The infinite-dilution OFB properties are compared with the bead-spring model (Zimm) theory which predicts quantitatively the frequency dependence of the observed properties for the PS and PMS solutions studied except for the high-frequency regime. The sensitivity and precision of the OFB experiment is such that the extrapolation curves–and the resulting infinite-dilution properties–show substantially less scatter than comparable visco-elasticity (VE) data. There is no evidence of a change in the character of the concentration dependence for the range of concentrations studied.     

Nonlinear viscoelasticity of polystyrene solutions. II. Non-Newtonian viscosity

The steady shear viscosity η(k) and the stress decay function \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \eta \left({t,k} \right)$\end{document} (the shear stress divided by the rate of shear k after cessation of steady shear flow) were measured for concentrated solutions of polystyrene in diethyl phthalate. Ranges of molecular weight M and concentration c were 7.10 × 10⁵ to 7.62 × 10⁶ and 0.112–0.329 g/cm³, respectively. Measurements were performed with a rheometer of the cone-and-plate type in the range 10^?4 < k < 1 sec^?1. The Cox–Merz relation η(k) = |η^*(ω)|_ω=k was tested with the experimental result (|^*(ω)| is the magnitude of the complex viscosity). It was found to be applicable to solutions of relatively low M or c but not to those of high M and c. For the latter η(k) began to decrease at a lower rate of shear than |η^*(ω)|_ω=k did; the Cox–Merz law underestimated the effect of rate of shear. The stress decay function was assumed to have a functional form \documentclass{article}\pagestyle{empty}\begin{document}$\tilde \eta \left( {t,k} \right) = \sum {\eta _p \left( k \right)e^{ - t/\tau p\left( k \right)} } $\end{document} where τ₁ > τ₂ > …, and the values of τ₁, τ₂ η₁ and η₂ were determined for some solutions. The relaxation times τ₁ and τ₂ were found to be independent of k and equal to the relaxation times of linear viscoelasticity. At the limit of k → 0, η₁ and η₂ were approximately 60 and 20–30%, respectively, of η and the non-Newtonian behavior was due to large decreases of η₁ and η₂ with increasing k. It was shown that η₁(k) may be evaluated from the relaxation strength G₁(s) for the longest relaxation time of the strain-dependent relaxation modulus with a constitutive model for relatively high c–M systems as well as for low c–M systems.     

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.     

Rheological studies of moderately concentrated polystyrene solutions in the vicinity of the θ temperature. II. Shear-rate dependence for different thermodynamic conditions

The viscosity data of moderately concentrated polystyrene solutions in trans-decalin (TD) (θ solvent, θ temperature 21°C) and toluene (TL) (good solvent) reported in Part I are discussed in terms of Graessley's entanglement theory. Under good solvent conditions, Graessley's master curve provides an excellent fit up to high shear rates, whereas in the vicinity of the θ conditions the data have to be modified by a parameter η_fric introduced by Ito and Shishido. The characteristic time of mechanical response to flow of chains approximately given by the shift factor τ₀ is found in good solvents to be on the order of the Rouse relaxation time. In poor solvents, close to demixing, τ₀ tends to much higher values, indicating a reduced chain mobility. The influence of temperature on the viscosity decreases with increasing shear. The resulting apparent energy of activation of flow shows very small or even negative values at high shear rates. This behavior can be explained by the modified Graessley theory, however, in a quite natural way.     

Mobility of anionic spin probes in water-containing nylon 6 film

The rotational mobility of anionic spin probes in water-containing nylon 6 film was investigated by means of electron spin resonance (ESR) measurements for comparison with the results for nonionic spin probes reported previously. The extrema separation of the ESR spectra, 2Az′ increased with time owing to the evaporation of water. In the higher temperature region, 2Az′ increased steeply with time at first, and then more slowly, whereas for the nonionic spin probes, 2Az′ increased gradually and monotonically with time. This fact suggests that the anionic probe molecules are more strongly affected by water than the nonionic ones, i.e., the former probes are located in hydrophilic regions and the latter in hydrophobic regions. T_50G, which can be empirically correlated with the glass transition temperature of the polymer T_g decreased with increasing water content. The decreasing tendency for the anionic spin probes was stronger than that for the nonionic ones. This fact also indicates that the local environment around the probe molecules varies from probe to probe. The rotational correlation time τ_R decreased markedly with an increase in water content. The Arrhenius plots of τ_R showed two crossover points. The crossover points in the higher temperature region T_n decreased greatly with increasing water content. The difference in T_n between dried and water-containing films was larger than that for T_50G. The activation energy for rotation, E, also decreased with increasing water content. It is suggested that water concentrates around the anionic spin probes and makes their rotation much easier.     

Rheological studies of moderately concentrated polystyrene solutions. I. A new method for the extrapolation of the zero-shear viscosity

Measurements of the viscosity coefficient η of solutions of polystyrene (M_w = 6.0 × 10⁵ and 1.77 × 10⁶) in trans-decalin (TD, θ solvent) and toluene (TL, good solvent) as function of shear rate (11?10⁴ s^?1), concentration (4.24?11.21 wt %), and temperature (10–50°C) are reported. As a new theoretically grounded method for the determination of the zero-shear viscosity η₀ it is proposed to plot η as a function of $\left({\eta \dot \gamma} \right)^3$. The intercepts of the straight lines obtained by this procedure give η₀ in good agreement with directly measured values.     

Exploring rheological properties of aqueous polyaniline‐PVP dispersion

Stable aqueous dispersion of polyaniline (PAn) stabilized by a hydrophilic polymer poly(vinyl pyrrolidone) (PVP) exhibits interesting rheological properties different from its components. Shear thinning observed for both PVP and PAn–PVP colloid (PP) indicates partially entangled nature of the later. Linear viscoelastic response of PVP solution exhibit strong frequency dependence of elastic (G′) and viscous (G″) modulus over the whole frequency range (0.1–100 ras/s) where G′ never exceeds G″ indicating the applicability of the Rouse‐Zimm model to this system. On the other hand, there is a crossover of G′ and G″ in the rheological profile of PP dispersion so that a single relaxation time model can be applicable. Therefore, PVP presents an entangled polymeric system and supposed to have a spectrum of relaxation times, whereas PP resembles to a physically crosslinked system with a single relaxation time. Increasing the extent of hydrogen bonding within the system (by raising the fraction of PAn or by leaving the solution undisturbed for long) relaxation time also becomes longer. The large difference in values of steady and complex shear viscosity (η and η*) within LVE regime reflects that original Cox‐Merz rule is obviously inapplicable to these systems. But at larger strain amplitude, η and η* are satisfactorily coincident that indicates a broader applicability of the modified Cox‐Merz rule. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 2443–2455, 2008     

Moderately concentrated solutions of polystyrene. I. Viscosity as a function of concentration,temperature, and molecular weight

Data on the viscosity η of moderately concentrated solutions of polystyrene are reported. Several solvents were investigated, including cyclopentane solutions over a temperature span between θ_U = 19.5°C and θ_L = 154.5°C. The data were analyzed in terms of a relation giving η as a function of αφM, where α_φ is the expansion factor for the chain dimensions in a solution with volume fraction φ of polymer with molecular weight M. It is shown that values of α_φ so determined decrease as ? lnα_φ/? lnφ = (1 ? 2μ)/6μ for φ greater than φ^* = 0.2M/s³ for moderately concentrated solutions, where s is the root-mean-square radius of gyration and μ = ? ln[η]/? lnM with [η] the intrinsic viscosity.     

Dynamic light scattering of polystyrene single chains in a semidilute solution of poly(methyl methacrylate) and benzene

Translational diffusion and internal motion have been observed by dynamic light scattering of optically labeled single chains of polystyrene (PS) in a semidilute solution of poly(methyl methacrylate) and benzene for the case in which the dimension R_g of the PS chain is comparable to the correlation length of the matrix solution. The molecular weight M_w dependence of the hydrodynamic radius R_h is expressed as R_h ～ M, while R_h ～ M in pure benzene. The average linewidth Γ for internal motions (KR_g > 1) appears to depend on the magnitude K of the scattering vector approximately as Γ ∝ K⁴ at higher KR_g ( > 1), in contrast with the fact that Γ ∝ K³ approximately for KR_g > 1 in pure benzene. The scaling law for the K dependence of Γ does not hold in low-molecular-weight PS owing to the K dependence of Γ /K² for KR_g < 1.     

Dynamic birefringence of oligostyrene: A symptom of “polymeric” mode

The dynamic birefringence and the dynamic viscoelasticity of an oligostyrene, A1000, whose molecular weight (M_w = 1050) was comparable to the Kuhn segment size, M_K, were examined near and above the glass‐transition temperature in order to characterize polymeric features of very short chains with M ∼ M_K. The complex shear modulus, G*(ω), was similar to that for supercooled liquids: No polymeric modes such as the Rouse mode were detected at low frequencies of viscoelastic spectrum. On the other hand, the strain‐optical coefficient was found to be negative in the terminal flow zone and positive in the glassy zone. Because the negative birefringence of polystyrene is originated by polymeric modes associated with chain orientation, the present results indicate that polymeric modes exist and become dominant for birefringence in the terminal flow. The data were analyzed using a modified stress‐optical rule: The modulus and the strain‐optical ratio were separated into polymeric (rubbery) and glassy components. The total modulus, G*(ω), was mostly due to the glassy component, G_G*(ω), resulting in the positive birefringence. G_G*(ω) for A1000 agreed with that for high M polystyrenes when compared at a comparable reduced frequency scale. The polymeric component, G_R*(ω), giving rise to the negative birefringence was lower than G_G*(ω) over the whole frequency range but its contribution to the birefringence exceeded that of the glassy component at low frequencies because of the larger optical anisotropy and longer characteristic relaxation time of the former. The limiting modulus of G_R* at high frequencies was about 3 times lower than that for high M polystyrenes, indicating that the main‐chain orientation of the oligostyrene on instantaneous deformation was reduced compared with that of high M polystyrenes. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 954–964, 2000     

Effect of concentration of the crosslinking agent and diluent during polymerization on the viscoelastic spectra of poly(2-hydroxyethyl methacrylate) networks

The effect of concentration of the crosslinking agent (ethylene dimethacrylate) and diluent (water) during the crosslinking copolymerization on the shape and position of retardation spectra in the dry state has been investigated for poly(2-hydroxyethyl methacrylate) networks. With increasing water content during network formation, the maxima of the retardation spectra, L_m, increase and the position of the spectra is shifted toward shorter retardation times, τ. The results are in quantitative agreement with the modified Rouse–Mooney (R–M) molecular theory and suggest the influence of deformation due to the diluent during network formation on the viscoelastic behavior. With increasing content of the crosslinking agent, the retardation spectra are shifted toward longer times. At a constant reference temperature T₀ = 115°C the retardation time, τ_m, at the maxima of the spectra increases with increasing content of effective chains in the network, ν_e. However, after a correction for the effect of the monomeric frictional coefficient, ξ, τ_m/ξ decreases with increasing v_e at a rate which agrees quantitatively with the R–M theory. The slope of the retardation spectra in the main transition region and the value of their maxima decrease with increasing v_e; a comparison of these dependences with theory leads to the most probable distribution of submolecules in the chains. The contribution of long retardation times to the equilibrium compliance, J_e, of the systems under investigation was estimated; it was shown that the application of the Thirion–Chasset extrapolation method for the determination of J_e of loose networks requires a certain type of dependence of the retardation or relaxation spectra on τ.