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.     

Nonlinear viscoelasticity of polystyrene solutions. I. Strain-dependent relaxation modulus

Strain-dependent relaxation moduli G(t,s) were measured for polystyrene solutions in diethyl phthalate with a relaxometer of the cone-and-plate type. Ranges of molecular weight M and concentration c were from 1.23 × 10⁶ to 7.62 × 10⁶ and 0.112 to 0.329 g/cm³. Measurements were performed at various magnitudes of shear s ranging from 0.055 to 27.2. The relaxation modulus G(t,s) always decreased with increasing s and the relative amount of decrease (i.e.,–log[G(t,s)/G(t,0)]) increased as t increased. However, the detailed strain dependences of G(t,s) could be classified into two types according to the M and c of the solution. When cM < 10⁶, the plot of log G(t,s) versus log t varied from a convex curve to an S-shaped curve with increasing s. For solutions of cM > 10⁶, the curves were still convex and S-shaped at very small and large s, respectively, but in a certain range of s (approximately 3 < s < 7) log G(t,s) decreased rapidly at short times and then very slowly; a peculiar inflection and a plateau appeared on the plot of log G(t,s) versus log t. The strain-dependent relaxation spectrum exhibited a trough at times corresponding to the plateau of log G(t,s). The longest relaxation time τ₁(s) and the corresponding relaxation strength G₁(s) were evaluated through the “Procedure X” of Tobolsky and Murakami. The relaxation time τ₁(s) was independent of s for all the solutions studied while G₁(s) decreased with s. The reduced relaxation strength G₁(s)/G₁(0) was a simple function of s (The plot of log G₁(s)/G₁(0) against log s was a convex curve) and was approximately independent of M and c in the range of cM <10⁶. This behavior of G₁(s)/G₁(0) was in agreement with that observed for a polyisobutylene solution and seems to have wide applicability to many polymeric systems. On the other hand, log G₁(s)/G₁(0) as a function of log s decreased in two steps and decreased more rapidly when M or c was higher. It was suggested that in the range of cM < 10⁶, a kind of geometrical factor might be responsible for a large part of the nonlinear behavior, while in the range of cM > 10⁶, some “intrinsic” nonlinearity of the entanglement network system might be important.     

Nonlinear rheology of highly entangled polymer solutions in start‐up and steady shear flow

Orientation angle and stress‐relaxation dynamics of entangled polystyrene (PS)/diethyl phthalate solutions were investigated in steady and step shear flows. Concentrated (19 vol %) solutions of 0.995, 1.81, and 3.84 million molecular weight (MW) PS and a semidilute (6.4 vol %) solution of 20.6 million MW PS were used to study the effects of entanglement loss on dynamics. A phase‐modulated flow birefringence apparatus was developed to facilitate measurements of time‐dependent changes in optical equivalents of shear stress (n₁₂ ≈ Cσ) and first normal stress differences (n₁ = n₁₁ ? n₂₂ ≈ CN₁) in a planar‐Couette shear‐flow geometry. Flow birefringence results were supplemented with cone‐and‐plate mechanical rheometry measurements to extend the range of shear rates over which entangled polymer dynamics are studied. In slow (τ > ) steady shear‐flow experiments using the ultrahigh MW polymer sample (20.6 × 10⁶ MW PS), steady‐state n₁₂ and n₁ results manifest unusual power‐law dependencies on shear rate [n_12,ss ～ ^0.4 and n_1,ss ～ ^0.8]. At shear rates in the range τ < < τ, steady‐state orientation angles χ_SS are found to be nearly independent of shear rate for all but the most weakly entangled materials investigated. For solutions containing the highest MW PS, an approximate plateau orientation angle χ_p in the range 20–24° is observed; χ_p values ranging from 14 to 16° are found for the other materials. In the start‐up of fast steady shear flow (γ˙ ≥ τ), transient undershoots in orientation angle are also reported. The molecular origins of these observations were examined with the help of a tube model theory that accommodates changes in polymer entanglement density during flow. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2275–2289, 2001     

Stress relaxation upon cessation of steady flow and the overshoot effect of polymer solutions

The basis for obtaining the steady-state compliance from stress relaxation upon cessation of steady flow (SRUCSF) data is derived. Measurements on three polymer solutions of differing molecular weight, polydispersity, and degree of entanglement coupling show good agreement between results from SRUCSF and creep-recovery experiments in both linear and nonlinear viscoelastic regions. The stress overshoot phenomenon is interpreted in terms of a change in entanglement spacing upon imposition of a shearing field. The phenomenon is analyzed in terms of a relaxation time for re-entanglement, which is found to be much longer than the relaxation time of the shear stress upon cessation of steady flow.     

Restricted swelling and its orientation effect on copolymer micellar solutions of hexagonal-packed cylinders under steady shear flow

We investigate the structure of polystyrene-b-poly(acrylic acid, sodium salt) copolymer in aqueous solutions. We offer detailed characterization of the micellar solutions by scattering techniques and show that they form hexagonal-packed cylinders. The micelles follow the swelling law expected for cylinders, upon addition of water, down to a concentration threshold below which the distance cylinder-cylinder remains constant. The results suggest that the cylinders are aggregated. It is proposed that this micellar association is the cause of the unusual orientation of the cylinders in a steady shear flow field, where the cylinder axis is found to be parallel to the velocity gradient.     

The start of the development of an European viewpoint on education in analytical chemistry at Euroanalysis III,Dublin

Analytical and Bioanalytical Chemistry -     

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     

Flow birefringence studies of a concentrated polystyrene solution in a two-roll mill. I. Steady flow and start-up of steady flow

We have studied a 4.8 volume percent solution of a narrow distribution polystyrene with molecular weight 3.84 × 10⁶ in flows generated by a co-rotating two-roll mill. These flows have a stagnation point at the midpoint between the roller axes. Further, they are linear, two-dimensional, and the magnitudes of the strain-rates are greater than the vorticity. The overall objective of our studies is to explore the dynamics of concentrated polymer solutions which are in the highly deformed state that is generated in the two-roll mill. Birefringence data are presented for both steady flow and start-up of steady flow in the two-roll mill. The steady and transient data are used to analyze the linear and nonlinear viscoelastic regions of material behavior. In the nonlinear regime, the birefringence upon start-up shows an initial overshoot followed by a strong undershoot that is enhanced as the ratio of elongation to rotation is increased (i.e., the flow becomes increasingly extensional in character.) We attribute this undershoot, which does not seem to appear in simple shear flows, or flows close to simple shear flow, to polymer segment stretching following an initial period of segment reorientation. Model studies are currently underway to test this notion. © 1992 John Wiley & Sons, Inc.     

Incompatibility of polymer solutions. III. Critical opalescence in the system polystyrene + polyisobutylene + toluene

The angular dependence of light scattering from the homogeneous ternary system polystyrene + polyisobutylene + toluene was measured at a fixed (critical) concentration as a function of temperature. On approaching the critical temperature the scattering becomes very large and strongly angle-dependent. The experiments show that the interaction range r_G is of the same order of magnitude as the molecular radius of gyration of the polymer, but cannot be set equal to it as in the case of a binary system of one polymer in a single solvent. The temperature dependence of the various interaction parameters does not depend on the molecular weights of the polymers.     

Role of ionic species and valency on the steady shear behavior of partially hydrolyzed polyacrylamide solutions

Polyacrylamides are anionic polymers with a large number of charges along the polymer chains. The rheological properties of aqueous polyacrylamide solutions can be significantly modified by varying the solvent environment with the addition of salt. The presence of cations substantially reduces the inter- and intra-molecular interactions of the macroions. It was found that the valency of the cation has a strong effect on the rheological behavior of polyacrylamide solutions, but the size and type of salt have a negligible effect.The reduction in the solution viscosity with di-valent salts (e.g., CaCl₂, MgCl₂, BaCl₂, and MgSO₄) can be as high as an order of magnitude compared with mono-valent salt (KI, KC1, NaCl, and NaBr), depending on the salt concentration and shear rate. An identical viscosity function can be obtained for different types of polyacrylamide solutions by varying the salt content in solution. This interesting feature provides a useful means in the development and preparation of certain ideal fluids for simulation studies of complex flow problems.     

Effect of shear flow on the phase‐separation behavior in a blend of polystyrene and polyvinyl methyl ether

The phase behavior and phase‐separation dynamics of polystyrene/polyvinyl methyl ether (PS/PVME) blend with a critical composition of 70 vol % PVME were examined with a light scattering technique under a shear‐rate range of 0.1–40 s^?1. If the shear rates were less than 8 s^?1 and the starting temperatures of the measurement were 343 and 383 K, respectively, two cloud points were observed, whereas after the shear rate was higher than 8 s^?1, only one cloud point existed, 20 K higher than that of the static state of the blend. Investigation of the phase‐separation dynamics at 443 K suggested that in the vorticity direction the phase‐separation behavior at the early stage and the later stage can be explained by Cahn–Hilliard linearized theory and the exponent growth law, respectively. Phase separation occurs after a shearing time, which was called a delay time τ_d. The delayed time τ_d, the apparent diffusion coefficient, and the exponent term of the blend show strong dependence on shear rates. A theoretical prediction of the phase behavior of PS/PVME under a shear flow field by introducing an elastic energy term into Flory's equation‐of‐state theory was made, and the prediction was consistent with the experimental results. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 661–669, 2003     

Stress relaxation of polybutadiene at large deformation. Measurements of stress and birefringence in shear and elongation

The rheological behavior of an uncrosslinked polybutadiene on sudden application of finite strain was examined. The shear stress σ, two components of birefringence, and the extinction angle were measured in shear (magnitude of shear γ ≤ 3.5) and tensile stress and the birefringence were measured in uniaxial elongation (elongation ratio λ ≤ 3.8). Measurements were performed at 30°C with a tensile tester equipped with appropriate sample holders. The stress-optical coefficient was 3.01 × 10^?9Pa^?1. The first and second normal-stress differences v₁ and v₂ were separately evaluated with the use of stress-optical law. The Lodge—Meissner relation v₁ = γσ held good. The ratio v₂/v₁ was independent of time and varied from about ?0.3 to ?0.2 with increasing γ in the range of measurements. Each of the stress components was factored into a function of strain and one of time, and the latter was common to all the stress components. Simple formulas were proposed to represent stress components in step deformations.     

Nonlinear viscoelasticity of entangled polymer chains. II. Comparison of the slip-link model with stress measurements at constant rate of elongation on narrow-distribution polystyrenes

Stress measurements have been performed at constant-strain-rate stretching of narrow distribution polystyrenes with various molecular weights elongated up to 300% at about 30°C above the glass transition temperature. The experimental stress-strain curves are compared with those calculated on the basis of a new treatment of Doi's molecular model (Part I). Satisfactory agreement is obtained, leading to a rather precise estimate of the equilibration time. The latter has a theoretical dependence on the square of molecular weight.     

Intramolecular and intermolecular association during chemical cross-linking of dilute solutions of different polysaccharides under the influence of shear flow

Effects of shear flow on intramolecular and intermolecular associations of dilute aqueous alkali solutions of dextran, hydroxyethylcellulose (HEC), and a hydrophobically modified analogue (HM-HEC) in the presence of a chemical cross-linker agent were characterized with the aid of viscometry and rheo-small-angle light scattering (rheo-SALS) methods. The picture that emerges at short times in the course of cross-linking of the polymer solutions under the influence of a constant shear rate is that HEC coils contract because of intramolecular cross-linking, whereas the HM-HEC species show an incipient association and the dextran molecules are unaffected. At longer times, interchain cross-linking of the polymers promoted the growth of large flocs, which were disrupted by shear forces when they were sufficiently large. These findings are novel, and both the building up of aggregates and disaggregation are well substantiated by the SALS results.     

Incompatibility of polymer solutions. I. Binodals and spinodals in the system polystyrene + polyisobutylene + toluene

The incompatibility of polystyrene (PS) and polyisobutylene (PIB) in toluene is characterized by binodals obtained from phase separation and spinodals obtained from light scattering at zero scattering angle. From these the interaction parameter χ₂₃, between unlike polymer segments was calculated. It appears to be dependent on molecular weight and composition. The nature of χ₂₃, especially its low value, is discussed.     

Stress driven shear bands and the effect of confinement on their structures--a rheological, flow visualization, and Rheo-SALS study

An equimolar mixture of a cationic surfactant, cetylperidinium chloride (CPyCl), and salt sodium salicylate (NaSal) forms wormlike micelles in aqueous solutions. Under shear, the solution shows a pronounced shear-thickening behavior, which is coupled with oscillations in shear rate and the apparent viscosity. In this shear-thickening regime shear bands form, which also oscillate in position and intensity. These shear bands are visualized by direct imaging and Rheo-small angle light scattering methods. Temporal intensity fluctuations of the shear bands were evaluated using image analysis. Fourier transformations (FT) of the oscillating shear rate and intensity of the shear bands showed a single dominating frequency in the power spectrum analysis. This characteristic frequency as well as the amplitude of shear rate fluctuation was found to increase with stress. From the rheological and optical measurements, we propose that a stress driven mechanism is responsible for the formation of shear bands. Experiments performed in transparent parallel-plate geometry show dampening of the shear rate oscillations and increase in the characteristic frequency with decrease in the gap. Power spectrum analysis and the SALS measurements confirm the formation of different structures as a function of gap size in the parallel-plate geometry.