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.     

Driven translocation of semiflexible polyelectrolyte through a nanopore

The influence of the chain stiffness on the translocation of semiflexible polyelectrolyte through a nanopore is investigated using Langevin dynamics simulations. Results show that the translocation time τ increases with the bending modulus k_θ because of the increase of viscous drag forces with k_θ. We find that the relation between τ and k_θ, the asymptotic behavior of τ on the polyelectrolyte length N, and the scaling relation between τ and the driving force f are dependent on k_θ and N. Our simulation results show that the semiflexible polyelectrolyte chain can be regarded as either a flexible polyelectrolyte at small k_θ or large N where its radius of gyration R_G is larger than the persistence length L_p or a stiff polyelectrolyte at large k_θ or short N where R_G < L_p. Results also show that the out‐of‐equilibrium effect during the translocation becomes weak with increasing k_θ. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 912–921     

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     

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.     

Dynamic viscosity of dilute aqueous solutions of poly(acrylic acid) at frequencies of 2–500 kHz

The dynamic viscosity of aqueous solutions of poly(acrylic acid) at a polymer concentration of ca. 0.15 g/100 ml has been measured at frequencies from 2 to 500 kHz as a function of degree of polymerization P, degree of neutralization α, and salt (NaCl) concentration C_s. Relaxation spectra have been obtained from the dynamic viscosity. The spectra in the short relaxation time region can be approximated by the Zimm theory for the conformational relaxation of nonionic polymers. The maximum relaxation time τ₁ of the Zimm spectra is proportional to P² and depends rather moderately on α and C_s. Increased deviation is found, however, in the long relaxation time region, in particular for high values of P and α and low values of C_s. The major part of the deviation is interpreted in terms of rotational relaxation of a molecule as a whole. The rotational relaxation time τ_R is proportional to P³ and increases with increasing α and decreasing C_s. The remaining part of the excess spectra located between τ₁ and τ_R is ascribed to the deviation of the conformational relaxation from the Zimm theory arising from ionization of the polymer.     

Strain criterion in nonlinear creep and recovery in concentrated polymer solutions

Transient and steady-state rheological data are reported for several anionic polystyrene solutions in tritolylphosphate (1. 6 < cM/ρM_c < 7). Here c is the concentration of the solution, M is the molecular weight, ρ the density of the undiluted polymer, and M_c the molecular weight between entanglements as determined from zero-shear viscosity. The polystyrene used had M_w = 410,000 and M_w/M_n < 1.06. Data are also given for solutions of polyisobutylene and poly(vinyl acetate) with larger M_w/M_n. The results give a critical strain γ′ ∝ c⁻¹ such that linear viscoelastic behavior was obtained in a simple shear deformation with shear less than γ′. A simplified version of the constitutive equation of Bernstein, Kearsley, and Zapas is used with an empirical strain function F (γ) which contains γ′ as a parameter to discuss transient and steady-state behavior in terms of the distribution of relaxation (or retardation) times determined for linear viscoelastic responce. Features of the dependence of the steady-state viscosity η_k, recoverable compliance R_k, the first-normal stress function N_k⁽¹⁾ on shear rate k are discussed in terms of F (γ) and the distribution of relaxation times to conclude that the latter plays a dominant role in the behavior observed in the range of k usually studied. The results predict that the reduced functions η_k/η₀, R_k/R₀, and N_k⁽¹⁾/N₀⁽¹⁾ should depend on η₀R_0k, and that the functional form depends markedly on the distribution of relaxation times, at least in the range η₀R₀k < 10². Comparison with the mechanistic model of Doi and Edwards shows a similar F (γ) but substantial differences in the reduced functions caused by a very narrow distribution of relaxation times in the model.     

Huggins constant k′ for flexible chain polymers

The Huggins constant k′ in the expression for the viscosity of dilute nonelectrolytic polymer solutions, η = η(1 + [η] c + k′[η]²c² + …), is calculated. For polymers in the theta condition, k′ is estimated to be 0.5 < k_θ′ ≤ 0.7. For good solvent systems, the Peterson-Fixman theory of k′ has been modified; the equilibrium radial distribution function in the original theory is replaced with a parametric distribution for interpenetrating macromolecules in the shear force field. Comparison of the modified theory with experimental k′ for polystyrenes and poly(methyl methacrylates) of different molecular weights in various solvents shows good agreement. An empirical equation which correlates the Huggins constant k′ and the viscosity expansion factor α_η for polymers has been found to coincide well with the modified theory.     

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     

Conformational changes of a polyelectrolyte in mixtures of water and acetone

Samples of a polyelectrolyte poly(methacryloylethyl trimethylammonium methylsulfate), PMETMMS, with molar masses M_w = 22−25 × 10⁶ were examined with viscosity, static light scattering, and conductivity measurements in a water–acetone solvent. Because acetone is a nonsolvent for this polymer the measurements were performed to determine the influence of the solvent composition, the polymer concentration, and the presence of added ions on the conformation of the polyelectrolyte in mixed solvents. The possible influence of a hydrodynamic field on the polymer conformation was also studied. The viscosity of the polymer solutions as a function of polymer concentration, as well as of the solvent composition, was studied using a broad range of shear rates. When the mass fraction of acetone in the solvent, γ, is below 0.5, the solutions show a usual polyelectrolyte behavior. When γ ≥ 0.80, the polymer adopts a compact conformation. This is observed as a decrease of the radius of gyration, R_g, second virial coefficient, A₂, the viscosity, and also as a change in the conductivity of the solution. The change in the polymer conformation may be induced also by dilution. When 0.60 ≤ γ < 0.80, a gradual decrease in the polymer concentration leads to a sudden decrease of the reduced viscosity, which indicates a decrease in the particle size. The values of M_w measured by static light scattering were constant in all experiments. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1107–1114, 1998     

A theoretical approach to the kinetics of polymer degradation in solution

After main-chain scission in a polymer, the frequency of encounter between segments in the different fragments is related to the separating process between the fragments. The relationship obtained shows that the separating time is proportional to M ^½, where M is the molecular weight, when the excluded volume disappears. When good solvent is used, the half-time for the separation is obtained as τ_½ = const. M ^0.16–0.22, which is approximated to the experimental data obtained previously (τ_½ = const. M ^0.34 and τ_½ = const. M ^0.22) for the degradations of polyisobutene and poly(phenyl vinyl ketone), respectively. The increase of the half-time with increasing coil density can be explained by the excluded volume. The inverse proportionality of the diffusion of segments to solvent viscosity explains the proportionality of the half-time to microviscosity. The above separating process reverses the reaction between polymer radicals. From their dependence on the chain length, τ_½/k_D = const. M ^½ (where k_D is the specific rate for the reaction), is estimated. Such an approximation holds, regardless of the type of solvent.     

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.     

Doi-Edwards theory evaluation in double-step strain flows

It is widely accepted that for reversing double-step strain deformations, predictions based on the Doi-Edwards (DE) molecular theory without the independent alignment approximation (IAA) are superior to predictions obtained with the IAA, or equivalently, the Kaye-Bernstein-Kearsley-Zapas (K-BKZ) theory. This summation, however, is based on data obtained over limited ranges of strain and time: the time both between the step strains (t₁) and following the second step strain (t–t₁). In this study, a thorough evaluation of the DE theory is carried out using a comprehensive double-step strain flow data set. The results of this study indicate that the DE theory is an improvement over the K-BKZ theory in flows with strain reversal but only for cases when the criteria t₁, t–t₁ ? τ_k is satisfied. The constant τ_k, defined as the time beyond which the stress relaxation modulus is factorable: G(γ, t) = h(γ)G(t), is believed to represent the end of the chain retraction process in the DE theory. It appears that the dynamics of chain retraction have an important influence on double-step strain behavior and, therefore, should be accounted for in molecular-based theories devised to have general validity in this important deformation history. © 1994 John Wiley & Sons, Inc.     

Rheology of concentrated solutions of helical polypeptides

Rheological studies were carried out on concentrated m-cresol solutions of two helical synthetic polypeptides; poly-γ-benzyl-L -glutamate (PBLG; molecular weight, 150,000) and poly-?-carbobenzyloxy-L -lysine (PCBZL; molecular weight, 200,000). Steady shear measurements were made over a range of 0.01–16,000 sec^?1 to obtain steady shear viscosity and first normal stress difference. Dynamic viscosity and dynamic storage modulus were measured both by oscillatory shear between cone and plate and also by an eccentric rotating disk device over frequency ranges of 0.1–400 and 0.1–63 rad/sec, respectively. The concentration ranges were such that both liquid crystalline and isotropic solutions were investigated. The previously reported observations of an apparent negative first normal stress difference within a defined range of shear rate for liquid crystalline solutions were confirmed for the PBLG and PCBZL solutions. At high shear rates the peaks in plots of steady shear viscosity against concentration were profoundly suppressed but peaks in first normal stress difference versus concentration were not. The observation of liquid crystalline order in PCBZL/m-cresol solutions at room temperature constitutes evidence that the inverse coil-helix transition temperature is lower in concentrated solutions than in dilute solutions. The critical concentration for formation of the liquid crystalline phase was higher for PCBZL than for PBLG, despite a higher axial ratio, due to helix flexibility.     

Correction for instrument time constant and baseline in determination of reaction kinetics

Rates of reactions can be expressed as dn/dt = kf(n), where n is moles of reaction, k is a rate constant, and f(n) is a function of the properties of the sample. Instrumental measurement of rates requires c(dn/dt) = ckf(n), where c is the proportionality constant between the measured variable and the rate of reaction. When the product of instrument time constant, τ, and k is ? 1, the reaction is much slower than the time response of the instrument and measured rates are unaffected by instrument response. When τ k < 1, = 1, or >1, the reaction rate and instrument response rate are sufficiently comparable that measured rates are significantly affected by instrument response and correction for instrument response must be done to obtain accurate reaction kinetics. This paper describes a method for simultaneous determination of τ, k, c, and instrument baseline by fitting equations describing the combined instrument response and rate law to rates observed as a function of time. When τ cannot be neglected, correction for instrument response has previously been done by truncating early data or by use of the Tian equation. Both methods can lead to significant errors that increase as τk increases. Inclusion of instrument baseline as a fitting parameter significantly reduced variability in k and c compared with use of measured instrument baselines. The method was tested with data on the heat rate from acid‐catalyzed hydrolysis of sucrose collected with three types of calorimeters. In addition, to demonstrate the generality of this method of data analysis, equations including τ, k, c, and instrument baseline are derived for the relation between the reaction rate and the observed rate for first order, second order (first in each reactant), nth order in one reactant, autocatalytic, Michaelis–Menten kinetics, and the Ng equation. © 2010 Wiley Periodicals, Inc. Int J Chem Kinet 43: 53–61, 2011     

Viscometric properties of dilute polystyrene/dioctyl phthalate solutions

We report viscometric data collected in a Couette rheometry on dilute, single‐solvent polystyrene (PS)/dioctyl phthalate (DOP) solutions over a variety of polymer molecular weights (5.5 × 10⁵ ≤ M_w ≤ 3.0 × 10⁶ Da) and system temperatures (288 K ≤ T ≤ 318 K). In view of the essential viscometric features, the current data may be classified into three categories: The first concerns all the investigated solutions at low shear rates, where the solution properties are found to agree excellently with the Zimm model predictions. The second includes all sample solutions, except for high‐molecular‐weight PS samples (M_w ≥ 2.0 × 10⁶ Da), where excellent time–temperature superposition is observed for the steady‐state polymer viscosity at constant polymer molecular weights. No similar superposition applies at a constant temperature but varied polymer molecular weights, however. The third appears to be characteristic of dilute high‐molecular‐weight polymer solutions, for which the effects of temperature on the viscosity curve are further complicated at high shear rates. The implications concerning the relative importance of hydrodynamic interactions, segmental interactions, and chain extensibility with increasing polymer molecular weight, system temperature, and shear rate are discussed. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 787–794, 2006     

Rate constants in the photopolymerization of H2O2–isoprene systems

The polymerization reactions of isoprene with the use of H₂O₂ as the photoinitiator have been studied in acetone solutions. The lifetime of chain radicals, τ_s, was evaluated to be 0.29 by the rotating sector technique. The rate constants k_p and k_t were found to be 19.1 and 3.40 × 10⁷ L/mol- s, respectively.     

Hole growth as a microrheological probe to measure the viscosity of polymers confined to thin films

We review recent hole growth measurements performed at elevated temperatures in freely-standing polystyrene (PS) films, using optical microscopy and a differential pressure experiment (DPE). In the hole growth experiments, which were performed at temperatures close to the bulk glass-transition temperature of PS, T = 97 °C, we find evidence for nonlinear viscoelastic effects, which markedly affect the growth of holes in freely-standing PS films. The hole radius R initially grew linearly with time t before undergoing a transition to exponential growth characterized by a growth time τ. The time scale τ₁ for the decay of the initial transient behavior prior to reaching steady state was consistent with the convective constraint release mechanism of the tube theory of entangled polymer dynamics, while the characteristic hole growth times τ of the holes were consistent with significant reductions in viscosity of over eight orders of magnitude with increasing shear strain rate due to shear thinning. DPE measurements of hole growth on very thin freely-standing films revealed that hole formation and growth occurs only at temperatures that are comparable to or greater than T, even for films for which the T_g value was reduced by many tens of degrees Celsius below the bulk value. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B:Polym Phys 44: 3011–3021, 2006     

Ionic conduction in LiScn/Dimethylformamide/Poly(propylene oxide) system II. Mobility of lithium and thiocyanate ions

Ionic conductivity σ, shear viscosity η, and glass transition temperature T_g were measured on systems composed of lithiumthiocyanate LiSCN, N,N-dimethylformamide (DMF), and poly (propylene oxide) (PPO) over wide ranges of LiSCN concentration C and DMF content Φ. Using the dissociation constant of LiSCN reported in Part I, we have determined the concentration n of Li⁺ and SCN^? ions and then the mobility μ from σ. Data indicate that in the binary system of LiSCN/PPO, the σ versus C curve exhibits a maximum ca. C = 0.3 mol/L. In low C range, μ is independent of C but decreases with C in the range of C > 0.3 mol/L. Similar n dependence of μ is seen in the ternary systems containing DMF. The ratio of μ₀/μ (C) is lower than the ratio of viscosity η(C)/η₀ where μ₀ and η₀ indicate the values at infinite dilution of LiSCN. Thus the friction coefficient ?_ion for the translational diffusion of the ions is not proportional to the macroscopic viscosity. Relationship between μ and the monomeric friction ?_p for the segmental motion of the PPO chains is also discussed based on the data of T_g and the Williams-Landel-Ferry equation. ©1995 John Wiley & Sons, Inc.     

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     

MATRIX EFFECTS ON THE RADIATIONLESS TRANSITION' S1→T1 AND T1→S0 FOR BENZENE AND N, N, N', N'-TETRAMETHYLPARAPHENYLENE-DIAMINE MOLECULES

Abstract— The present study attempts to correlate the phosphorescence life time τ_p at 77°K of a definite solute: tetramethylparaphenylenediamine (TMPD) with various solvents viscosity and polarity. A few experiments with benzene in the same solvents are also reported. The following results have been obtained:

• 1 The measured τ_p vary regularly with the sample immersion time in liquid N₂, reaching a constant value after a few hours. This effect is related to the glass matrix relaxation. The rate constant K_isc (S, ₁→T₁) is also found to vary during relaxation of the solvent.
• 2 In the expression giving the nonradiative rate constant K_nr (T₁→S₀), the bimolecular quenching term appears negligible for high viscosity matrices i.e. for η= 10⁹ poises for benzene and for TMPD. K_nr is found to vary linearly with log η, as well as the intersystem crossing S₁→T₁ rate constant K^isc.
• 3 Both K_nr (T₁→S₀) and K^isc (S₁→T₁), increase with decreasing polarity of the solvent.
• 4 From our own observations and literature data[6] for C₆H₆ it appears that solvent viscosity does not contribute appreciably to the observed temperature effect on the solute τ_p when only a monomolecular triplet deactivation is operative.