Dependence of viscosity of concentrated polymer solutions upon molecular weight and concentration

Molecular weight M and concentration c dependencies of the zero-shear viscosity (η) were measured over wide ranges of M and c for concentrated solutions of linear and branched poly(vinyl acetate) as well as of polystyrene under θ conditions. The log η versus log M and log η versus log c curves for a given system can be superposed by the horizontal shift along the abscissa, giving smooth master curves. From the shift factors the ratio of two exponents β and α, which appear in the following equation, can be evaluated: η = K′(cρ)^αM^β, where ρ is the density of the solution and K′ is a constant at constant temperature. The evaluated values of β/α for the systems under θ conditions are equal to or very close to 0.50 as was anticipated from the previous work. The above superposition method was also applied to available viscosity data, and it was found that β/α had a good correlation with a in [η] = KM^a. This indicates that the individual molecules in concentrated solutions maintain the same individuality as in dilute solutions, and might be a positive support to the packed sphere model proposed previously by the authors. The effect of solvent on the molecular weight and the concentration dependencies of viscosity was also discussed.     

Generalized correlations for molecular weight and concentration dependence of zero-shear viscosity of high polymer solutions

Data are presented to show that two correlations of viscosity–concentration data are useful representations for data over wide ranges of molecular weight and up to at least moderately high concentrations for both good and fair solvents. Low molecular weight polymer solutions (below the critical entanglement molecular weight M_c) generally have higher viscosities than predicted by the correlations. One correlation is η_sp/c[η] versus k′[η], where η_sp is specific viscosity, c is polymer concentration, [η] is intrinsic viscosity, and k′ is the Huggins constant. A standard curve for good solvent systems has been defined up to k′[η]c ≈? 3. It can also be used for fair solvents up to k′[η]c ≈? 1.25· low estimates are obtained at higher values. A simpler and more useful correlation is η_R versus c[η], where η_R is relative viscosity. Fair solvent viscosities can be predicted from the good solvent curve up to c[η] ≈? 3, above which estimates are low. Poor solvent data can also be correlated as η_R versus c[η] for molecular weights below 1 to 2 × 10⁵.     

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.     

Relaxation characteristics and intrinsic birefringence and viscosity of polystyrene solutions for a wide range of molecular weights

A comparison is made between measurements on polystyrene solutions and the relaxation characteristics and intrinsic birefringence and viscosity given by the theory for the flexible Gaussian chain of variable number of segments and with internal viscosity and internal hydrodynamic interaction. This is done in order to determine the applicability of the theory to polymers over a wide range of molecular weights, including the low molecular weight range in which there may be conflict with the theoretical assumption of chains having a large number of segments. The longest, terminal relaxation time and the number of chain segments are determined from measurements of the frequency dependence of oscillatory flow birefringence while the intrinsic birefringence and viscosity are determined from steady flow measurements. The range of molecular weights studied is from approximately 900 at 10⁶. It is found that the segment weight is approximately 1000 and the number of segments is in direct proportion to the molecular weight for the range from 1 to 1000 segments. The terminal relaxation time has a molecular weight dependence of the type given by the theory but with better agreement for higher molecular weights. While the measured dependences of the intrinsic viscosity and birefringence are in agreement with theory for molecular weights greater than 5 × 10⁴, they deviate significantly for molecular weights below 1 × 10⁴. The ratio of the intrinsic birefringence to intrinsic viscosity, which in theory is a constant independent of molecular weight, is found to change at the lower molecular weights.     

Moderately concentrated solutions of polystyrene. II. Integrated-intensity light scattering as a function of concentration,temperature, and molecular weight

Integrated-intensity light scattering data are reported for moderately concentrated solutions of polystyrene in benzene and in cyclopentane. The benzene system is one for which the second virial coefficient A₂ is large; data obtained over the range 0.5 < A₂Mc < 30, with c the polymer concentration, are analyzed in terms of the (extrapolated) intensity at zero angle and the angular dependence of the intensity. The former is discussed in terms of power law representations based on scaling relations, which are found to represent the data. The latter is discussed in terms of the dependence of the chain dimensions on concentration. With cyclopentane, the behavior is similar for temperatures for which A₂ is near its maximum, but for T near either Θ_U or Θ_L, for which A₂ is zero or small, the angular dependence of the scattering is distinctly different, with the intensity exhibiting a maximum as a function of scattering angle.     

Rotational diffusion of 9,10-diphenylanthracene in di-n-butyl phthalate solutions of polystyrenes by the fluorescence polarization method: Dependence on polymer concentration and molecular weight

A versatile double-beam polarization fluorimeter has been constructed for measuring the polarization of fluorescence from polymer solutions, melts, and glasses. Polarizations can be determined over a range of temperatures from ?20 to +80°C in a controlled atmosphere with a precision of ±0.001 to ±0.005 for the studies reported herein. Data collected at different temperatures for 1.5 × 10^?5M solutions of 9,10-diphenylanthracene (PA) in di-n-butyl phthalate (BP) fit a relation of the Perrin type, 1/P = (1/P₀) + (ST/_η1), where P is the polarization, T is the absolute temperature, and _η1 is the solvent viscosity. The constants P₀ and S were 0.400 ± 0.005 and (7.4 ± 0.3) × 10^?3 P/°K, respectively. Polarizations were also determined at 25.0 ± 0.1°C for BP solutions containing 1.5 × 10^?5M PA and polystyrenes at various weight fractions w₂ and molecular weights M. Rotational friction coefficients ζ_r deduced from these data showed no dependence on M from 5.1 × 10⁴ to 8.6 × 10⁵ g/mole, and a gradual increase as w₂ was varied from 0 to 0.1. It is concluded from these results that PA is an especially attractive emitter for rotational diffusion studies in nonaqueous systems, and that the abrupt changes in ζ_r with w₂ and M observed for some other emitter–polymer systems and attributed to onset of coil overlap are not universal characteristics of such systems.     

Dilute solutions of nylon 12—II. Relationship between intrinsic viscosity and molecular weight

Samples of poly(12-dodecanelactam) (polylaurolactam, nylon 12) with $\text{M}$_n 1 × 10³?33 × 10³ were prepared. Polymerizations initiated with water or with lauric acid proceeded under conditions for minimum changes in end-group concentration. Values of $\text{M}$_n were calculated from the end-group content and $\text{M}$_n from light scattering in the mixture m-cresol/60 vol% of 2.2,3,3-tetrafluoropropanol. From measurements of intrinsic viscosity in m-cresol, the relationship [n] ? $\text{M}$_n was established in the given range of $\text{M}$_n. The relationship [n] ? $\text{M}$_w for $\text{M}$_w from 3.3 × 10³ to 125 × 10³ has been established.     

Viscoelastic behavior of low molecular weight polystyrene

The shear creep and creep recovery behavior of narrow molecular weight distribution polystyrene samples of low molecular weight, 1.1 × 10³, 3.4 × 10³, and 1.57 × 10⁴ are reported as a function of temperature, near and above the glass temperature. Time-temperature equivalence for the total creep compliance is found to be nonapplicable, and in fact the steady-state recoverable compliance, J_e, is a strong function of temperature. The time-scale shift factors for the recoverable compliance are analyzed in the light of free volume theory. Viscosity data are presented for samples with molecular weights between 1.1 × 10³ and 6.0 × 10⁵. The temperature dependence of the characteristic time constant ηJ_e can be explained in terms of free volume concepts whereas that of viscosity η cannot. Effects of residual molecular weight heterogeneity are demonstrated.     

Solution properties of high molecular weight polystyrene

A series of polystyrenes with weight-average molecular weight M?_w up to 1.3 × 10⁷ was prepared by anionic polymerization in tetrahydrofuran (THF). Each sample was characterized by gel-permeation chromatography, light scattering, and viscometry. It was found that each sample had an almost symmetrical and very narrow molecular weight distribution (M?_w/M?_n < 1.07). The mean-square unperturbed radius of gyration 〈S²〉₀ was determined in trans-decalin at 20.4°C as 〈S²〉₀ = 7.8₆ × 10^?18M?_w (cm²). The particle scattering factor was well represented by the Debye equation irrespective of solvent in the range of M?_w < 4 × 10⁶, and only a small deviation was observed in benzene at higher molecular weights. The penetration function Ψ ≡ A₂M²/4π^3/2N_A〈S〉^23/2 was found to approach a relatively low asymptotic value of 0.21–0.23 at molecular weights above 2 × 10⁶ in benzene at 30°C, where A₂ is the second virial coefficient and N_A is Avogrado's number. It was also found that the theta temperature in trans-decalin was affected by the nature of polymer samples. A difference of about 3°C in the theta temperature was observed between two series of anionic polystyrenes, one prepared in THF and the other in benzene, but there was practically no difference in unperturbed chain dimension.     

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.     

Effect of temperature and molecular weight on enthalpy relaxation in polystyrene

Isothermal enthalpy relaxation in polystyrene was measured as a function of temperature and molecular weight on a differential scanning calorimeter. Relaxation spectra were derived from the data and expressed as a distribution of relaxation times. For a given molecular weight the relaxation spectra at different temperatures could not be superimposed by a shift in time. The relaxation curves of samples of different molecular weights could be superimposed only when the difference between the temperature at which the relaxation was monitored (T_a) and their respective T_g was the same. The relaxation spectrum at any temperature for a given molecular weight was also expressed as a distribution of energies. The average energy represented by this distribution was associated with an activation energy required for the motion of a chemical repeat unit. The activation energy extracted from the temperature shift in the relaxation spectra corresponded to the motion of a statistical unit (Kuhn's segment) in polystyrene.     

The effect of polyethylene oxide molecular weight on determination of its concentration in aqueous solutions

The influence of the molecular weight of polyethylene oxide (PEO) on the results from several methods for determining its concentration in aqueous solutions has been investigated. Modified versions of the complexation reactions of PEO with molybdophosphoric acid and with tetraiodobismuthate give results that are independent of molecular-weight effects for the range ca. 400-10(6). The reaction with KBiI(4) is the less accurate. Oxidative digestion is also independent of molecular-weight, but impurities easily obscure the measurements. Other methods studied included interferometry, viscometry, and complexation with tannic acid. Interferometry and the reaction with tannic acid were only independent of molecular-weight for M > 4000. With viscometry very poor results were obtained.     

Pressure dependence of the viscosity of dilute polystyrene solutions in toluene

The effect of pressure on the viscosity of dilute solutions of anionically polymerized polystyrene (M?_w = 209,000; M_w/M_n = 1.12) in toluene has been studied at different temperatures and concentrations using a falling-body viscometer. Measurements were performed in the concentration range from 0.0025 to 0.02 g/mL and at temperatures from 25 to 45°C under pressure up to 1057 bars. The viscosity coefficient η increases exponentially with pressure at a given temperature and concentration, while the apparent volume of activation V^? decreases with increasing temperature. The hypothesis that the pressure dependence of η is given by the pressure dependence of the activation energy holds true under the prevailing thermodynamic conditions. Log η increases linearly with increasing concentration at a given pressure. Intrinsic viscosity increases with increasing pressure, whereas the Huggins constant decreases.     

Intrinsic viscosity of polymers of low molecular weight

Experimental evidence concerning the dependence of the intrinsic viscosity [η] on molecular weight M in the low molecular weight range (from oligomers to M = 5 × 10⁴) has been collected in a variety of solvents for about ten polymers, i.e., polyethylene, poly(ethylene oxide), poly(propylene oxide), polydimethylsiloxane, polyisobutylene, poly(vinylacetate), poly(methyl methacrylate), polystyrene, poly-α-methylstyrene, and some cellulose derivatives. In theta solvents, the constancy of the ratio [η]Θ/M^0.5 extends down to values of M much lower than those predicted by current hydrodynamic theories. In good solvents, and on decreasing M, the polymers examined, with the exception of polyethylene and some cellulose derivatives, show a decrease in the exponent a of the Mark-Houwink equation [η] = KM^a. This upward curvature gives rise to the existence of a more or less extended linear region where the equation [η] = K₀M^0.5 is obeyed. Below the linear range, i.e., for even shorter chains, the exponent a can increase, i.e., polydimethylsiloxane, or decrease below 0.5, i.e., poly(ethylene oxide), depending on the particular chain properties. These different dependences have been discussed in terms of: (a) variations of thermodynamic interactions with molecular weight; (b) variations of conformational characteristics (as for instance the ratio) 〈r0²/nl²〉, where 〈r0²〉 is the unperturbed mean square end-to-end distance and n is the number of bonds each of length l; (c) hydrodynamic properties of short chains.     

Dependence of the volume and viscosity of naphthalene-ethanol-octane solutions on composition at 298 K

The solubility of naphthalene in ethanol-octane mixtures was measured by the isothermal saturation method. The solution densities were determined and the partial and apparent molar volumes of naphthalene were calculated. The viscosity was measured with an Ubbelohde viscosimeter with a suspended level. All measurements were performed at 298.15 K. The results were discussed based on interactions in solution.