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.     

The molecular weight dependence of intrinsic viscosity of rod-like micelles of dodecyldimethylammonium chloride

It is shown that the non-linear logarithmic dependence of the intrinsic viscosity on the molecular weight for rod-like micelles of dodecyldimethylammonium chloride (as reported by Ozeki and Ikeda [1]) can be interpreted in terms of the Yamakawa-Fujii theory of worm-like chains. Characteristic parameters of the micelles are estimated: persistence length (a=14 nm), linear mass density (M _L=4800 nm^–1), diameter (d=3 nm), molecular pitch (b=0.052), and the number of surfactant chains in a layer of rod-like micellen=12. The results are compared with those derived from light-scattering measurements.     

Corresponding state relations for the Newtonian viscosity of polymer solutions. II. Further systems and concentrated solutions

In earlier work we have indicated a superposition principle for moderately concentrated mixtures (c ? 2/[η]) in good and poor solvents. By an examination of data on a number of vinyl polymers and cellulose derivatives in good as well as poor solvents, the validity of this principle is extended to concentrated solutions (c ? 50%). The characteristic concentration factor γ is proportional to M over the whole concentration range, with 0.47 ≤ a₁ ≤ 1.10 being larger for good than for poor solvents, the result obtained earlier. Significant deviations from this relationship are noted in good solvents for those low molecular weights at which deviations from the usual intrinsic viscosity relationship occur. This may be related to the expansion factor of the polymer coil. On the basis of these results, the concentration and molecular weight dependence of the viscosity in the concentrated solution can be related to each other in terms of the parameter a₁ and thus to thermodynamic characteristics. In this manner a bridge between the relatively dilute and concentrated regions is established. Currently used semiempirical expressions are analyzed in terms of these results. For the polystyrene–cyclohexane systems and θ ? 9 ≦ T ≦ θ + 3, γ can be identified with the critical concentration for phase separation. Provided an “entanglement” concentration c_e exists, in the neighbourhood of which the concentration dependence of the viscosity changes reapidly, γ can alternatively be shown to be proportional to c_e, or c_e ∝ M. The temperature reduction scheme suggested earlier remains to be investigated.     

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.     

The skewness of polymer molecular weight distributions

A new parameter α₃ for characterization of the skewness of a polymer's molecular weight distribution (MWD) based on the statistics is introduced. For α₃ > 0, the skewness is positive, characterizing the MWD with a tail at higher-MW side. For α₃ = 0, the MWD is symmetric. For α₃ < 0, the skewness is negative, characterizing the MWD with a tail at the lower-MW side. A relationship between α₃ and the first four (from zeroth to third) moments of the MWD is developed which allows calculation of the skewness without detailed calculation of the MWD. An example of polymerization of styrene with n-butyllithium is given to demonstrate the characteristics of α₃.     

Anionic polymerization of caprolactam. XLIII. Relationship between osmometric molecular weight,viscosity, and endgroups of a polymer

For unfractionated anionic polymers, the following relationship between the osmometric molecular weight and intrinsic viscosity is valid: M?_n = 13200[η]^1.115 (cresol), or M?_n = 13000[η]^1.021 (93.8% H₂SO₄). A comparison of the osmometric and viscometric data with the number of endgroups of a polymer confirmed the finding that under certain conditions, moderately branched molecules can be formed; the above parameters depend on the type of the activator used.     

Aging in a free-energy landscape model for glassy relaxation. II. Fluctuation-dissipation relations

Several fluctuation-dissipation relations are investigated for a simple free-energy landscape model designed to describe the primary relaxation in supercooled liquids. The calculations of the response and of the correlation functions are performed for a quench from a high temperature to a low temperature. In the model, all dynamical quantities reach equilibrium after long times, but for times shorter than the re-equilibration time they do not exhibit time-translational invariance and the fluctuation-dissipation theorem is violated. Two measures for these violations are considered. One such measure is given by the slope in a plot of the integrated response versus the correlation function and another one by the so-called fluctuation-dissipation ratio. It is found that these measures do not coincide and furthermore are not independent of the dynamical variable considered in the calculation. We propose to determine the fluctuation-dissipation ratio experimentally via measurements of the deuteron spin-lattice relaxation rate and the dielectric loss.     

The relation between the molecular weight and intrinsic viscosity of polyvinyl alcohol

Summary A series of unfractionated and fractionated samples of polyvinyl acetate were prepared by homogeneous solution polymerization of vinyl acetate in N,N-dimethylformamide at 90 °C. in presence of , -azo-(-cyano-n-valeric acid) as initiator under a variety of conditions. A portion of each sample was hydrolysed to polyvinyl alcohol. The number average molecular weights have been determined by end-group titrations. The following molecular weight-intrinsic viscosity relationships for polyvinyl alcohol have been obtained. [] = 33.88 X 10^–5 M ^0.716 — for unfractionated samples in water at 30 °C. [] = 29.51 X 10^–5 M ^0.716 — for fractionated samples in water at 30 °C.With 1 figure and 2 tables     

The relation between the molecular weight and intrinsic viscosity of polymethyl acrylate

Summary A series of unfractionated and fractionated samples of polymethyl acrylate of different low molecular weights have been prepared by homogeneous solution. polymerization in dimethyl formamide in presence of , '-azo-(-cyano-n-valeric acid) as initiator under a variety of conditions. The number-average molecular weights have been determined by end-group titrations and vapour pressure osmometry. The following []-M relationships for polymethyl acrylate have been obtained.[] = 33.5 x 10^–5M^0.63 for unfractionated samples in benzene at 25 °C. [] = 3.89 x 10^–5M^0.843 for fractionated samples in benzene at 35 °C.With 2 tables     

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.     

Characterization of polymer molecular weight distributions by ratios of molecular weight averages

The molecular weight (MW) distribution of a polymer is characterized by a hierarchy of average MWs and their appropriate combinations. For example, the ratio of the weight-average to the number-average MW is the most frequently used measure of the polydispersity of a polymer. As is well known the lower bound to this ratio is unity, and it has been shown that the upper bound is (m + 1)²/4m, where m = M_max/M_min is the ratio of the highest to the lowest MW of the MW species present in a given polymer. This upper bound corresponds to an extremely bimodal MW distribution of one half weight fraction with M_min and the other half with M_max. The behavior of the upper bound for two special unimodal distributions is investigated: one is the triangular distribution, the other the quadrilateral. The results suggest that the upper bound for all possible unimodal distributions is considerably less than the corresponding general case, especially for large values of m. For example, the maximum ratios for the quadrilateral distribution and the general upper bound are 1.04 and 1.125 for m = 2; 1.43 and 3.205 for m = 10; 2.56 and 25.5 for m = 100; 3.99 and 250.5 for m = 1000, respectively.     

Effect of polymer molecular weight on the polymer/surfactant interaction

A thermodynamic analysis of the interaction between fourteen different molar mass poly(ethylene oxide)s (PEO) and sodium dodecyl sulfate (SDS) based on the measured surfactant-binding isotherms is given. The surfactant-binding isotherms were determined by the potentiometric method in the presence of 0.1 M inert electrolyte (NaBr). It was found that there is no PEO/SDS complex formation if M(PEO) < 1000. In the molecular weight range 1000 < M(PEO) < 8000, the critical aggregation concentration (cac) and the surfactant aggregation number are decreasing as the polymer molecular weight increases. The saturated bound surfactant amount is proportional to the number concentration of the polymer in this molecular weight range. If M(PEO) exceeds approximately 8000, the cac does not depend on the polymer molar mass, and the saturated bound amount of the surfactant becomes proportional to the mass concentration of the polymer. It was also observed that independently of the polymer molecular weight the surfactant aggregation number increases as the equilibrium surfactant monomer concentration increases from the cac to the critical micellar concentration (cmc). Finally, it was demonstrated that only one polymer molecule is involved in the complex formation independently of the polymer molecular weight.     

Temperature dependence of viscosity for sucrose laurate/water micellar systems

The viscous behavior of sucrose laurate aqueous systems of high hydrophilic-lipophilic balance up to a 50% (wt) surfactant concentration at temperatures between 5°C and 60°C has been studied. Systems up to a 45% (wt) surfactant concentration show Newtonian behavior. The influence of temperature was studied using the activated diffusive relaxation model described by Goodwin. A maximum specific viscosity that appears at lower temperature as sucrose laurate concentration increases can be observed. These results are related to the micellar growth of the sucrose laurate aggregates as temperature rises. More concentrated systems show complex viscous response. Thus, a limit viscosity at low shear rates and a shear-thinning behavior after a critical shear rate are observed. Limit viscosity decreases and critical shear rate increases as temperature rises. This behavior is related to the threshold micelle concentration for entanglement of rod-like micelles.Nomenclature A Parameter of the equation that relatesE and temperature - B Pre-exponential factor of the Arrhenius equation - C Sucrose ester concentration (kg · m^–3) - CMC Critical micelle concentration - E Activation energy for long-range diffusive motion (Goodwin model) - E _a Activation energy of the viscous flow (Arrhenius equation) - E ₀ Parameter of the equation that relatesE and temperature - HLB Hydrophilic/lipophilic balance of the sucrose ester - J Constant that depends on the aqueous phase viscosity and mean micellar radius - k Boltzmann's constant - k ₁ Parameter of the Goodwin equation - k ₂ Parameter of the Goodwin equation - q _rel Contribution of the hydrodynamic interactions - R _e External radius of the sensor system - R _i Inner radius of the sensor system - T Temperature - T _max Temperature at the maximum viscosity - Newtonian viscosity - _i Intrinsic viscosity - _rel Relative viscosity = _solution/_water - _red Reduced viscosity = _sp/C - _sp Specific viscosity = _rel – 1 - ₀ Zero-shear-rate viscosity     

Molecular rheology of concentrated polymer systems. I

The primitive chain model of Doi and Edwards is generalized to include the short-time relaxation process. Stress relaxation after a sudden imposition of strain is studied in detail. It is shown that in the linear region (small strain) stress relaxation occurs in two steps, the relaxation of chain segments between the fixed entanglement points, and the relaxation of the entanglement points, in accordance with the conventional picture, whereas in the nonlinear region (large strain) there appears a new relaxation process between the above two. The characteristic time of this process is the Rouse relaxation time which the entire chain would have if there were no entanglements, and increases with the square of the molecular weight. This result is consistent with experimental observations.