Intrinsic viscosity in branched free-radical polymers

The intrinsic viscosity ratio [η]_B/[η]_L was calculated as a function of average branching density for trifunctionally branched, free-radical polymers. Calculations were made for the g^1/2, g^3/2, and h³ rules, using realistic distributions of molecular weights and branches. Experimental data on branched poly(vinyl acetate) lay between the curves obtained from the g^1/2 and h³ relations.     

Determination of polymer branching with gel-permeation chromatography. II. Experimental results for polyethylene

The recently developed methods of characterizing branching in polymers from gelpermeation chromatography and intrinsic viscosity data are verified experimentally. An iterative computer program was written to calculate the degree of branching in whole polymers. Long-chain branching in several low-density polyethylene samples was determined by both the fraction and whole polymer methods. The two methods gave consistent ranking of the branching in the samples although absolute branching indices differed. Effects of various experimental errors and the particular model used for branching were investigated. For polyethylene, the data show that the effect of branching on intrinsic viscosity is best described by the relation 〈g₃〉_W^1/2 = [η]_br/[η]₁ where 〈g₃〉_w is the weight-average ratio of mean-square molecular radii of gyration of linear and trifunctionally branched polymers of the same weight-average molecular weight.     

Intrinsic viscosity of polydisperse branched polymers

The relationships between molecular weight distribution and structure in polymerizations with long-chain branching were reviewed and extended. Results were applied to an experimental examination of intrinsic viscosity in polydisperse, trifunctionally branched systems. Several samples of poly(vinyl acetate) were prepared by bulk polymerization under conditions of very low radical concentration. The relative rate constants for monomer transfer, polymer transfer, and terminal double-bond polymerization were established from the variation of M _n and M _w with the extent of conversion. Average branching densities were then calculated for each sample and ranged as high as 1.5 branch points/molecule. Intrinsic viscosities [η]_B were measured in three systems: a theta-solvent, a good solvent, and one that was intermediate in solvent interaction. These results were compared with calculated viscosities, [η]_L, which would have been observed if all the molecules had been linear. The values of [η]_B/[η]_L were substantially the same in all three solvents. The variation of this ratio with branching density was compared with the theory of Zimm and Kilb as adapted to polydisperse systems. Discrepancies were noted, and the adequacy of present model distribution functions for branched polymers was questioned.     

Properties of cellulose acetate in solution. I. Light scattering,osmometry, and viscometry on dilute solutions

Light-scattering, osmotic pressure, and viscometric studies on fractions of cellulose acetate (degree of substitution 2.45) in three solvents are described. The data yield the dependence of the mean-square radius of gyration 〈s²〉, the second virial coefficient Γ₂, and the intrinsic viscosity [η] on molecular weight M and temperature. The results are interpreted to show that excluded volume effects on 〈s²〉 are negligible, even though Γ₂ is large and dΓ₂/dT is positive. The large experimental value of d In [η]/d In M is interpreted in terms of partial draining effects. Data on 〈s₂〉 and [η] for other cellulose esters in the literature are similarly interpreted. Significant aggregation found in solutions of cellulose acetate in many solvents is discussed.     

Synthesis and solution properties of linear,four-branched,and six-branched star polyisoprenes

A number of linear, four- and six-branched regular star polyisoprenes were synthesized by anionic polymerization techniques in benzene using lithium as the counterion and polyfunctional silicon chloride compounds as the coupling agents. Light-scattering measurements in dioxane were performed in order to establish Θ solvent conditions. Determinations of the radius of gyration of the polymers of different structure indicate that g = 〈S²〉_0,br/〈S²〉_0,lin agree closely with random flight calculations for the ratios. Intrinsic viscosities determined in a Θ solvent establish g′ = [η]_br/[η]_lin to be 0.77₃ and 0.62₅ for the four- and six-branched polyisoprenes, respectively. In a good solvent g′ values are slightly lower. These values are compared with theoretical estimates. Viscosities of 19.29% (w/w) solutions of the polyisoprenes in n-decane at 25°C are correlated with the intrinsic viscosities of the polymers under Θ conditions.     

Temperature dependence of limiting viscosity number and radius of gyration of cellulose diacetate in acetone

Acetone solutions of a cellulose diacetate fraction were studied by viscosity and light scattering methods over the range 12.6–50.3². The temperature dependences of the limiting viscosity number [η], the mean-square radius of gyration 〈s²〉, and the second virial coefficient A₂ were determined. The unperturbed mean-square radius of gyration 〈s²〉_o and the expansion factor α, were estimated by using theoretical relations to the interpenetration function. It was found that dln 〈s²〉_o/dT is ?6.4 × 10^?3 deg^?1, while α, is close to unity over the whole temperature range studied. The viscosity results are interpreted to show that the draining effect is not negligible and the Flory viscosity parameter Φ slightly increases with increasing temperature. It is finally concluded that the value of ?6.9 × 10^?3 deg^?1 for dln [η]/dT can be ascribable to the rapid decrease in 〈s²〉_o.     

Hydrodynamic interaction effects on intrinsic viscosity of perturbed chains: Experimental results for polystyrene

Experimental measurements of intrinsic viscosity and radii of gyration of monodisperse samples of polystyrene of molecular weights 2.33 X 10⁵, 4.11 X 10⁵, 6.70 X 10⁵, and 2.3 X 10⁶ dissolved in the homologous series of 1-chloroparaffins from butane to undecane are reported. The dependence of the viscosity expansion coefficient α_η on the expansion factor α_s for the radius of gyration is discussed in the light of the results obtained by an expansion of the Fixman theory for perturbed chains to include partial draining. These results give support to the finite chain effect on the hydrodynamics of expanded coils in the usual range of molecular weight. The exponent a in the relation α³_η = α^a_s depends on molecular weight and agrees with recent nondraining theoretical calculation for exceedingly high polymers.     

Evaluation of K⊖ from [η]-M-data of binary and ternary polymer systems

Unperturbed dimensions of flexible linear macromolecules can be obtained from [η]-M-data in any solvent, good or poor, single or mixed. Usually K_θ is estimated by a relationship between [η]/M_W^0.5 and M_w^0.5 first proposed by Burchard and by Stockmayer and Fixman. But, it is well-known that the Burchard-Stockmayer-Fixman-plot shows downward curvature, especially for good solvent systems. Various efforts have been made to achieve relations with better linearity. One of the first was the semi-empirical relation between ([η]/M_w^0.5)^0.5 and M_w/[η] by Berry. Predicting a relationship of the excluded volume parameter z to the viscosity expansion factor by α⁵_η instead of α⁵_η Tanaka obtains that ([η]/M_w^0.5)^5/3 is linear in M_w^0.5. By allowing for the dependence of the viscometric interaction parameter B, which is correlated to the second virial coefficient A₂, on molar mass, Gavara, Campos and Figueruelo predict a linear dependence of [η]/M_w^0.5 against A₂.M_w^0.5. It is not our intention here to discuss the validity of these theories, but to compare them with experimental data.     

Polyelectrolyte behavior and macromolecular properties of sodium amylopectin xanthate in dilute solution

Sodium amylopectin xanthate was prepared by xanthation of potato amylopectin in alkaline medium. The pure product was characterized by I₂ solution and ultraviolet spectra of the xanthate groups. The polyelectrolyte behavior of Na amylopectin xanthate in aqueous and salt solutions was investigated by viscometry and light scattering. Its polyelectrolyte behavior in aqueous solution as studied viscometrically was completely different from that of Na amylose xanthate, which is characteristic of linear polyelectrolyte molecules. This difference in behavior could be partly due to the branched structure of amylopectin molecule. A dissymmetry study of Na amylopectin xanthate in aqueous and salt solutions, however, showed that Na amylopectin xanthate molecule underwent expansion in water by about 1.3 times its linear dimension in 0.5M NaCl (unperturbed value). Light-scattering measurements confirmed that the Na amylopectin xanthate molecule had a polydisperse random-coil chain configuration in 0.25M NaCl. Its molecular weight, end-to-end length, and other parameters in salt and alkali solutions were also determined, and the data were then compared with those of Na amylose xanthate in the same media. The solution behavior of Na amylopectin xanthate in 1M NaOH was further investigated and linear expansion factor α, excluded volume factor A₂M _W/[η], and Flory's hydrodynamic constant φ were evaluated.     

Combination of nitroxide‐mediated polymerization and SET‐LRP for the synthesis of high molar mass branched and star‐branched poly(n‐butyl acrylate) characterized by size exclusion chromatography and rheology

Branched and star‐branched polymers were successfully synthesized by the combination of two successive controlled radical polymerization methods. A series of linear and star poly(n‐butyl acrylate)‐co‐poly(2‐(2‐bromoisobutyryloxy) ethyl acrylate) statistical copolymers, P(nBA‐co‐BIEA)_x, were first synthesized by nitroxide‐mediated polymerization (NMP at T > 100 °C). The subsequent polymerization of n‐butyl acrylate by single electron transfer‐living radical polymerization (SET‐LRP at T = 25 °C), initiated from the brominated sites of the P(nBA‐co‐BIEA)_x copolymer, produced branched or star‐branched poly(n‐butyl acrylate) (PnBA). Both types of polymerizations (NMP and SET‐LRP) exhibited features of a controlled polymerization with linear evolutions of logarithmic conversion versus time and number‐average molar masses versus conversion for final M_n superior to 80,000 g mol^?1. The branched and star‐branched architectures with high molar mass and low number of branches were fully characterized by size exclusion chromatography. The Mark–Houwink Sakurada relationship and the analysis of the contraction factor (g′ = ([η]_branched/[η]_linear)_M) confirmed the elaboration of complex PnBA. The zero‐shear viscosities of the linear, star‐shaped, branched, and star‐branched polymers were compared. The modeling of the rheological properties confirmed the synthesis of the branched architectures. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

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.     

Chain dimensions and transport coefficients of sodium poly(isoprenesulfonate) in aqueous sodium chloride

Sodium poly(isoprenesulfonate) (NaPIS) fractions consisting of 1,4‐ and 3,4‐isomeric units (0.44:0.56) and ranging in molecular weight from 4.9 × 10³ to 2.0 × 10⁵ were studied by static and dynamic light scattering, sedimentation equilibrium, and viscometry in aqueous NaCl of a salt concentration (C_s) of 0.5‐M at 25 °C. Viscosity data were also obtained at C_s = 0.05, 0.1, and 1 M. The measured z‐average radii of gyration 〈S²〉_z^1/2, intrinsic viscosities [η], and translational diffusion coefficients D at C_s = 0.5‐M showed that high molecular weight NaPIS in the aqueous salt behaves like a flexible chain in the good solvent limit. On the assumption that the distribution of 1,4‐ and 3,4‐isomeric units in the NaPIS chain is completely random, the [η] data for high molecular weights at C_s = 0.5 and 1 M were analyzed first in the conventional two‐parameter scheme to estimate the unperturbed dimension at infinite molecular weight and the mean binary cluster integral. By further invoking a coarse‐graining of the NaPIS molecule, all the [η] and D data in the entire molecular weight range were then analyzed on the basis of the current theories for the unperturbed wormlike chain combined with the quasi‐two‐parameter theory. It is shown that the experimental 〈S²〉_z, [η], and D are explained by the theories with a degree of accuracy similar to that known for uncharged linear flexible homopolymers. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2071–2080, 2001     

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.     

Excluded volume study of a sulfonate-containing polyelectrolyte in salt solutions

Chain characteristics of a linear sulfonate-containing homopolymer, sodium poly(3-methacryloyloxypropane-1-sulfonate), in aqueous salt solutions (ionic strength, C_s = 0.01N to 5N NaCl) have been investigated by light scattering and intrinsic viscosity. The molecular weight (M?_w)–viscosity relation can be well described by the Mark–Houwink and the Stockmayer–Fixman equations. The coil is highly expanded even in the most concentrated NaCl solution (6N), and no 1:1 electrolyte was found to precipitate this polymer. A linear relation was observed between the viscosity expansion factor, α^3η, and (M?_w/C_s)^1/2. Examination of the data in terms of theories for excluded volume and hydrodynamic interaction suggests that the coil experiences dominant hydrodynamic interaction, corresponding to a nondraining coil, and the second virial coefficient and coil expansion at high C^s can be correlated by the Flory–Krigbaum–Orofino equation. Results for this polymer are compared with those for other polyelectrolytes, and are discussed in terms of chain structure, flexibility, and hydrophobicity.     

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.     

Dilute solution properties of a polybenzimidazole

Light scattering and viscometric studies have been carried out on dilute solutions of a polybenzimidazole in N,N-dimethylacetamide. The data, which span the molecular weight range 2.9 ≦ 10^?4M_w ≦ 23.3, and the temperature range 290 ≦ T/K ≦343, yield the dependence of the mean-square radius of gyration 〈s²〉_LS, the second virial coefficient A₂, and the intrinsic viscosity [η] on molecular weight M_w and temperature. The unperturbed mean-square radius 〈s〉_LS was calculated using experimental values of 〈s²〉_LS and A₂. It was found that excluded volume effects on 〈s²〉_LS are very small. The unperturbed hydrodynamic chain dimension 〈s〉_η was estimated by considering draining effects. A small value of the draining parameter was obtained. Analysis of the temperature dependence of A₂ and [eta;] leads to the conclusion that this system approaches a lower theta temperature with increasing temperature. The steric factor σ = 〈s〉/〈s〉f, based on the value of 〈s〉_f calculated for the polymer chain with free rotation, is nearly unity. Most of these properties can be interpreted in terms of long rotational units within the main chain.     

Perturbation expansion of polymers in shear flow

The effect of shear flow on the excluded volume properties of bead spring chains is calculated via the perturbation expansion of the mean-square end-to-end distance with respect to the excluded volume parameter z. The coefficient of the series 〈R²〉/〈R²〉₀ = 1 + C₁ · z − … vanishes for large flow rates but shows a strong dependence on the shear rate in the intermediate regime.     

Hydrodynamic properties of regular star-branched polymer in dilute solution

We calculate quantities such as g = [η]_b/[η]_l and h = (f_t)_b/(f_t)_t for regular star-branched polymer with and without excluded volume. We have applied a numerical method introduced by Barrett for the linear chain and have solved the integral equations which are conducive to calculate the translational coefficient friction and the intrinsic viscosity in the Kirkwood-Riseman theory. We utilize preaveraging but avoid other approximations. In general, we obtain values which have a better accord with experimental data than traditional Kirkwood-Riseman and Zimm-Kilb formulas. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys, 35: 563–567, 1997     

Organometallic polymers I. Solution polymerization of α-ferrocenylmethylcarbonium fluoborate

The self-condensation of α-ferrocenylmethylcarbonium ion in nitroethane yielded polymers of M_n up to 20,000. The change of [η] and M_n with the reaction time indicated that the process consisted of a rapid primary growth stage, an induction period, a second growth stage, and a crosslinking stage. The [η]–M_n correlation for a series of polymeric fractions in the M_n = 0.1–7.2 × 10⁴ range points to a highly branched structure.     

Excluded volume effect of linear polymer molecules

An approximate closed expression for the excluded volume effect of linear polymer molecules is developed with the aid of a uniform expansion model of perturbed chains. The linear expansion factor α for the end-to-end distance is given by (α³ ? 1) + (3/8) (α⁵ ? α³) = (5/2)z where z is the excluded volume parameter. This equation is numerically close to the Ptitsyn equation in the ordinary range of α; i.e., for 1 ≤ α ≤ 2.