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.     

Dilute solution properties of branched polymers

For calculating the ratio of the intrinsic viscosities of branched and linear polymers of the same molecular weight, [η]_B/[η]_L, a new theory taking into account the excluded volume effect is presented. By using the modified Flory equation, the excluded volume effect of branched polymers is predicted with the aid of the first-order perturbation theory. The linear expansion factor α_s is converted to the hydrodynamic expansion factor α_η by using the Kurata-Yamakawa theory. Our calculated results, i.e., [η]_B/[η]_L and 〈s²〉_B/〈s²〉_L, agree well with experiment for various type branched polymers, i.e., randomly branched and comb-shaped polymers of poly(vinyl acetate).     

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 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.     

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.     

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.     

Molecular weight influence on viscosimetric parameters of poly(4-vinylpyridine) polymers

This work describes the effect of the molecular weight on the viscosimetric parameters of poly(4-vinylpyridine) (P4VP) polymers in ethanolic solution. Numerous studies concerning this question have been reported in very separate intervals of molecular weight. We have observed a discordance (discontinuity) in the variation of the intrinsic viscosity as a function of the molecular weight of these polymers ([η]=f(M_w)). In order to establish a general relationship between viscosimetric parameters and M_w, we have considered 10 P4VP samples in a wide interval of molecular weights: 0.75×10⁴ to 153×10⁴. These results have been compared and completed with that of the literature. We have observed that:

(i)

All viscosimetric parameters (intrinsic viscosity [η], Huggins constant k_H, second virial coefficient, viscosimetric expansion coefficient α_η, and critical concentration) change according to a continuous function without a break.

(ii)

The lower is the molecular weight of P4VP; the higher are the variations of the expansion coefficient and the interaction effects.

(iii)

The variation of the intrinsic viscosity versus the molecular weight follows a unique relation in the whole M_w range. In fact, the Berkowitz equation (1), described for a limited range of relatively high M_w (10⁵ to 18.5×10⁵) is extended for all M_w interval values.

(iv)

Empiric laws for [η], k_H, A₂ and C^* and variations as a function of molecular weight were proposed for the P4VP in ethanol.

     

Structural rigidity control in arborescent graft polymers

Successive chloromethylation-anionic grafting sequences on polystyrene have led to well-defined hyperbranched macromolecules. Polymers with branching functionalities f > 5000 and molar masses over 10⁷ g/mol were thus prepared while maintaining a ratio of mass- to number-average molar masses M_w/M_n = 1,1-1,3. Three series of styrene polymers were prepared by varying the density of grafting sites along the chain, resulting in controllable stiffening of the molecular structure. Characterization of the branched molecules by size-exclusion chromatography/low-angle laser light scattering and light scattering confirmed a geometric increase in molar mass for successive generations, and a low ratio of mass- to number-average molar masses. Static and dynamic light scattering experiments showed that the molecules behave like hard spheres in dilute solutions. Measurements in the semidilute range, however, showed a progressive structural stiffening effect as the branching density increases.     

Dynamic birefringence and rigidity of poly-p-benzamide molecules in solutions

Flow birefringence (FB) in solutions of a number of samples of poly-p-benzamide (PPBA) in sulphuric acid has been investigated; intrinsic viscosities [η] of the same solutions have been measured.Characteristic values of FB, [n], and orientation angles [χ/g] of the solutions were determined. Molecular weights of all the PPBA samples were calculated by using the values of [χ/g] and [η]. The dependence [η] = 1.6 × 10^?5 M^1.7 shows that the conformation of PPBA molecules in solution is close to that of a straight rod.The use of characteristic values of [n] and [χ/g] and the theory of optical anisotropy of persistent chains gives quantitative data on the equilibrium rigidity of PPBA molecules. It was shown that the number of molecular units in a statistical segment of the PPBA chain is 320 and the corresponding length of the segment is 2000 Å.     

Extension of the chain‐end,free‐volume theory for predicting glass temperature as a function of conversion in hyperbranched polymers obtained through one‐pot approaches

Compared with linear polymers, more factors may affect the glass‐transition temperature (T_g) of a hyperbranched structure, for instance, the contents of end groups, the chemical properties of end groups, branching junctions, and the compactness of a hyperbranched structure. T_g's decrease with increasing content of end‐group free volumes, whereas they increase with increasing polarity of end groups, junction density, or compactness of a hyperbranched structure. However, end‐group free volumes are often a prevailing factor according to the literature. In this work, chain‐end, free‐volume theory was extended for predicting the relations of T_g to conversion (X) and molecular weight (M) in hyperbranched polymers obtained through one‐pot approaches of either polycondensation or self‐condensing vinyl polymerization. The theoretical relations of polymerization degrees to monomer conversions in developing processes of hyperbranched structures reported in the literature were applied in the extended model, and some interesting results were obtained. T_g's of hyperbranched polymers showed a nonlinear relation to reciprocal molecular weight, which differed from the linear relation observed in linear polymers. T_g values decreased with increasing molecular weight in the low‐molecular‐weight range; however, they increased with increasing molecular weight in the high‐molecular‐weight range. T_g values decreased with increasing log M and then turned to a constant value in the high‐molecular‐weight range. The plot of T_g versus 1/M or log M for hyperbranched polymers may exhibit intersecting straight‐line behaviors. The intersection or transition does not result from entanglements that account for such intersections in linear polymers but from a nonlinear feature in hyperbranched polymers according to chain‐end, free‐volume theory. However, the conclusions obtained in this work cannot be extended to dendrimers because after the third generation, the end‐group extents of a dendrimer decrease with molecular weight. Thus, it is very possible for a dendrimer that T_g increases with 1/M before the third generation; however, it decreases with 1/M after the third generation. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1235–1242, 2004     

Poly(2,6-diphenyl-1,4-phenyl oxide): Characterization by gel-permeation chromatography,light scattering,and solution viscosity

Ten unfractionated poly(2,6-diphenyl-1,4-phenylene oxide) samples were examined by gel permeation chromatography (GPC) and intrinsic viscosity [η] at 50°C in benzene, by intrinsic viscosity at 25°C in chloroform, and by light scattering at 30°C in chloroform. The GPC column was calibrated with ten narrow-distribution polystyrenes and styrene monomer to yield a “universal” relation of log ([η]M) versus elution volume. GPC-average molecular weights, defined as M?gpc = \documentclass{article}\pagestyle{empty}\begin{document}$\Sigma w_i [\eta ]_i M_i /\Sigma w_i [\eta ]_i$\end{document}, w_i denoting the weight fraction of polymer of molecular weight M_i, were computed from the GPC and [η] data on the polyethers. The M?GPC were then compared with the weight-average M?_w from light scattering. The intrinsic viscosity (dl/g) versus molecular weight relations for the unfractionated poly(2,6-diphenyl-1,4-phenylene oxides) determined over the molecular weight range 14,000 ≤ M?_w ≤ 1,145,000 are log [η] = ?3.494 + 0.609 log M?_w (chloroform, 25°C) and log [η] = ?3.705 + 0.638 log M?_w (benzene, 50°C). The M?_w(GPC)/M?_n(GPC) ratios for the polymers in the molecular weight range 14,000 ≤ M?_w ≤ 123,000 approximate 1.5 according to computer integrations of the GPC curves with the use of the “universal” calibration and the measured log [η] versus log M?_w relation. The higher molecular weight polymers (326,000 ≤ M?_w ≤ 1,145,000) show slightly broadened distributions.     

Electric birefringence and conformation of polychlorohexylisocyanate in solutions

Electric birefringence was investigated for solutions of polychlorohexylisocyanate fractions for molecular weights 30·6 × 10⁴–1·2 × 10⁴ in tetrachloromethane.Experimentally found dispersion of the Kerr effect is used for estimating the coefficients of rotatory diffusion D_r of molecules. A comparison of rotatory diffusion D_r values with molecular weights M and intrinsic viscosities [η] of fractions shows that the value of D_rM[η] decreases with M. This illustrates the change in the conformation of molecules from a random coil to a rod.On the basis of experimental dependences of D_r and the Kerr constants K on M, the main structural parameters of the polymer investigated were determined: the number of monomer units in a segment, the projection of the length of the monomer unit on the axis of the molecule, the value of the dipole moment μ₀ of the monomer unit and the angle formed by μ₀ and the chain direction.     

Calculation of topological parameters of hyperbranched macromolecules synthesized via living radical polymerization

Numerical calculations of the kinetic model of synthesis of hyperbranched polymers in the living radical polymerization mode were performed. Analytical expressions were obtained that make it possible to predict the maximum yield of hyperbranched polymers and their topological parameters, such as the branching frequency; the numbers of living ends, monomer units and multiple bonds per macromolecule; and the degree of conversion at the gel point. The model is based on the use of a branching monomer M_m that contains m ≥ 2 polymerizable bonds in its molecule in combination with a monomer M₁ capable of forming linear chains only.     

Generalized Universal Molecular Weight Calibration Parameter in GPC

Abstract

In this report we show by experimental and theoretical investigations that the commonly used GPC universal calibration parameter, the intrinsic viscosity multiplied by the weight average molecular weight ([η] M^w) is incorrect. The error which can arise by using [η] M to calculate the molecular weight across the GPC chromatogram for nonuniformly branched polymers [poly(vinyl acetate) and low density polyethylene] and copolymers with compositional drift, could be very large. We also show conclusively that the number average molecular weight M_n is the correct average to use for the universal calibration parameter. We therefore recommend that our general universal Calibration parameter [η] M_n be used for calculating the molecular weight across the chromatogram for all polymer systems (linear and branched homopolymers, copolymers with or without compositional drift and for polymer blends).     

Structural and conformational properties of hyperbranched copolymers based on perfluorinated germanium hydrides

Two series of hyperbranched copolymers based on perfluorinated germanium hydrides of various topological structures are studied in dilute chloroform solutions by the methods of molecular hydrodynamics and optics. The first series is composed of copolymers with various molecular masses (from 2.3 × 10⁴ to 31 × 10⁴) that contain rigid linear chains between branching points and various amounts of branching points in cascades of the dendritic fragment, while the second series is comprised of copolymers that, at close branching degrees, on average, are characterized by a looser structure owing to a large amount of linear units at the periphery of macromolecules with M = (2.5 × 10⁴)?(23 × 10⁴). Macromolecules of the studied polymers have compact dimensions and a high density of the polymer substance; their shape asymmetry is low. In terms of these characteristics, they approach dendrimers. At a fixed molecular mass, the copolymers with a loose structure have higher dimensions of macromolecules and higher intrinsic viscosities.     

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.     

Chromatographic mass of a polymer and its use for determining the molecular mass characteristics

The method has been proposed for determining the molecular characteristics of flexible-chain polymers that obey the universal calibration principle and for which there are available experimental data on the intrinsic viscosity. This method is based on studying the dependence of the hydrodynamic volume M _n[η], M _w[η], M _z[η], and M _η[η] on parameter a in the Mark-Kuhn-Houwink equation. It has been found that, for parameter a varying in the range from 0.5 to 0.8, the weight-average hydrodynamic volume M _w[η] remains almost unchanged. This allows estimation of M _w based on a single intrinsic viscosity value. The notion of the chromatographic mass of a polymer is advanced and is employed for determining other molecular mass parameters.     

Hydrodynamic and conformational properties of silicon-substituted addition-type polynorbornene macromolecules

Seven polynorbornene samples containing trimethylsilyl side groups that were prepared by the addition polymerization of 5-trimethylsilyl-2-norbornene in the presence of catalytic systems (π-C₅H₉NiCl)₂-methylaluminoxane and nickel naphthenate-methylaluminoxane have been studied by translational isothermal diffusion and viscometry. The molecular masses of the polymer samples are measured. Kuhn-Mark-Houwink equations for diffusion coefficient D and intrinsic viscosity [η] are determined in toluene at 25°C: D = 6.94 × 10^?4 M ^?0.61 and [η] = 1.53 × 10^?3 M ^0.82. The equilibrium rigidity of polymers chains is estimated as A = 47 ± 9 Å. The conformational features of the silicon-containing polynorbornene are analyzed by the PM3 quantumchemical semiempirical method on the basis of simulation of its decamer chain fragments. In terms of microstructure and equilibrium rigidity, the above-described addition poly(trimethylsilylnorbornene) is close to poly(trimethylsilylpropyne) synthesized using niobium pentachloride as a catalyst. This finding explains similar membrane gas-separation properties of these polymers.     

Proposed GPC and intrinsic viscosity analyses of randomly crosslinked polymers having primary distributions of the schulz-zimm form

A mathematical treatment is presented for the gel-permeation chromatographic and intrinsic viscosity behavior of randomly crosslinked polymers having primary molecular weight distributions of the Schulz-Zimm form. Kimura's serial solution of the integro-differential equation derived by Saito for randomly crosslinked polymers is employed for the distribution function. The intrinsic viscosity of a molecule containing i crosslinks is assumed related to that of a linear molecule of the same number of units through [η]_br/ = g_i^½[η]_l where g_i = (R_br²)_i/R_l² = {[1 + (i/6)]^½ + (4i/3π)}^?½. R_brand R_l denoting the root-mean-square radii of gyration of branched and linear chains of the same mass. It is also assumed that GPC elution is controlled by the hydrodynamic volumes of the molecules. Representative calculation results are displayed for polymers with a narrow primary distribution and the “most probable” primary distribution. Results for the latter polymers are compared with those previously obtained by a somewhat different mathematical approach.     

Synthesis and characterization of well-defined poly(tert-butyl acrylate) star polymers

Star polymers with different numbers and lengths of poly(tert-butyl acrylate) (PTBA) arms were obtained via atom transfer radical polymerization. Aliphatic alcohols with different number of hydroxyl groups varying from 3 to 6 and calix[4]arenes based on pyrogallol with 12 and 16 phenol groups were transformed to bromoester derivatives to prepare multifunctional ATRP initiators used as the cores of the stars. The star polymers were characterized by GPC with refractive index, multiangle laser light scattering and viscosimetric detectors. The molar masses of the stars reached 70,000 g/mol and the molar mass dispersities did not exceed 1.2. To elucidate the compact structure of the stars, their true molar masses were determined by GPC with triple detection (RI-MALLS-Visco) and compared with the apparent molar masses obtained from the calibration with linear poly(tert-butyl acrylate) standards. The intrinsic viscosities of the PTBA stars of the same molar mass decreased with the number of star arms but were always lower than the intrinsic viscosities of the analogue linear PTBA polymers. The values of the branching ratio g′ decreased with increasing number of arms indicating more compact structure of stars. The branching ratio g′ was correlated to the empirical predictions.