Evaluation of steady-state flow curves from model molecular weight distributions

Graessley's theory of entanglement was applied to several model distributions. The distribution functions chosen were such that the differential weight distributions were triangular with respect to a log molecular weight axis. The molecular weight level, breadth of the molecular weight distribution, skewness, and blending of simple distributions were studied in their effect on the steady-state flow curve. The governing factor for the shape of the reduced flow curve was shown to be M?_z+1·M?_z/M?_w². Other general features of the flow curve predicted by Graessley's theory were discussed.     

Dispersion in gel permeation chromatography. A rapid numerical solution to Tung's integral equation

A rapid iteration method has been developed to correct the molecular weight averages calculated from raw GPC data for dispersion. Though simple in its performance, it covers the general case that the instrumental spreading characteristics (Tung's resolution factor h) depend on the elution volume. Moreover, it is irrelevant whether the calibration curve, being the logarithmic plot of the molecular weight versus the elution volume, is linear or not. The method has been applied to a number of well-characterized polystyrene mixtures and yields molecular weight averages which agree with those predicted theoretically. The effect of asymmetry exerted by the dispersion on both molecular weight averages M?_n and M?_w is also discussed.     

Shear degradation of polyisobutene

A polyisobutene of M?_w 1.98 × 10⁶, M?_w/M?_n 1.8, was extruded in an Instron capillary rheometer. Shear degradation occurred at high shear stresses, approaching melt fracture, and was more prominent at lower extrusion temperatures for tests at 60–140°C. The capillary was 2.0 in. long with a length/diameter ratio of 66.7 and a 90° entrance angle. Repetitive extrusions at constant shear rate caused a decrease in a molecular weight and a simultaneous narrowing of the molecular weight distribution. Extrudate expansion was measured after each successive capillary pass for tests at 80°C. Extrudate swelling correlated well with (M?_z+1) M?_z/M?_w, except for the two first passes, where melt fracture was pronounced. The correlation with equilibrium extrudate expansion was almost as good for (M?_z/M?_w)^3.7 (Mill's correlation) and for M?_z+1 alone. The efficiency of bond rupture is low, with the energy required to rupture 1 mole of bonds being about 200,000 kcal at 80°C.     

Randomly branched styrene/divinylbenzene copolymers. I. Preparation,molecular weight characterization,and GPC analysis

A series of randomly branched copolymers of styrene and divinylbenzene were prepared using a benzoyl peroxide-initiated free-radical bulk polymerization at 78°C. DVB contents were varied from 0.01% to 2%. Two samples were polymerized with 0.4% DVB to different conversions: series 9A at 6% conversion and series 9B at 15% conversion (just short of the gelation point). Both samples were fractionated and the fractions characterized by ultracentrifugation, light scattering, osmometry, viscometry, and gel permeation chromatography. The data indicated that the fractions were not of narrow MWD and that the breadth of the MWD of the fractions from series 9B were greater than those of 9A. GPC calibration curves of M, [η], and M [η] were generated for both 9A and 9B fractions by employing curve-fitting techniques to the GPC data. For all of the fractions 9B, the molecular weight calibration provided accurate values of M?_z, M?_w, and M?_n, suggesting that no serious peak spreading had occurred in the GPC experiments. The universal calibration parameter M[η] for the 9A fractions agreed with that of linear polystyrene, while that of the high-conversion series 9B did not. It will be shown in a later paper that series 9B is highly branched, while 9A is lightly branched. Consequently, it is recommended that any GPC analysis of branching units make an allowance for the deviation of highly branched polymers from the linear M[η] calibration curve.     

Influence of molecular weight and molecular weight distribution on the tensile properties of amorphous polymers

Tensile property data for polystyrene samples of varying polydispersity are correlated with various parametric measures of molecular weight. Traditional measures of molecular weight, such as M?_n, M?_w, and M?_z, are shown to be unable to account for the variation of tensile properties with molecular weight. However, a new molecular weight parameter, termed the failure property parameter, is able to provide a single relationship between tensile strength and the parameter for both the broad and narrow distribution polymers. The form of this parameter is consistent with its having origins in the view that it is the entanglement network in an amorphous polymer that provides the observed strength properties. Specifically for polystyrene, the failure property parameter results indicate that material below 60,000 molecular weight does not contribute to polymer strength. Although the results of this investigation are specifically for polystyrene, the arguments used to develop the failure property parameter are not dependent on polymer chemical structure. Consequently, we believe that both the concepts and definition of this new parameter are applicable to all amorphous polymers.     

Comparaison des méthodes de fractionnement par chromatographie de perméation et par gradient d'elution pour la détermination des répartitions moléculaires des polypropylénes

Molecular weight distributions for polypropylene samples have been determined by a permeation fractionation method (GPC). Porous silica beads were used as a packing material for the columns. The set of columns allows a good separation of the polypropylene macromolecular chains in a range of molecular weights from 5000 to 1.5 × 10⁶, and the thermal and mechanical stabilities of these beads are very good. The calibration has been carried out with fractions of polypropylene of narrow molecular weight distribution prepared by a large-scale column fractionation. The molecular weights M?_w and M?_n and the ratios M?_w/M?_n calculated from the GPC curves show, in general, good agreement with the ones calculated from the column fractionation curves. However, the M?_w/M?_n ratios are always highter in the case of GPC fractionation. This could be due to diffusion phenomena.     

Silylation of water glass. IV. Synthesis and characterization of silylated silicic acids

Trimethylsilylated (TMSS) and dimethylphenylsilylated silicic acids (DMPS) were prepared via extraction with tetrahydrofuran, started from aqueous sodium silicate. Characterization of these silylates based on the ¹³C and ²⁹Si NMR spectra, X-ray diffraction, gel permeation chromatography, and solution viscosity were conducted to study their molecular structures. It was found that the GPC elution volume was significantly deviated from the calibration curve and also the exponents in solution viscosity vs. molecular weight (M?_n) relationship were very low, almost 0.3. These solution properties suggest that such silylates have compact spherical structures with a high degree of branching, also supported from the other data.     

Estimation of Mw from dilute solution viscosity

The viscosity-average molecular weight, M_v, of a polymer is given operationally through its limiting viscosity number [η] and the Mark-Houwink equation [η] = KM_v^α, where K and α are empirical constants. If [η] is measured under different conditions, α and M_v will vary for the same sample. M_v^α is the α-order moment about the origin of the differential weight distribution of the polymer. Practically, the results of a series of M_v measurements on the same polymer are equivalent to a cluster of fractional moments of the weight distribution, with orders between 0.55 and 0.80. It is shown that the first moment of this distribution, M_w, may be estimated reliably by a straightline plot of M_v against α-extrapolated to α equals 1. This simple expedient is effective although there are probably no molecular weight distributions in which the relation is strictly linear and there are no mathematical distributions for which the αth root of the αth moment is a linear function of α for all α. The deviation from linearity is small enough, however, that the real curve can be represented by a straight line over a short range of α. Thus, M_w can be measured accurately, but M_n, M_z, or the breadth of the distribution is not accessible by this method. Experimental and literature examples show that the precision of M_w estimated by this method compares well with that of primary methods for measuring this molecular weight average. If a linear relationship is observed with reliable α values, this appears to be a sufficient condition for estimation of a valid M_w.     

Number‐average molecular weight of branched polymers from SEC with viscosity detection and universal calibration

BACKGROUND: Number‐average molecular weight, M?_n, is an important characteristic of synthetic polymers. One of the very few promising methods for its determination is size‐exclusion chromatography (SEC) using on‐line viscometric detection and assuming the validity of the universal calibration concept. RESULTS: We have examined the applicability of this approach to the characterization of statistically branched polymers using 22 copolymers of styrene and divinylbenzene as well as 3 homopolymers of divinylbenzene with various degrees of branching. SEC with three on‐line detectors, i.e. concentration, light scattering and viscosity, enables us to evaluate experimental data by various computational procedures yielding M?_n and weight‐average molecular weight, M?_w. Analysis of the results has shown that the universal calibration theorem has limited validity, apparently due to the dependence of the Flory viscosity function on the molecular shape, the molecular weight distribution and the expansion of molecules. CONCLUSION: For complex polymers, the universal calibration, i.e. the dependence of the product of intrinsic viscosity and molecular weight, [η]M, on elution volume, can differ in values of [η]M from those obtained for narrow molecular weight standards by 10–15%. The method studied is helpful for the determination of M?_n of polymers, in particular of those with very broad molecular weight distribution, such as statistically highly branched polymers. Copyright © 2008 Society of Chemical Industry     

The numerical evaluation of parameters in distribution functions of polymers from their molecular weight distributions

It has been shown that the parameters in the log-normal (LN) and generalized exponential (Gex) distributions can be evaluated if molecular weight ratios H and H_z, or their equivalents, are known for a linear, unimodal homopolymer. It is suggested that better checks of observed and calculated values of M_z can be obtained for such polymers as are characterized by m ≤ 0.15, by treating them as Gex rather than LN distributions.     

Molecular weight distribution and swell properties of some low-pressure polyethylenes

The molecular weight distribution of some low-pressure homopolymers of ethylene intended for use in blown bottles have been determined by using a modified sand column technique. A correlation is indicated between the Z-average molecular weight M?_z and the postextrusion swelling of the resin as measured by the weight of a bottle blown from that polymer.     

The relation between melt flow properties and molecular weight of polyethylene

The non-Newtonian behavior of commercial linear polyethylene samples and their fractions were studied at 190°C. The viscosity η versus shear rate \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \gamma $\end{document} curves of whole polymers could be superimposed onto a single master curve despite the variations of their molecular weights and molecular weight distributions. For fractions, however, the same master curve was inapplicable, and the sensitivity of the viscosity to shear rate was found to be greater than those of the whole polymers. The zero-shear viscosities η₀ of fractions were related to the 3.42 power of the weight-average molecular weight M_u as follows: For whole polymers, the zero-shear viscosities were found to be considerably higher at the same M_w and markedly lower at the same z-average molecular weight M_z than those of the fractions. Thus, it was concluded that η₀ corresponds to an average of molecular weight between M_w and M_z. It was found that the molecular relaxation time τ is proportional to M_z^5.3 for whole polymers and to η₀M_w for fractions. Using these relations it was possible to relate the flow ratio, the ratio of flow rates at two different shear stresses, with the molecular weight distribution.     

Size-exclusion chromatography with two-angle laser light-scattering (SEC-TALLS) of high molecular weight and branched polymers

The accuracy and precision of results obtained from light-scattering detection at two angles (TALLS) for size-exclusion chromatography (SEC) are examined for linear narrow molecular weight distribution polystyrenes between 1,290,000 and 20,000,000 MW and for branched polyesters. The ratio of light-scattering intensities at 15° and 90° is used to calculate weight-average molecular weight, M?_w, and an average root-mean-square radius, r?_gu, equivalent to the z-average radius. A shape for the polymer molecule is assumed and an analytical relationship for the particle-scattering function is required. It is shown that analysis of the data using the particle-scattering function for a random coil is valid for both high molecular weight, linear polystyrenes and long-chain branched polyesters. The radius, r?_gu, is determined with high precision by using the ratio of light-scattering signals, which is insensitive to errors in sample concentration and changes in the eluent flow rate. The correct average radius for the whole polymer is obtained despite using low-efficiency, large-particle diameter SEC columns; however, axial dispersion significantly affects molecular weights and radii calculated at each retention volume that can limit the utility of plots used to deduce polymer conformation. © 1995 John Wiley & Sons, Inc.     

Determination of average molecular weights of a polyelectrolyte by meniscus-depletion sedimentation equilibrium

Long-column meniscus-depletion sedimentation equilibrium has been used to determine M?_n, M?_w and M?_z for the Na⁺ and (CH₃)₄N⁺ salts of a high molecular weight poly(styrene sulphonate). Consideration has been given to ways of avoiding the need for dialysis when dealing with polyelectrolyte solutions.     

Molecular weights of styrene–maleic anhydride copolymers

A dual-calibration method for the determination of molecular weights and molecular weight distribution of styrene–maleic anhydride copolymers (S/MA) by gel permeation chromatography (GPC) is introduced. It might be applicable to copolymers of other type. A linear relationship of intrinsic viscosity [η] and weight-average molecular weight (M?_w) for unfractionated S/MA in tetrahydrofuran (THF) at 25°C can be expressed by the equation The maleic anhydride content of the copolymers ranges from 5 to 50 mole-%, and the M?_w range is from 2 × 10⁴ to 7 × 10⁶. The plot of log [η] M?_w versus GPC elution volume of the S/MA copolymers falls on the same curve as that of the polystyrene standards in THF.     

Statistics of star-shaped molecules III. Stars with branched nuclei

Number and weight average molecular weights and scattering behaviour of star molecules with extended branched nuclei are calculated by application of cascade theory. The nuclei considered arise from random polycondensation of monomers of the A₃ or of the A—B/C type. Nuclei of the first type are characterised by very large molecular polydispersities (M_w/M_n α M_w), while nuclei of the second type have less broad molecular weight distributions (M_w/M_n α M_n). The rays of the stars are assumed to be either monodisperse or to obey the Schulz—Flory “most probable” length distribution. Analytic expressions are given for M_w, M_n, 〈S²〉_z and the particle scattering factor P_z(h) which was averaged over the ensemble. The results are compared with stars of spherical and uniform nuclei whose molecular weights and mean square radii of gyration equal M_wN and 〈S²〉_zN from the other two types of nuclei. In the limit of very large ray lengths the scattering behaviour is determined solely by the number of branches z. At shorter chain length of the rays structure and polydispersity of the nuclei have marked influence. This influence is still easily noticed from the angular dependence of scattered light at chain lengths where no differences in the 〈S²〉_z versus M_w plot are detectable. The mean square radius of gyration depends only weakly on the number of rays and eventually becomes independent of it if z 15. In that limit 〈S²〉_z depends virtually on the length of the rays alone and its distribution. Stars whose rays have a most probable length distribution exhibit 〈S²〉_z values twice as large as stars with monodisperse rays. A procedure is suggested and discussed for the determination of the number and length of the rays if the scattering behaviour of the isolated nucleus and the isolated linear chains is known.     

Characterization of branched polyethylene fractions from the elution column

Fractions from several elution column runs on samples of up to 6 g. of a well-characterized high-pressure polyethylene were analyzed by absolute molecular weight methods and several other techniques. The M_n and M_w integral distribution curves are free from any reversal, as was the viscosity distribution curve. Fractions with M_w as high as 8 × 10⁶ were recovered, more than 20 times higher than the original sample's M_w. The polydispersity of the fractions increases from M_w/M_n = 1.5 or less in the low molecular weight fractions to a nearly constant value of 4.5–5.0 in fractions above 60% cumulative sample weight. Nonetheless, refractionation on the elution column shows that the fractions are narrowly distributed in terms of solubility, while GPC analysis reveals that the fractions have an extremely narrow size distribution. It is concluded from the combined results that long-chain branching plays an important role in determining the equilibrium solubility and, further, that long-chain branching increases the polymer solubility. Sample calculations are provided, which illustrate the effect of fraction polydispersity on calculated original sample molecular weights and the fit of the fractionation results to several model distribution functions.     

Melting and crystallization behavior of poly(tetrafluoroethylene). New method for molecular weight measurement of poly(tetrafluoroethylene) using a differential scanning calorimeter

Melting and crystallization behavior of virgin polytetrafluoroethylene have been studied using a differential scanning calorimeter. Following quantitative relationship was found between number average molecular weight of polytetrafluoroethylene and the heat of crystallization in the molecular weight range of 5.2 × 10⁵ to 4.5 × 10⁷: M?_n = 2.1 × 10¹⁰ ΔH_c^?5.16, where M?_n is number average molecular weight and ΔH_c is the heat of crystallization in cal/g. The heat of crystallization is independent of cooling rate ranging from 4 to 32°C/min. This relationship provides a simple rapid and reliable method for measuring the molecular weight of polytetrafluoroethylene.     

Melt fracture of polystyrene

The results of an experimental study of melt fracture using polystyrene samples of narrow and broad molecular weight distribution are presented. The weight average molecular weight (M?_w) ranged from 9.72 × 10⁴ to 1.80 × 10⁶ and the distribution breadth M?_w/M?_η from 1.06 to 9.21. The critical shear stress varies linearly with 1/M?_w, increases slightly with temperature and is independent of the distribution of molecular weights. This type of behavior is satisfactorily explained in terms of Graessley's entanglement theory.     

Structure and properties of cellulose graft copolymers. II. Cellulose–styrene graft copolymers synthesized by the ceric ion method

Styrene was graft-copolymerized onto wood cellulose by the ceric ion method of Mino and Kaizerman. The grafting reaction was found to depend strongly on the concentration of ceric ion in the grafting system and maximum grafting occurred in a narrow range of concentration of initiator, 1.0 × 10^?3-1.8 × 10^?3 mol/l, at 58 ± 1°C. A pretreatment technique, developed to enhance the monomer diffusion into cellulose, was found to increase the grafting considerably. The structures of the cellulose-styrene graft copolymers were studied by hydrolyzing away the cellulose backbone to isolate the grafted polystyrene branches. The molecular weight and the molecular weight distributions of the grafted polystyrene were determined using gel permeation chromatography. The number-average molecular weight (M?_n) ranged from 23,000 to 453,000 and the polydispersity ratios (M?_w/M?_n) varied from 2.5 to 8.0. The grafting frequencies calculated from the per cent grafting and molecular weight data were of the order of 0.05–0.4 polystyrene branches per cellulose chain.