Determination of molecular weight distributions of novolac resins by gel permeation chromatography

The number- and weight-average molecular weights of several statistical and high ortho novolac resins were determined using gel permeation chromatography (GPC). The standards used were pure compounds having between 2 and 12 phenol units bridged via methylene linkages. Three series of compounds were studied: (i) those with methylene linkages substituted in only the ortho positions relative to the phenolic hydroxyl group; (ii) those in which all para positions, together with sufficient ortho positions, were used to synthesize the compounds; and (iii) those in which the methylene linkages were substituted at a mixture of ortho and para positions. Such compounds, having known molecular architecture and units similar to the segments of industrial novolac resins, provide for a more exact measurement of the molecular weight than do the commonly used poly(styrene) standards. Using these new standards the number average molecular weights of the resins determined by GPC were in good agreement with the average molecular weight obtained by ¹H-NMR spectra of the resins, particularly for low molecular weight resins. GPC analysis of higher molecular weight resins tends to underestimate the molecular weights because of complications introduced by hydrogen bonding. © 1997 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 35: 1399–1407, 1997     

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.     

Determination of optimized blending fractions for particle size distributions or molecular weight distributions involving various physical properties

It has been shown in this study that the effects of particle size distribution or molecular weight distribution on selected physical properties can be related by a generalized blending approach that involves similar equations. The blending equations developed involve different z-fractions where z = 3 for volume blending of spherical particles, or z = 2 for surface blending of spherical particles, or z = 1 for the weight blending of molecular weights. This new analysis approach addresses the magnitude of the ratios of particle size averages, D_x/D_y, or ratios of molecular weight averages, M_x/M_y, as well as the location of this maximum, the level of distribution information available for the starting materials, and the type of z-fraction blending. To illustrate this approach suspension viscosity/concentration data was used to show how the D_x/D_y ratio could be introduced successfully to analyze latex volume blending where z = 3. In addition, the maximum steady-state elastic compliance, J_e, as a function of weighted blends (z = 1) of two different molecular weights of polyisobutylene was shown to fit the simple equation J_e = 1.187 (M₃/M₂) (M₄/M₁) reasonably well. © 1995 John Wiley & Sons, Inc.     

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.     

Ratios of average molecular weights and molecular weight distributions in polymers

If M_min and M_max are lower and upper bounds, respectively, to the molecular weights of different molecular weight species contained in a polymer, the weight-average to number-average molecular weight ratio M _w/M _n cannot exceed (1 + M_max/M_min)²/(4M_max/M_min). The ratio attains this maximum possible value if the masses of the two species with molecular weights M_min and M_max are equal and the masses of all the other species are negligibly small, corresponding to maximum spread in the molecular weight distribution within the specified bounds. Also for a given value of M _w/M _n = α, the M_max cannot be smaller than [2α ? 1 + 2α^1/2(α ? 1)^1/2]M_min. The minimum possible value of M_max/M_min consistent with α given is obtained in the case of maximum spread described above. If only one species is predominant, then both M _w/M _n and M_max/M_min approach unity, as is well known. Similar relations hold for the ratios of higher-order average molecular weights for which the role of the mass fractions is replaced by higher-order distribution functions.     

Theory of polydispersity correction procedures for the determination of molecular weight dependences

Correction procedures are described for determining the molecular weight dependence of an arbitrary property X(M) = K_xM² for monodisperse samples from measured values of polydisperse samples. The required data may be various types of average values of the molecular weight and of arbitrary powers of X. These procedures can easily be applied numerically or represented graphically.     

Persulphate initiated polymerisation of methacrylamide—I: Kinetics and molecular weight dependences

A kinetic study of the aqueous persulphate initiated polymerisation of methacrylamide has shown that the rate of polymerisation is represented by the equation $\text{R}{}_{\mathrm{p}}\text{= k [}\text{M}\text{] [}\text{I}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ where the overall rate constant, $\text{k = 1.3 × 10}{}^9\text{exp}\text{(?18,400/}\text{RT}\text{) 1}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{mol}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{s}{}^{\mathrm{?1}}$. Chain transfer with monomer, where C_M = 5.4 × 10^?3 at 60°, is shown to be the dominant transfer step. Comparison with kinetic studies of the analogous acrylamide polymerisation shed doubt on the ‘cage effect’ interpretation of complex orders with respect to monomer. An alternative explanation is proposed.     

Prediction of molecular weight distributions in branched polymers

A theoretical treatment of long branching in radical polymerization in a continuous stirred tank reactor is presented. The treatment takes into account radical termination by disproportionation and/or combination, transfer to polymer and/or additive of low molecular weight, residence time, and injection of polymer. In the absence of added polymer, the molecular weight distribution depends upon four parameters which can be expressed as P?_n and ratios of the rate of polymerization to the rate at which radicals (a) leave the reactor, (b) combine, and (c) transfer to polymer. The kinetic equations were converted into a differential equation which was solved numerically to give polymer and radical moments. An analytical solution is presented for the case where combination is absent. P?_w/P?_n is predicted to increase smoothly in a markedly exponential manner with increasing polymer transfer, combination, P?_n, and mean residence time. At no stage do any of the moments become infinite unless the residence time is infinite. For polymers of wide distribution, the second and higher radical moments can exceed the corresponding polymer moments so that most of the large molecules leave the reactor as radicals. Beasley's and Nicolas's treatments are shown to be limiting cases at low polymer transfer.     

Self-seeded crystallization and its potential for molecular weight characterization. I. Experiments on broad distributions

The effect of varying molecular weight distribution on the self-seeding phenomenon was investigated by using high molecular weight polyethylene fractions prepared by the stirring-induced crystallization method of Pennings. The numbers of self-seeding nuclei per gram were determined by measuring crystal dimensions, and were found to increase with increasing molecular weight of the polymer, in accord with previous findings. In another experiment, self-seeded single crystals were grown from materials of various molecular weights, prepared by blending two samples of differing molecular weight in various proportions. The concentration of nuclei varies linearly with the weight fraction of high molecular weight polymer in the mixture. This result is shown to be consistent with the proposition that each nucleus contains on the average an identical number (most plausibly one) of molecules of very high molecular weight. The application of this finding to the determination of molecular weights is discussed, and it is shown that the technique provides a method of unprecedented sensitivity for the characterization of the very high end of the molecular weight spectrum. Some morphological results are also presented. In particular, direct observations of the nuclei were found to be consistent with the loosely connected multiple nucleus structure, which had been proposed previously to account for certain light-scattering results.     

Kinetics of chain collapse in dilute polymer solutions: molecular weight and solvent dependences

The molecular weight and solvent dependences of the characteristic time of chain collapse were studied for poly(methyl methacrylate) (PMMA) of the molecular weight Mw=6.4x10(6) and 1.14x10(7) in pure acetonitrile (AcN) and in the mixed solvent of AcN+water (10 vol %). The size of PMMA chains was measured as a function of the time after the quench by static light scattering and the chain collapse processes were expressed by the plot of the expansion factor alpha2 vs ln t. The chain collapse in the mixed solvent AcN+water (10 vol %) was found to occur much faster than that in pure AcN, though the measurement of the former collapse process required several hours. In order to make a comparison between the rates of chain collapses, the fast chain collapse process was superposed on the slow one by scaling the time of the fast process as gammat. The scale factor gamma was determined by comparing the chain collapse processes of nearly the same equilibrium expansion factor with each other. Accordingly, the superposition of the collapse for Mw=6.4x10(6) on that for Mw=1.14x10(7) yielded gammam=4.0+/-0.6 for the process in AcN+water and 5.5+/-0.6 in AcN. The superposition of the chain collapse process in AcN+water on that in AcN yielded gammas=9.5+/-1.4 for Mw=6.4x10(6) and 12.0+/-1.8 for Mw=1.14x10(7). This analysis suggests that gammam and gammas are constant independent of each other. Thus, by assuming the molecular weight dependence of gammam approximately Mz, the characteristic time tauexp of chain collapse was conjectured as tauexp approximately kappaMz, where kappa reflects the nature of solvent species. The ratio of kappa for PMMA in AcN to that in AcN+water is given by gammas. The exponent was estimated to be z=2.4+/-0.7 for AcN+water and 3.0+/-0.7 for AcN. These values are compatible with the theoretical prediction z=3 based on a phenomenological model, though the observed characteristic times are longer by several orders of magnitude than those of the theoretical prediction.     

Sedimentation transport method for determination of molecular weight distributions

The sedimentation of the system polystyrene-cyclohexane at the Flory temperature has been studied with emphasis on the effects of pressure as well as concentration. The relation between the molecular weight M and the limiting sedimentation coefficient s₀⁰, is found to be s₀⁰ = 1.50 × 10^?15 M^1/2 (sec.) The concentration dependence parameter k_s has the form, k_s = k′M^1/2 = k″s₀⁰ with k′ = 4.5–5.5 × 10^?4. However, a rather unexpected dependence of k_s on the rotor speed is also found. A procedure is proposed for deducing solute molecular weight distributions from boundary spreading data in sedimentation transport experiments, a so-called “single concentration” method, requiring only one sedimentation run. Application to several polystyrenes (in cyclohexane at 35°C) with narrow, broad, and very broad distributions demonstrates the feasibility of the procedure. Comparisons are made with data from elution chromatography and gel permeation chromatography. The GPC method predicts somewhat broader distributions than those obtained by the other two methods.     

Bimodal molecular weight distributions of a branched condensation polymer

Gel permeation chromatograms of a hindered-phenolic, branched condensation polymer display pronounced bimodality at high conversion. The true molecular weight distributions, obtained by means of a GPC calibration curve based on narrow-distribution fractions, exhibit corresponding anomalous high-molecular-weight “shoulders.” These results are discussed in terms of preferential aggregation and reaction of the higher-molecular-weight species during the polymerization, promoted by intermolecular hydrogen bonding in the apolar reaction medium.     

Electrochemical preparation of polymers with multinodal molecular weight distributions

A direct electrochemical preparation of polybutadiene to a predetermined molecular weight distribution has been achieved. Butadiene is a good model compound, inasmuch as the rate of electrochemical initiation is high relative to the rate of propagation. Rate constants for the temperature interval of interest and for systems of comparable composition were reported and permitted a priori prediction of the weight fractions and the number average molecular weight of each fraction. Polymer was formed by a series of pulses which successively initiated polymerization, permitted growth in the absence of current, and electrochemically terminated polymerization. The polymers produced showed excellent agreement with calculated composition and distribution in the range of molecular weights 10,000 to 50,000.     

Derivation of continuous molecular weight distributions from the generating function

A method described in the statistical literature for the numerical inversion of Laplace Transforms of real functions has been adapted for the derivation of molecular weight distributions from calculations of the generating function for a polymeric system. It has been shown that a third-order refinement is sufficiently accurate for molecular weight distributors broader than 4.0. This allows the calculations to be carried out with a precision of 16 decimal digits which is commonly used in Fortran. Where higher precision is available, the treatment is applicable to narrower MWDs.     

Influence of preparative conditions on molecular weight and stereoregularity distributions of polypropylene

The influence of preparative conditions on the molecular weight and stereoregularity distributions of polypropylene was investigated. The stereoregularity distribution is narrowed by using a highly stereospecific catalyst, by decreasing the polymerization temperature, and for the three-component catalyst by keeping the mole proportion of the electron-donating third component at 0.5. The molecular weight distribution can be narrowed by using a highly stereospecific catalyst, a high monomer concentration, and a high polymerization temperature, and by having a lower conversion, particularly at low monomer concentration. The possibility of long-chain branching in polypropylene was indicated by data from the fractionation of tritium-labeled polymers.     

Calculation of molecular weight distributions for polymers undergoing random crosslinking and scission

A simple, practical calculation procedure has been developed for predicting the changes in molecular weight distribution of a polymer undergoing random crosslinking and/or degradation. Simulations of the random crosslinking and degradation of narrow and broad Poisson-type distributions have been made. The results agree with those calculated from Kimura's analytical solutions to Saito's general equations after a correction has been made for a mathematical error in Kimura's solution. This method can be applied to determining the probabilities of crosslinking and scission for any arbitrary molecular weight distribution expressed in tabular form. The importance of using narrow distribution samples to estimate crosslinking from changes in molecular weight distribution is graphically demonstrated.