Schulz-Zimm Distribution Research Articles

Comparison of different polymolecularity correction formulae for the burchard‐stockmayer‐fixman relationship

Polymolecularity correction of the original and the modified procedure according to Burchard, Stockmayer and Fixman, especially useful for the determination of the unperturbed dimensions of macromolecules, is derived via the general “principle of corresponding averages” and compared with the polymolecularity correction methods published by Sutter et al. and Rosenvasser et al. It is shown that the polymolecularity correction factor to be used with the mass-average molar mass in the Burchard-Stockmayer-Fixman relationship is identical to that derived from the corresponding averages of this relationship. Numerical calculations with series of hypothetical monomolecular and polymolecular (exhibiting Schulz-Zimm distributions of the molar mass) polymer-homologous polymer samples in three different thermodynamically good solvents also demonstrate the superiority of the correction method according to Sutter et al. as compared to the correction method of Rosenvasser et al.

The radiation crosslinking and scission of some polyphosphazenes

The radiation behaviour of four aryloxy and one arylamino polyphosphazenes has been studied. In particular the gel formation has been measured as a function of the absorbed dose. The data have been interpreted in the form of Charlesby-Pinner plots, modified using the Dole-Inokuti relationship for an initial Schulz-Zimm molecular weight distribution. All the samples fitted the model well with no concurrent scission. The G( X) values were determined from the corresponding D gel values. The latter were obtained by extrapolation to S+;S 1 2 = 2.0. A wide range of G( X) values was found depending on the molecular structure of the polyphosphazenes. Possible reasons for the results are discussed.

Determination of the size distribution of bovine casein micelles using photon correlation spectroscopy

Photon correlation spectroscopy may be used to determine particle diffusion coefficients. By calculations involving two parameter Schulz-Zimm size distributions contaminated with trace amounts of larger particles, it has been demonstrated that the effects of the large particle component on the measured diffusion coefficient are observed first at small angles and that diffusion coefficients at wider angles depart from their zero contaminant values only at much higher levels of contamination. Provided the particles described by the Schulz-Zimm distribution are large enough to exhibit intramolecular interference, it has been confirmed that size distribution parameters can be extracted from the slope of a linear fit to this wide-angle data and its intercept at zero-angle. The casein micelle system from skim-milk contaminated by much larger fat globules exemplifies the systems described in these model calculations. Diffusion coefficients have been measured as a function of the scattering angle for casein micelles subjected successively to a series of fat-removal techniques. Integrated scattering measurements in the same angular range confirmed the loss of fat and also showed micellar scattering unaffected by the procedures. Though still causing the diffusion coefficients measured at angles less than 70° to increase, the final stages of fat removal had no effect on the values obtained at angles ⩾80°. These wide-angle data were then used to compute Schulz-Zimm distribution parameters to describe the casein micelle size distribution.

Correction for axial dispersion in gel permeation chromatography with a detector of molecular masses

Procedures suggested for data correction of a concentration detector in gel permeation chromatography for axial dispersion may be used in data correction of a detector of molecular masses, e.g., viscometer or light scattering cell. The procedure is demonstrated using model chromatograms calculated for two polymers of different polydispersity obeying the Schulz-Zimm distribution function. A procedure is outlined for determination of the spreading factor h by comparing molecular masses calculated from data obtained by both detectors with values provided by the calibration dependence of the column.

Determination of the number‐, weight‐, and z‐average dimensions of macromolecules by viscosity and GPC measurements

AbstractThe molecular weight averages are calculated which have to be inserted into the Fox‐Flory relationship \documentclass{article}\pagestyle{empty}\begin{document}$ {{\left[ \eta \right] = \Phi (\overline {r_{{\rm av}}^{\rm 2} } )^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern‐\nulldelimiterspace} 2}} } \mathord{\left/ {\vphantom {{\left[ \eta \right] = \Phi (\overline {r_{{\rm av}}^{\rm 2} } )^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern‐\nulldelimiterspace} 2}} } {M_{{\rm av}} }}} \right. \kern‐\nulldelimiterspace} {M_{{\rm av}} }} $\end{document} if the number‐, weight‐, or z‐average dimensions are to be determined for polymolecular polymer samples. The resulting complex molecular weight averages \documentclass{article}\pagestyle{empty}\begin{document}$ M_{r_{\rm n} } \eta _{\rm w},{\rm }M_{r_{\rm w} } \eta _{\rm w} $\end{document}, and \documentclass{article}\pagestyle{empty}\begin{document}$ M_{r_{\rm z} } \eta _{\rm w} $\en{document} are compared with simple molecular weight averages and for Schulz‐Zimm and logarithmic normal distributions of the molecular weight their numerical relations to the viscosity‐(Mη), number‐ (Mn), and weight‐average molecular weight (Mw) calculated. A simple straight‐forward method is outlined for the determination of the number‐, weight‐, and z‐average dimensions of polymolecular polymers from viscosity and gel permeation chromatography measurements.

Calculation of the molecular weight distribution of crosslinked polymer chains using the theory of branching processes

AbstractA numerical method for calculation of the weight‐fraction distribution resulting from random crosslinking of polydisperse primary chains is suggested. The method is based on the numerical inverse Laplace transform of the weight‐fraction generating function given by the theory of branching processes. The applicability of the method was demonstrated using the Schulz‐Zimm type distribution of primary chains. A good agreement was obtained by comparing the results with those following from the kinetic theory of crosslinking for the most probable distribution of primary chains at low degrees of crosslinking.

Quality control of clinical dextran by rapid gel-permeation chromatography

Clinical dextran can be fractionated by rapid gel-permeation chromatography on SpheronR with the aid of a high-performance liquid chromatograph. The linear calibration function is obtained by dextran fractions whose molecular weights Mw and Mn are determined by absolute methods (light scattering, membrane osmometry) and end group analysis. The transformation of elution diagrams into a molecular weight distribution is described. Characteristic data for dextran 40 and 70 were obtained from the integral distribution curves. The molecular weight distribution of dextran fractions was found to fit a Schulz-Zimm distribution which is typical for long chain polymers. For quality control purposes clinical dextran can be characterized by its molecular weight distribution with high accuracy within one hour.

Change of Molecular Size Distribution of Polymers Induced by Crosslinking. II. Poisson Distribution

The change of molecular size distribution of polymers induced by crosslinking is obtained in the case when the initial distribution is Poisson's type. The behaviour of it near the gel point when p is large is given by m ( p , t )∼ p -5/2 . For the Schulz-Zimm distribution, some conjecture about the asymptotic behaviour of the size distribution is given.

Change of Molecular Size Distribution of Polymers Induced by Crosslinking. I. Uniform Distribution

The change of molecular size distribution of polymers induced by crosslinking is obtained in the case when the initial distribution is Poisson's type. The behaviour of it near the gel point when p is large is given by m ( p , t )∼ p -5/2 . For the Schulz-Zimm distribution, some conjecture about the asymptotic behaviour of the size distribution is given.

Theoretical Depolymerization Kinetics. II. The Effect of Molecular-Weight Distribution in Degrading Polymers Undergoing End-Group Initiation

It is shown that the set of coupled rate equations describing the end-group-initiated degradation of a polymer containing various degrees of polymerization can be transformed into a set of coupled equations describing the rate of change of the moments of the molecular-weight distribution. These moment equations can be used as the basis for approximate but systematic treatments of degradation in systems with arbitrary initial molecular-weight distributions. In particular, in this paper it is shown that if the Schulz—Zimm distribution is assumed to be sufficiently general to represent the molecular-weight distribution through the course of the degradation, the moment equations allow the problem to be reduced to three simultaneous differential equations. These equations have been solved by digital computer and the sample weight, number-average molecular weight, and distribution-width parameter tabulated as functions of time for a wide range of kinetic parameters and initial distributions.

Theoretical Depolymerization Kinetics. III. The Effect of Molecular-Weight Distribution in Degrading Polymers Undergoing Random-Scission Initiation

Differential equations describing the rate of change of the moments of the molecular-weight distribution in the random-scission-initiated degradation of a heterodisperse polymer have been derived from the kinetic rate equations. These moment equations are used as the basis of an approximate treatment in which differential equations for the parameters in the Schulz—Zimm distribution are derived. These three equations have been solved by digital computer for a wide range of initial distributions and kinetic parameters. The effect of molecular-weight distribution on this type of degradation is discussed on the basis of the above equations.

Evaluation of the molecular weight distribution and crosslink density of the sol and gel components by intrinsic viscosity measurements

AbstractA theory is developed utilizing the partitioning of a partially crosslinked polymer into sol and gel fractions when the initial molecular weight may be expressed as a Schulz‐Zimm or sum of Schulz‐Zimm distributions. The results show that the initial molecular weight distribution may determined from the ratio of the initial and sol fraction intrinsic viscosities. The treatment is also extended to the case where chain degradation accompanies the cure. The theory also permits the evaluation of the gel, sol, and overall crosslink index and coefficient of each component distribution of the polymer as a function of the weight soluble fraction. The components may be the same kind or different kinds of polymers or copolymers. The computed results for a two‐component system clearly show how the differences between the molecular weights or the distributions of the two components influence the difference between the crosslink indices of the two components in the vulcanized gel.