An analytical solution of a newly proposed peak broadening equation and extension of polydispersities to higher molecular weight averages in size exclusion chromatography (SEC)

Summary Herein is reported an analytical solution to the peak broadening or peak dispersion/flattening equation based on the recently proposed Instrumental Spreading Shape Function and its application to correction for imperfect resolution (inadequate peak separation and/or excessive peak broadening) for higher molecular weight averages. The relationship of these higher MW averages with the familiar Weight Average and number average molecular weights is also discussed. Criteria for perfect resolution are specified and a true molecular weight calibration curve is accordingly defined.     

Poly(N-isopropylacrylamide). I. Interactions with sodium dodecyl sulfate measured by conductivity

Conductometric titration of poly(N-isopropylacrylamide) (polyNIPAM) with sodium dodecyl sulfate (SDS) gave two apparent transitions labeled C1 and C2. The C1 transition was independent of polyNIPAM concentration in the 0.05–0.3 wt % range, whereas C2 was proportional to the polymer concentration. C1 corresponded to the onset of binding of surfactant with polymer. Arguments based on a simple mass action model for micellization are presented to show that C2, the second transition, is not due to any simple explanation such as being the point above which only free micelles are formed with surfactant addition. The cloud point of polyNIPAM increased with the amount of bound surfactant. This was attributed to electrostatic contribution of bound sulfate groups to the increased solubility of polyNIPAM. © 1993 John Wiley & Sons, Inc.     

Modelling free-radical copolymerization kinetics—evaluation of the pseudo-kinetic rate constant method, 2. Molecular weight calculations for copolymers with long chain branching

The full moment equations and equations using pseudo-kinetic rate constants for binary copolymerization with chain transfer to polymer in the context of the terminal model have been developed and solved numerically for a batch reactor operating over a wide range of conditions. Calculated number- and weight-average molecular weights (M̄_n and M̄_w) were compared with those found using the pseudo-kinetic rate constant method (PKRCM). The results show that the weight-average molecular weights calculated using PKRCM are in agreement with those found using the method of full moments for binary copolymerization when polymeric radical fractions φ₁^˙ and φ₂^˙ of type 1 and 2 (radical centers are on monomer types 1 and 2 for a binary copolymerization) are calculated accounting for chain transfer to small molecules and polymer reactions in addition to propagation reactions. Errors in calculating M̄_w using PKRCM are not always negligible when polymer radical fractions are calculated neglecting chain transfer to small molecules and polymer. In this case, the relative error in M̄_w by PKRCM increases with increase in monomer conversion, extent of copolymer compositional drift and chain transfer to polymer rates. The errors in calculating M̄_w, however, vanish over the entire monomer conversion range for all polymerization conditions when chain transfer reactions are properly taken into account. It is theoretically proven that the pseudo-kinetic rate constant for chain transfer to polymer is valid for copolymerizations. One can therefore conclude that the pseudo-kinetic rate constant method is a valid method for molecular weight modelling for binary and multicomponent polymerizations.     

Modelling free-radical copolymerization kinetics, 3. Molecular weight calculations for copolymers with crosslinking

A comprehensive model for molecular weight calculations of free-radical crosslinking copolymerizations was developed using the pseudo-kinetic rate constants and the method of moments. The moments of copolymer chain distributions are defined in such a way so that the molecular weight averages of crosslinking copolymers can be calculated using the moments. The present model is based on a general crosslinking copolymerization scheme, accounting for chain transfer to small molecules and polymer, bimolecular termination, and crosslinking reactions. The influence of crosslinking reactions on molecular weight development is discussed. The effects of the reactivity of pendant double bonds on the moments development were further demonstrated using model simulations. The simulations results suggest that the higher-order molecular weight averages are very sensitive to the reactivity of pendant double bonds. It was found that chain transfer to polymer affects the gelation point significantly. The radical fractions must be calculated accounting for chain transfer reactions in addition to propagations in order to properly evaluate pseudo-kinetic rate constants. The present model was used to predict kinetic behavior and molecular weight development of styrene/m-divinylbenzene and styrene/ethylene dimethacrylate free-radical crosslinking copolymerizations in benzene solution at 60°C. It was found that the present model is in excellent agreement with the experimental data published in the literature. Model predictions and experimental data show that the reactivity of pendant double bonds is much lower than that of vinyl and divinyl monomers. The simulation results suggest that the assumption of the same reactivity of functional groups is likely not valid for many free-radical crosslinking copolymerizations. The present model based on a kinetics approach can be used to predict molecular weight development for vinyl/divinyl free-radical crosslinking copolymerizations and to estimate kinetic parameters in the pre-gelation period.     

Effect of reactor residence time distribution on the size distribution of polymer particles made with heterogeneous Ziegler-Natta and supported metallocene catalysts. A generic mathematical model

A generic mathematical model for analyzing the effect of ideal and non-ideal reactor residence time distributions on the size distribution of polymer particles produced with heterogeneous Ziegler-Natta and supported metallocene catalysts was developed. It was shown that the residence time distribution in polymerization reactors can have a significant effect on the size distribution of polymer particles and this can lead to imperfect replication of the catalyst particle size distribution.     

Molecular‐Weight‐Dependence on Domain Formation of Grafted Poly(ethylene‐co‐propylene) in a Poly(propylene) Matrix

Several novel poly(propylene)‐graft‐poly(ethylene‐co‐propylene) copolymers with isotactic poly(propylene) (PP) backbones and ethylene/propylene rubber (EPR) branches were synthesized. The thermomechanical properties of these samples were investigated using a dynamic mechanical analyzer. There appeared to be a critical EPR molecular weight above which a two‐phase system developed with EPR domains dispersed in a PP matrix. This domain formation gave an enhanced loss modulus compared to a commercial high impact PP product below 40°C.     

Polymerization of methyl methacrylate up to high degrees of conversion: Experimental investigation of the diffusion-controlled polymerization

The kinetic model given by Marten and Hamielec that describes the bulk polymerization of methyl methacrylate (MMA) and accounts for diffusion-controlled termination and propagation was modified to include termination by combination and reaction diffusion and was then tested using isothermal conversion/time and molecular weight data obtained dilatometrically at various temperatures and with three different initiators. For each series of measurements two adjustable parameters were fitted to the conversion/time data. Excellent fits were obtained and the adjustable parameters were found to be the same for all concentration levels of the three initiators and to vary in a simple manner with temperature. The predicted molecular weight averages and molecular weight distributions were in satisfactory agreement with those found experimentally considering the difficulty of measuring high molecular weight PMMA by GPC. This model thus satisfies the specifications for a polymer reactor model that can be used to optimize commercial production systems.     

Bivariate chain length and long chain branching distribution for copolymerization of olefins and polyolefin chains containing terminal double-bonds

Polyolefins containing long chain branches can be synthesized using certain metallocene catalysts such as Dow Chemical's constrained geometry catalyst. These polyolefins combine the excellent mechanical properties of polymers with narrow molecular weight distribution with the easy processability of polymers containing long chain branches. A mathematical model for the chain length distribution for these novel polyolefins was derived from basic principles and an analytical solution for the chain length distributions of the populations containing different number of long chain branches per polymer molecule was obtained. The analytical solution agrees with the direct solution of the population balances and with a Monte-Carlo simulation model. It is also shown that this solution applies for copolymers using pseudo-kinetic rate constants and Stockmayer's bivariate distribution.     

Analyzing TREF data by stockmayer's bivariate distribution

The Stockmayer bivariate distribution is used to qualitatively and quantitatively interpret TREF (temperature rising elution fractionation) spectra of polyolefins made by multiple site type catalysts. TREF spectra simulated by the proposed method can adequately fit experimental TREF data presented in the literature and can be used to help understand the mechanism of TREF separation and the nature of multiple site types catalysts.