Crosslinking index,molecular weight distribution and rubber equilibrium shear modulus during polyfunctional crosslinking of existing polymer: Part 4, recapitulation and presentation of general results of calculations for a hypothetical crosslinking process

In previous papers a statistical theory was presented concerning network formation by polyfunctional crosslinking of existing polydisperse (non-uniform) primary polymers. Relationships were derived between network parameters and the equilibrium shear modulus during crosslinking processes of polymers of various molecular weight distributions. In the present paper the various relationships obtained are compared. Moreover, results of calculations for a hypothetical crosslinking process are presented, such as the weight fractions of sol, ideal network and free or dangling ends and the molecular weights between crosslinks as functions of the equilibrium shear modulus for various molecular weight distributions. Furthermore, the results of fractionation of the primary polymer, as a consequence of the crosslinking process, are shown and also the crosslinking indexes as functions of the sol fraction.     

Crosslinking index,molecular weight distribution and rubber equilibrium shear modulus during polyfunctional crosslinking of existing polymer: Part 3, primary polymer with a cumulative flory distribution or a cumulative Schulz-Flory distribution of the molecular weights

In two previous papers theoretical and experimental results of a statistical theory were presented concerning gels formed by the polyfunctional crosslinking of an already existing polydisperse primary polymer of known, but relatively simple molecular weight distribution. The present paper deals with the calculation of network parameters during the crosslinking process of a primary polymer with a cumulative Flory or a cumulative Schulz-Flory distribution of the molecular weights. Relationships between the equilibrium shear modulus, the average crosslinking indices (in the system as a whole and the network fraction as well), the sol fraction, the fraction of ideal network, the fraction of dangling ends and the functionality of the crosslinks are given and also the molecular weights between crosslinks. With these relationships it is possible to calculate from the measured equilibrium shear modulus the network parameters, provided the number and weight average molecular weights of the polymer and the crosslink functionality are known.     

Crosslinking Kinetics: Partitioning According to Number of Crosslinks

Summary: To mathematically describe crosslinking kinetics for polymers, we have proposed a novel method that accounts for the number of crosslinks, that is, partitioning according to number of crosslinks (PANC). By contrast, the well‐known method of numerical fractionation tracks generations of crosslinked molecules, defined to include a range of crosslinks. The proposed crosslinking kinetics yield a population balance model that provides moments and hence measurable average properties such as number average molecular weight and weight average molecular weight, polydispersity and average crosslink number. The gel points for batch and continuous‐flow stirred‐tank reactors are derived. Because the usual closure methods do not yield satisfactory convergence, new representations for post‐gelation moments are proposed. The results realistically show how the moments change with time in the post‐gel region.

The average number of crosslinks in the bulk and sol versus time in a batch reactor; the gel point is the dashed line.     

Molecular Weight Distribution in Living Polymerization Proceeding with Reshuffling of Polymer Segments due to Chain Transfer to Polymer with Chain Scission, 2. Monte Carlo Simulation of Polymer Reshuffling Proceeding with Disproportionation of Chain Functionalities

The method for analyzing the reshuffling of polymer segments developed previously has been extended to systems involving the disproportionation of chain functionalities. The effect of interchain exchange reactions of this type, leading to the redistribution of chain lengths and of the chain functionalities (redistribution of living and dead chain ends), was analyzed by means of the Monte Carlo simulations. In the systems, in which no propagation occurs (monomer concentration equal to zero), a set of polymer chains containing one living and one dead end was taken as an initial material. A series of simulations were performed for systems with differing molecular weight distributions of the starting macromolecules. Uniform (no chain length distribution polymer – all chains are of the same length), Poisson, and the most probable (geometric) distributions were taken into consideration. Although the molecular weight distributions (MWDs) of functionally different chains of the same polymer were different apart from the eventual equilibrium conditions, the overall MWD was very close to that observed in analogous systems without disproportionation. The same was observed concerning MWDs in modeled polymerization systems, in which reshuffling and disproportionation accompanied propagation. Consequently, a method of estimating the ratio of rate constants of propagation and reshuffling (i. e. k_p /k _tr) in the relevant polymerization systems, using the observed polydispersity indexes, was proposed. The extent of disproportionation can be evaluated from the determined relationships of the polydispersity index and of the monofunctional chains fraction as functions of the average number of chain transformations.     

Molecular weight distribution in random crosslinking of polymer chains

The molecular weight distribution (MWD) of crosslinked polymer molecules formed during polymeric network formation is the sum of the fractional MWDs containing 0, 1, 2, 3, … crosslinkages. The MWD for polymer molecules containing ?? crosslinkages is investigated for the random crosslinking of polymer chains whose initial MWD is given by the Schulz-Zimm distribution. For a very narrow initial MWD, each fractional MWD with ?? = 0, 1, 2, … is independent and a multimodal distribution is obtained for the whole distribution. When the initial MWD is uniform, the average crosslinking density within the polymer fraction whose degree of polymerization is r, ρ_r is simply given by ρ_r = ρ_gel,c – 2/r irrespective of the extent of crosslinking reaction where ρ_gel,c is the crosslinking density within gel fraction at the gel point. On the other hand, the MWDs with ?? crosslinkages overlap each other with different ?? values significantly for the broader initial distributions, and ρ_r increases with the progress of crosslinking reactions. The value of ρ_r increases with increasing r but levels off asymptotically at large r. The average crosslinking density of polymer molecules containing ?? crosslinkages ρ_?? is an increasing function of k but soon reaches a plateau; sooner for the broader initial MWDs. For ?? ≥ 1, ρ_?? is always larger than the average crosslinking density of the whole reaction system ρ in the pregelation period, i.e., in terms of the crosslinking density, the difference between polymer molecules with and without crosslinkage is most significant. In general, the average crosslinking density ρ, which is convenient to use in describing the nature of the whole reaction system, cannot be considered as a characteristic degree of crosslinking for polymer molecules containing at least one crosslinkage. Consideration of the bivariate distribution of r and k reveals important aspects of the polymeric network formation that have been obscured in the conventional theories in which the averages including linear polymers are solely considered. © 1995 John Wiley & Sons, Inc.     

有关聚合物平均分子量和分子量多分散性的教学点滴

从相对数均分子量和重均分子量的标准偏差角度来解释聚合物分子量分布指数与分布宽度的关系，利于学生对分子量分布指数含义的理解。     

Polymer characterization by interaction chromatography

Liquid chromatography (LC) is a powerful tool for the characterization of synthetic polymers, that are inherently heterogeneous in molecular weight, chain architecture, chemical composition, and microstructure. Of different versions of the LC methods, size exclusion chromatography (SEC) is most commonly used for the molecular weight distribution analysis. SEC separates the polymer molecules according to the size of a polymer chain, a well‐defined function of molecular weight for linear homopolymers. The same, however, cannot be said of nonlinear polymers or copolymers. Hence, SEC is ill suited for and inefficient in separating the molecules in terms of chemical heterogeneity, such as differences in chemical composition of copolymers, tacticity, and functionality. For these purposes, another chromatographic method called interaction chromatography (IC) is found as a better tool because its separation mechanism is sensitive to the chemical nature of the molecules. The IC separation utilizes the enthalpic interactions to vary adsorption or partition of solute molecules to the stationary phase. Thus, it is used to separate polymers in terms of their chemical composition distribution or functionality. Further, the IC method has been shown to give rise to much higher resolution over SEC in separating polymers by molecular weight. We present here our recent progress in polymer characterization with this method. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 1591‐1607, 2005     

Modeling of Polymer Network Formation from Preformed Precursors

Statistical and kinetic methods of modeling of polymer networks formation from pre‐prepared precursors are described and illustrated. Such precursors are characteristic of distributions of molecular weight and numbers of reactive groups of distinguishable type and reactivity, and contain branch points which get elastically active in the crosslinking stage. The examples include telechelic polymers with functionality distribution, functional copolymers composed of functional and non‐functional units and groups of different reactivity. The basics of statistical and kinetic network formation methods are outlined and their adaptation to cover the non‐uniform collection of precursor molecules as one of the components. The distributions of properties of the precursors are expressed by probability generating functions which serve as input for the crosslinking stage.

     

Molecular weight distribution effects on the structure of strongly adsorbed polymers by Monte Carlo simulation

Monte Carlo simulations are reported to study the structure of polymers adsorbed from solution onto strongly attractive, perfectly smooth substrates. Six systems spanning a range of molecular weight distributions are investigated with a coarse-grained united atom model for freely rotating chains. By employing a global replica exchange algorithm and topology altering Monte Carlo moves, a range of monomer-surface attraction from weak (0.27kT) to strong (4kT) is simultaneously explored. Thus for the first time ever, equilibrium polymer adsorption on highly attractive surfaces is studied, with all adsorbed molecules displaying similar properties and statistics. The architecture of the adsorbed layers, including density profiles, bond orientation order parameters, radii of gyration, and distribution of the adsorbed chain fractions, is shown to be highly dependent on the polydispersity of the polymer phase. The homology of polymer chains, and the ergodicity of states explored by the molecules is in contrast to the metastable, kinetically constrained paradigm of irreversible adsorption. The structure of more monodisperse systems is qualitatively similar to experimental results and theoretical predictions, but result from very different chain conformations and statistics. The polydispersity-dependent behavior is explained in the context of the competition between polymers to make contact with the surface.     

Crosslinking index,molar mass distribution and rubber equilibrium shear modulus during polyfunctional crosslinking: Part II: Application to real systems

In a previous paper a theory was presented concerning gels formed by polyfunctional crosslinks in a polydisperse primary polymer. The equilibrium shear modulus as well as the crosslinking index were obtained as functions of the functionality of the crosslinks and of the sol fraction. In this paper the theory is applied to gelatin gels (known functionality of crosslinks) and to poly(vinyl chloride) gels (known number of crosslinks).     

Theoretical Derivation of the Molecular Weight Distribution of End‐Capped Linear Condensation Polymers

End‐capped, low molecular weight polymers have found numerous practical applications. By providing the end‐capper molecules with specific chemical functionality, the polymer material can be equipped with a desired chemical behavior for product application or polymer processing. Using probabilistic methods, formulas are derived for calculating the target molecular weight distribution and its averages for the case of linear condensation polymerization. The formulas are generally applicable, allowing for arbitrary amounts of monofunctional monomers or end‐capper molecules affecting either one or both functional groups involved in the polymerization process.

     

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.     

A Theoretical Study of the Star-Coupling Reaction of Polymer Chains

The set of kinetic differential equations for star-coupling of polymers have been solved rigorously both with and without the effect of steric hindrance. The relations between the molecular parameters of star-branched polymers and those of prepolymers are derived. When the star-coupling reaction goes quantitatively to completion, the same theoretical result was obtained, no matter whether there is steric hindrance or not. In this case, the more arms in the star-branched polymer, the more homogeneous is the molecular weight distribution; if the number of arms is large enough, the molecular weight distribution will become very narrow and almost be independent of the polydispersity of the arms.     

Review: mass spectrometry of step-growth polymers

For at least two decades different mass spectrometric techniques have been applied for polymer analysis, including the qualitative determination of chemical compositions, end group identification, functionality type distribution and the determination of the cyclisation extend at each degree of polymerisation. Molecular weights exceeding 100 KDa are provable by MALDI-ToF mass spectrometry. Further, molecular weight distributions are determined by MADLI-ToF. However, there are some hints that, there is a discrimination against higher molecular weight species in samples with polydispersity of more than 1.2. Furthermore, pathways for dendritic and hyper-branched polymers, supramolecular polymers and nano condensates are analysed by help of mass spectrometry.     

Manipulation of Molecular Weight Distribution Shape as a New Strategy to Control Processing Parameters

Molecular weight and dispersity (Ð ) influence physical and rheological properties of polymers, which are of significant importance in polymer processing technologies. However, these parameters provide only partial information about the precise composition of polymers, which is reflected by the shape and symmetry of molecular weight distribution (MWD). In this work, the effect of MWD symmetry on thermal and rheological properties of polymers with identical molecular weights and Ð is demonstrated. Remarkably, when the MWD is skewed to higher molecular weight, a higher glass transition temperature (T _g), increased stiffness, increased thermal stability, and higher apparent viscosities are observed. These observed differences are attributed to the chain length composition of the polymers, easily controlled by the synthetic strategy. This work demonstrates a versatile approach to engineer the properties of polymers using controlled synthesis to skew the shape of MWD.     

Mathematical Modeling of Atom‐Transfer Radical Polymerization Using Bifunctional Initiators

Summary: Bifunctional initiators can produce polymers with higher molecular weight at higher initiator concentrations than monofunctional initiators. In this study, we developed a mathematical model for ATRP with bifunctional initiators. The most important reactions in ATRP were included in the model. The method of moments was used to predict monomer conversion, average molecular weights and polydispersity index as a function of polymerization time in batch reactors. The model was used to understand the mechanism of ATRP and to quantify how polymerization conditions affect monomer conversion and polymer properties by examining the effect of several rate constants (activation, deactivation, propagation and chain termination) and of catalyst and initiator concentration on polymerization kinetics and polymer properties. When compared to monofunctional initiators, bifunctional initiators not only produce polymers with higher molecular weight averages at higher polymerization rates, but also control their molecular weight distributions more effectively.

Effect of initial catalyst concentration on polydispersity index as a function of time.     

Atom-transfer radical polymerization of styrene with bifunctional and monofunctional initiators: Experimental and mathematical modeling results

Bulk atom transfer radical polymerization (ATRP) of styrene was carried out at 110 °C using benzal bromide as bifunctional initiator and 1-bromoethyl benzene as monofunctional initiator. CuBr/2,2′-bipyridyl was used as the ATRP catalyst. The polymerization kinetic data for styrene with both initiators was measured and compared with a mathematical model based on the method of moments and another one using Monte Carlo simulation. An empirical correlation was incorporated into the model to account for diffusion-controlled termination reactions. Both models can predict monomer conversion, polymer molecular weight averages, and polydispersity index. In addition, the Monte Carlo model can also predict the full molecular weight distribution of the polymer. Our experimental results agree with our model predictions that bifunctional initiators can produce polymers with higher molecular weights and narrower molecular weight distributions than monofunctional initiators. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 2212–2224, 2007     

A Cleavable Crosslinking Strategy for Commodity Polymer Functionalization and Generation of Reprocessable Thermosets

Covalently crosslinked polymeric materials, known as thermosets, possess enhanced mechanical strength and thermal stability relative to the corresponding uncrosslinked thermoplastics. However, the presence of covalent inter-chain crosslinks that makes thermosets so attractive is precisely what makes them so difficult to reprocess and recycle. Here, we demonstrate the introduction of chemically cleavable groups into a bis-diazirine crosslinker. Application of this cleavable crosslinker reagent to commercial low-functionality polyolefins (or to a small-molecule model) results in the rapid, efficient introduction of molecular crosslinks that can be uncoupled by specific chemical inputs. These proof-of-concept findings provide one potential strategy for circularization of the thermoplastic/thermoset plastics economy, and may allow crosslinked polyolefins to be manufactured, used, reprocessed, and re-used without losing value. As an added benefit, the method allows the ready introduction of functionality into non-functionalized commodity polymers.     

A generalized theory for the glass transition temperature of crosslinked and uncrosslinked polymers

A generalized theory for the glass transition temperature of crosslinked and uncrosslinked polymers has been developed, which takes into account the influences of end groups, branching, and crosslinking, and their functionality distribution. DiBenedetto's theory was found to correctly characterize the influence of crosslinks on the glass temperature. Normalized to constant crosslink functionality, the crosslink constant is a universal parameter suggesting that the entropic theory of glasses is applicable to crosslinked systems. Data on linear polymers and networks from the crosslinking of polymer chains, vinyl/divinyl-copolymers and step-growth polymers, such as polyurethanes, amine-cured epoxies, or inorganic glasses, are presented.