Predicting the Change of MWD Caused by Interchange Reactions During Melt‐Mixing of Linear and Branched Polycondensates (AB2)

During mix melting of hyperbranched AB₂‐ and linear CD‐polycondensates distributive properties are changing by interchange reactions. Two mathematical modeling approaches are presented: (i) Simplified approach of monodisperse population of three‐arm stars undergoing interchange reactions, both analytical and by Monte Carlo simulations, assuming interchange as subsequent scission and recombination of fragments. (ii) Full system of interchange and polycondensation/hydrolysis reactions with Monte Carlo simulations and kinetic model describing reactions of free groups (A, B, C, D) and bonds (AB, CD, BC, AD). MC simulations show that the final molecular weight and branching distribution is attained after 10% of reaction time. The change of structure, from few large fragments to more, smaller ones, is slower.

     

Experimentally Calibrated Random Walk of Branched Polymers: A Pragmatic Approach

In order to control the branching behavior of polymers, the comparison of experimental and simulated data is important. The utilization of a nonlattice, self‐avoiding necklace‐bead random walk simulator is reported, which allows for the calculation of radii of gyration r _g of polymer molecules with branched structures. The focus is on sensitivity toward short‐chain branches, long‐chain branches (LCBs), and the copolymer composition. Using only two parameters—the size of monomer beads and the minimum angle between three subsequent beads—a fast and reliable parameter fit procedure based on experimental data is described. The procedure is exemplarily shown for copolymers of vinylidene fluoride and hexafluoropropene (HFP) with HFP contents in the copolymer of at most 0.3 and is easily transferable to other polymers that may be analyzed by size‐exclusion chromatography/multiangle laser light scattering close to θ conditions. Applying the Zimm–Stockmayer equation to simulated r _g data allows for comparing the “effective” number of LCBs with the number of LCBs given by kinetic simulations. A tool for better estimation of rate coefficients associated with the formation of short‐ and long‐chain branches is provided.

     

Computing the Trivariate Chain Length/Degree of Branching/Number of Combination Points Distribution for Radical Polymerization with Transfer to Polymer and Recombination Termination

Summary: Polymer molecules made by radical polymerization with transfer to polymer and recombination termination contain branch points (connecting branch arms to backbones) and combination points (connecting various molecular structures). The trivariate chain length/degree of branching/number of combination points distribution (CLD/DBD/CPD) was calculated using a two‐dimensional version of a previously used pseudo‐distribution approach. This yielded the CPD moments for given chain length n and number of branches i. Both DBD and CPD at given chain length resemble a binomial distribution. For the construction of the full DBD and CPD a set of orthogonal polynomials, the Krawtchouk polynomials, in combination with the binomial distribution was employed. A first‐order Krawtchouk approximation enabled to compute the full CLD/DBD/CPD from the three CPD moments as a function of n and i. Results agree well with those from a Monte Carlo (MC) simulation method. However, the large scatter due to the small numbers of molecules collected in the MC method at longer chain lengths prevents comparison in this range.

Solutions of 3D CLD/DBD/CPD: CPD at constant chain length (20 000) and number of branch points (20).     

Monte Carlo simulation of diepoxides and monoepoxides cured with amines

The diepoxide–monoepoxide–diamine curing systems are investigated with a Monte Carlo simulation. The dependence of the molecular weight distribution (MWD), gel fraction, and cycle rank of the polymers on the differences in the epoxy reactivities and the contents of the monoepoxide as a reactive diluent are discussed. Before gelation, the MWD of the curing systems with a lower content of the monoepoxide is broader than the MWD of the curing systems with a higher content, and it leads to a lower critical conversion. The gel fraction and cycle rank of the polymers decrease with an increasing amount of the diluent. Even fully cured, the system with a 0.6 epoxy molar fraction of the monoepoxide still has a large fraction of sol, about 49%. Although the various reactivities of the monoepoxide result in different ways of forming gels during curing, the final gel fractions are always near 100% as long as the epoxy molar fraction of the diluent is no more than 0.2. The profiles of the molecular weights of the polymers calculated by the model are in agreement with the experimental data. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1857–1868, 2002     

Monte Carlo Simulation of Controlled/Living Radical Polymerization in Emulsified Systems

A new MC simulation method is proposed for the controlled/living radical polymerization in a dispersed medium, assuming an ideal miniemulsion system. This tool is used to consider the effects of particle size on the polymerization rates and the molecular weight distributions. For NMP, the polymerization kinetics are basically governed by two conflicting factors, (i) the confined space effect that promotes the coupling reaction between a radical and a trapping agent and (ii) the isolation effect of radicals into different particles that suppresses the overall frequency of bimolecular termination. For RAFT polymerization, a significant rate enhancement by reducing the particle size could be observed only for the systems with fast fragmentation of adduct radicals.

     

Molecular Weight Development during Simultaneous Chain Scission,Long‐Chain Branching and Crosslinking, 1

A general matrix formula is proposed for the weight‐average molecular weights of the polymer systems formed through simultaneous scission, branching and crosslinking of N types of chains, assuming the chain connection statistics are Markovian. For the polymerization systems in which chains are generated consecutively, such as for free‐radical polymerization, the present theory can be applied by increasing the number of chain types N to infinity, by considering the chains formed at different times as different types of chains. The gel point determination reduces to the eigenvalue problem and the present theory extends the classical gelation theory to non‐random, history‐dependent reaction systems. From the mathematical point of view, this theory is capable of describing complex molecular build‐up processes through end‐linking, T‐ and H‐shaped chain connections, irrespective of reaction/reactor types used.

Schematic representation of the 0^th generation segment and the connection to the 1^st generation segments.     

The Quenched Instationary Polymerization System, 5. The Peak Broadness as an Invariant Quantity of the Number,Molar Mass and Hyper Distributions

The total number, molar mass and hyper distributions generated by quenched instationary polymerization techniques are dominated by the radical chain length distribution (RCLD) whereas the contribution from the polymer chain length distribution (PCLD) is in most cases negligible. For the determination of the rate constant of propagation (k_p) the location of different extraordinary points of the distribution curves is determined by the use of the first and second derivatives. For the number, molar mass and hyper distributions these points are related in an unambiguous way to k_p[M]t_x and can be used to extract k_p. The choice of t_x (duration of the dark period, or an initiation period, or the sum of different periods) depends on the experimental conditions (δ‐pulse, incomplete pre‐effect, combination of periods differing in initiation extent) and is essential for the proper determination of k_p. The broadness of appearing peaks (introduced as the difference between two successive points of inflections) turned out to remain the same irrespective which type of distribution curve was analyzed. Analytical expressions for the peak broadness were derived for different types of quenched instationary polymerization conditions. For δ‐pulse initiation the broadness of the Poisson peak depends simply on the number of propagation steps that occurred whereas for non‐δ‐pulse initiation conditions the peak broadness is governed by the corresponding duration of the initiation period.     

Monte Carlo Simulation of Ethylene Copolymers: A Look into the Monomer Sequence Distribution

A comprehensive mathematical model was developed using Monte Carlo simulation to describe the mechanism of ethylene and α-olefin copolymerization. The model studies the polymerization mechanism using coordination catalysts and is able to predict molecular weight and detailed chemical composition distributions. This work is considered to be a useful tool that enables us to understand and described the monomer sequence distribution as a function of chain length in semi-batch polymerization reactors.     

Synthesis of Branched Polymer Architectures from Molecular Weight and Branching Distributions for Radical Polymerisation with Long‐Chain Branching,Accounting for Topology‐Controlled Random Scission

Branched polymers like LDPE are known to possess a wide range of architectures. In this paper a modelling approach is developed, describing the relation between architectures, chemistry and reactor conditions with the general objective of improving characterisation and controlling visco‐elastic properties. More specifically, the particular scission kinetics of branched molecules as strongly contrasting with linear scission is described. A new method to synthesise branched architectures is developed as an alternative to full Monte Carlo (MC) sampling. It employs MC sampling for coupling of primary polymers only. Graph theory is used as an efficient storage method containing all topological information of individual molecules. The algorithm synthesises molecules for any given combination of chain length (n) and number of branches (N). The explicit and detailed knowledge of branched architectures allows finding the correct topological scission kinetics. Distributions of fragment lengths and numbers of branches on fragments after scission are obtained, showing a preference for short and long fragments. Approximate functions describing this have been implemented in another model, predicting molecular weight (MWD) and degree of branching (DBD) distributions using a Galerkin finite element method. Topological scission is seen to give MWD broadening and a higher branching density for long chains. Distributions of longest end‐to‐end distances could be computed for all architectural alternatives for given n, N. In conclusion, it is demonstrated that this method yields distributions of architectures consistent with MWD/DBD for radical polymerisation with long‐chain branching and random scission.     

Bimodal molecular weight distribution formed in the emulsion polymerization of ethylene

It is known that the molecular weight distribution (MWD) formed in an emulsion polymerization of ethylene can be bimodal. However, the origin of the bimodality has not been elucidated. In this article, a Monte Carlo simulation is conducted, mostly with parameters reported in the literature. The simulated MWDs are bimodal because of the limited volume effect; that is, the high molecular weight profiles are distorted by the small particle size, which is comparable to the size of the largest branched polymer molecule in a particle. The simulated MWDs agree reasonably well with the experimentally obtained MWDs. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 3426–3433, 2002     

On the Contraction Factor of Star‐Branched Polymers with Flory‐Distributed Arms

More than fifty years ago, Zimm and Stockmayer calculated the average contraction factor of star‐branched polymers (stars) with uniformly distributed arms to be 6f/{(f + 1)(f + 2)}. Since then this contraction factor has also been used for stars with other arm distributions. In this paper we determine the (probability) density function of the contraction factor of stars with arms with a Flory (most probable) distribution and conclude that this function is equal to that for stars with uniformly distributed arms. Other arm distributions, however, lead to different contraction factor density functions. The moments of the contraction factor distribution were precisely determined with the aid of a recursion method. The stochastical behavior of the contraction factor of stars was applied to size‐exclusion chromatography (SEC) analysis and showed that upward correction of the crude SEC data is necessary to determine the proper molecular‐mass distribution of stars.     

On the theory of radical depolymerization: A rigorous solution

A rigorous solution to a model of radical depolymerization is presented. The model includes random chain scission, depropagation, radical transfer, and first‐order radical termination. The evolution of the molecular weight distribution in the course of depolymerization has been determined under the condition that the initial polymer is characterized by Flory's distribution. The kinetic equations consider the presence of chains with two radicalized ends, which are usually neglected. The commonly used simplifying assumption of the steady‐state radical concentration is not employed, and this makes the obtained results valid at any ratios between the rate constants. The predictions of the steady‐state approximation are compared to those of the rigorous approach in the case of depolymerization accompanied by volatilization of monomeric species. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 965–982, 2003     

Chain Transfer and Efficiency of End‐Group Introduction in Free Radical Polymerization of Methyl Methacrylate in the Presence of Poly(methyl methacrylate) Macromonomer

Summary: Experimental and modeling studies of addition–fragmentation chain transfer (AFCT) during radical polymerization of methyl methacrylate in the presence of poly(methyl methacrylate) macromonomer with 2‐carbomethoxy‐2‐propenyl ω‐ends (PMMA‐CO₂Me) at 60 °C are reported. The results revealed that AFCT involving PMMA‐CO₂Me formed in situ during methyl methacrylate polymerization has a negligible effect on the molecular weight distribution.

     

Chain transfer to solvent in the radical polymerization of N‐isopropylacrylamide

Chain transfer to solvent has been investigated in the conventional radical polymerization and nitroxide‐mediated radical polymerization (NMP) of N‐isopropylacrylamide (NIPAM) in N,N‐dimethylformamide (DMF) at 120 °C. The extent of chain transfer to DMF can significantly impact the maximum attainable molecular weight in both systems. Based on a theoretical treatment, it has been shown that the same value of chain transfer to solvent constant, C_tr,S, in DMF at 120 °C (within experimental error) can account for experimental molecular weight data for both conventional radical polymerization and NMP under conditions where chain transfer to solvent is a significant end‐forming event. In NMP (and other controlled/living radical polymerization systems), chain transfer to solvent is manifested as the number‐average molecular weight (M_n) going through a maximum value with increasing monomer conversion. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011     

Accelerated iterative strategy for the divergent synthesis of dendritic macromolecules using a combination of living radical polymerization and an irreversible terminator multifunctional initiator

Our laboratory has reported the elaboration of an iterative strategy for the synthesis of dendritic macromolecules from conventional monomers. This synthetic method involves a combination of self‐regulated metal‐catalyzed living radical polymerization initiated from arenesulfonyl chlorides and an irreversible terminator multifunctional initiator (TERMINI). The previous TERMINI, (1,1‐dimethylethyl)[[1‐[3,5‐bis(S‐phenyl‐4‐N,N′ diethylthiocarbamate)phenyl]ethenyl]oxy]dimethylsilane, was prepared in nine reaction steps. The replacement of the previous TERMINI with one that requires only three steps for its synthesis, diethylthiocarbamic acid S‐{3‐[1‐(tert‐butyl‐dimethyl‐silanyloxy)‐vinyl]‐5‐diethylcarbamoylsulfanyl‐phenyl} ester, and the use of the more reactive Cu₂S/2,2′‐bipyridine rather than the Cu₂O/2,2′‐bipyridine self‐regulated catalyst have generated an accelerated method for the synthesis of dendritic macromolecules. This method provides rational design strategies for the synthesis of dendritic macromolecules with different compaction by the use of a single monomer. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 4894–4906, 2005     

Simulation of Molecular Weight Distributions Obtained by Pulsed Laser Polymerization (PLP): New Analytical Expressions Including Intramolecular Chain Transfer to the Polymer

A new approach for the simulation of PLP (pulsed laser polymerization) is presented. This approach allows one to obtain new analytical solutions for different polymerization schemes, including either chain transfer to the monomer or intramolecular chain transfer to the polymer. The first results of the simulation of PLP experiments on n‐butyl acrylate at 20 °C and ambient pressure are presented.

MWDs simulated for PLP of n‐butyl acrylate, in bulk at 20 °C and ambient pressure using three models: the model with intramolecular chain transfer to the polymer (solid line), the model with chain transfer to monomer (dashed line), and the classical model (dotted line).     

Dispersion copolymerization of methyl methacrylate and n-butyl acrylate

In the dispersion copolymerization of methyl methacrylate (MMA) and n-butyl acrylate (BA), the particle size increases with an increasing MMA fraction in the comonomer. The power dependence of the particle size on the initiator concentration also increases with an increasing MMA concentration. Similar to what can be found in the homopolymerizations, two populations can be observed in the molecular weight distributions of the copolymers. Core–shell structured particles with a poly(methyl methacrylate)-rich core and a poly(n-butyl acrylate)-rich shell result from the copolymerizations because of the significantly different reactivity ratios. The reaction rates of the dispersion copolymerization are lower than those of the homopolymerization of BA and close to or lower than those of the homopolymerization of MMA, depending on the ratio of the monomers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 2105–2112, 2007     

Polymer Distribution Change During Irreversible Depolymerization by Chain‐End Scission

The change in polymer distribution during depolymerization in which monomer molecules are severed one by one from the chain ends, is considered. Assuming irreversible depolymerization and equal reactivity of all the chain ends, a general formula to calculate the MWD is proposed. After the removal of monomers severed from the chains, the MWDs of depolymerized polymers always approach the most probable distribution whose PDI is practically equal to 2. It may appear to be counterintuitive, but the average MWs of polymers may increase during chain scission when the initial distribution is broader than the most probable (PDI > 2). The interpretation of the MWD data of polymers during depolymerization, such as those obtained by GPC, are not straightforward especially for the initial broad MWDs.

     

Evolution of the termination rate coefficient in styrene polymerization

Styrene bulk polymerization was conducted at 70 °C with a high initiator concentration, and this ensured that the dominant chain‐stopping mechanism was the combination of free radicals. The evolution of the molecular weight distribution (MWD) of the polymer was measured via the periodic removal of samples during the course of the reaction and their analysis with gel permeation chromatography. The overall termination rate coefficient was independent of the conversion in the dilute regime, as observed from cumulative MWDs. In the middle of the conversion range, the observed trend was compatible with a translational‐diffusion‐controlled mechanism for the termination step. A bimodal distribution of the molecular weights was also found at high conversions and could be explained in terms of an increase in the free‐radical concentration and a very low termination rate coefficient. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 178–187, 2005     

Molecular Weight Distribution of Living Radical Polymers

Summary: The molecular weight distribution formed in an ideal living radical polymerization is considered theoretically. It was found that the hypergeometric function that combines the most probable and the Poisson distribution represents a fundamental distribution of the living radical polymers. The number‐ and weight‐average molecular weights are derived for this fundamental distribution, together with those for polymerizations in a batch and in a continuous stirred tank reactor. These average molecular weight functions are obtained based on the arithmetic calculations without deriving the distribution functions. The effect of the monomer transfer reactions on the formed MWD is also considered. The present study clarifies the relationship between the reaction mechanism and the formed molecular weight distribution as well as the fundamental characteristics of living radical polymers.

Calculated number fraction distribution N(r) development with (dashed) and without (solid) the monomer transfer reactions.