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.     

Synthesis of a variety of star‐shaped polybenzamides via chain‐growth condensation polymerization with tetrafunctional porphyrin initiator

A variety of well‐defined tetra‐armed star‐shaped poly(N‐substituted p‐benzamide)s, including block poly(p‐benzamide)s with different N‐substituents, and poly(N‐substituted m‐benzamide)s, were synthesized by using porphyrin‐cored tetra‐functional initiator 2 under optimized polymerization conditions. The initiator 2 allowed discrimination of the target star polymer from concomitantly formed linear polymer by‐products by means of GPC with UV detection, and the polymerization conditions were easily optimized for selective synthesis of the star polybenzamides. Star‐shaped poly(p‐benzamide) with tri(ethylene glycol) monomethyl ether (TEG) side chain was selectively obtained by polymerization of phenyl 4‐{2‐[2‐(2‐methoxyethoxy)ethoxy]ethylamino}benzoate ( 1b ′) with 2 at ?10 °C in the case of [ 1b ′]₀/[ 2 ]₀ = 40 and at 0 °C in the case of [ 1b ′]₀/[ 2 ]₀ = 80. Star‐shaped poly(p‐benzamide) with 4‐(octyloxy)benzyl (OOB) substituent was obtained only when methyl 4‐[4‐(octyloxy)benzylamino]benzoate ( 1c ) was polymerized at 25 °C at [ 1c ]₀/[ 2 ]₀ = 20. On the other hand, star‐shaped poly(m‐benzamide)s with N‐butyl, N‐octyl, and N‐TEG side chains were able to be synthesized by polymerization of the corresponding meta‐substituted aminobenzoic acid alkyl ester monomers 3 at 0 °C until the ratio of [ 3 ]₀/[ 2 ]₀ reached 80. However, star‐shaped poly(m‐benzamide)s with the OOB group were contaminated with linear polymer even when the feed ratio of the monomer 3d to 2 was 20. The UV–visible spectrum of an aqueous solution of star‐shaped poly(p‐benzamide) with TEG side chain indicated that the hydrophobic porphyrin core was aggregated. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011     

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.

     

Deconvolution of Molecular Weight Distributions Using Dynamic Flory‐Schulz Distributions

Summary: The deconvolution of molecular weight distributions (MWDs) may be useful for obtaining information about the polymerization kinetics and properties of catalytic systems. However, deconvolution techniques are normally based on steady‐state assumptions and very little has been reported about the use of non‐stationary approaches for the deconvolution of MWDs. In spite of this, polymerization reactions are often performed in batch or semi‐batch modes. For this reason, dynamic solutions are proposed here for simple kinetic models and are then used for deconvolution of actual MWD data. Deconvolution results obtained with dynamic models are compared to deconvolution results obtained with the standard stationary Flory‐Schulz distributions. For coordination polymerizations, results show that dynamic MWD models are able to describe experimental data with fewer catalytic sites, which indicates that the proper interpretation of the reaction dynamics may be of fundamental importance for kinetic characterization. On the other hand, reaction dynamics induced by modification of chain transfer agent concentration seem to play a minor role in the shape of the MWD in free‐radical polymerizations.

This Figure illustrates that MWDs obtained at unsteady conditions should not be deconvoluted with standard steady‐state Flory‐Schulz distributions.     

Exploring Reaction Kinetics of a Multi‐Site Ziegler‐Natta Catalyst Using Deconvolution of Molecular Weight Distributions for Ethylene‐Hexene Copolymers

Industrial ethylene‐hexene copolymer samples produced using a supported Ti‐based Ziegler‐Natta catalyst were deconvoluted into five Flory molecular weight distributions (MWDs). Relationships between reactor operating conditions and deconvolution parameters confirmed that temperature and hydrogen and hexene concentrations influenced the MWD. The two sites that produced low‐molecular‐weight polymer responded similarly to changes in reactor operating conditions, as did the three sites that produce high‐molecular‐weight polymer. Increasing hexene concentration resulted in relatively more polymer being produced at the two low‐molecular‐weight sites and less at the high‐molecular‐weight sites. The information obtained will be useful for making simplifying assumptions during kinetic model development.

     

Calculation of Molecular Weight Distributions in Non‐Linear Free‐Radical Polymerization Using the Numerical Fractionation Technique

Mathematical Modeling of non‐linear polymerization systems subject to gel formation is a challenging endeavor. At the gel point, the second and higher molecular weight moments diverge to infinity making it impossible to obtain the molecular weight distribution (MWD). The numerical fractionation (NF) technique utilizes a refinement of the method of moments to model non‐linear polymerization systems that form gel. Since the method of moments yields results in terms of average quantities, some information is lost when reconstructing the MWD using NF. As a consequence, a broad shoulder appears at the high chain length end of the MWD tail. This study demonstrates that the validity of the gamma distribution deteriorates for the broader branched polymer generations and evaluates the performance of various alternative model distributions. Proper selection of the model distribution enhances the NF‐reconstructed MWD.     

Synthesis of poly(naphthalenecarboxamide)s with low polydispersity by chain‐growth condensation polymerization

Condensation polymerization of 6‐(N‐substituted‐amino)‐2‐naphthoic acid esters ( 1 ) was investigated as an extension of chain‐growth condensation polymerization (CGCP). Methyl 6‐(3,7‐dimethyloctylamino)‐2‐naphthoate ( 1b ) was polymerized at ?10 °C in the presence of phenyl 4‐methylbenzoate ( 2 ) as an initiator and lithium 1,1,1,3,3,3‐hexamethyldisilazide (LiHMDS) as a base. When the feed ratio [ 1a ]₀/[ 2 ]₀ was 10 or 20, poly(naphthalenecarboxamide) with defined molecular weight and low polydispersity was obtained, together with a small amount of cyclic trimer. However, polymer was precipitated during polymerization under similar conditions in [ 1a ]₀/[ 2 ]₀ = 34. To increase the solubility of the polymer, monomers 1c and 1d with a tri(ethylene glycol) (TEG) monomethyl ether side chain instead of the 3,7‐dimethyloctyl side chain were synthesized. Polymerization of the methyl ester monomer 1c did not proceed well, affording only oligomer and unreacted 1c , whereas polymerization of the phenyl ester monomer 1d afforded well‐defined poly(naphthalenecarboxamide) together with small amounts of cyclic oligomers in [ 1d ]₀/[ 2 ]₀ = 10 and 29. The polymerization at high feed ratio ([ 1d ]₀/[ 2 ]₀ = 32.6) was accompanied with self‐condensation to give polyamide with a lower molecular weight than the calculated value. Such undesirable self‐condensation would result from insufficient deactivation of the electrophilic ester moiety by the electron‐donating resonance effect of the amide anion. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011     

Polyphosphazene Based Star‐Branched and Dendritic Molecular Brushes

A new synthetic procedure is described for the preparation of poly(organo)phosphazenes with star‐branched and star dendritic molecular brush type structures, thus describing the first time it has been possible to prepare controlled, highly branched architectures for this type of polymer. Furthermore, as a result of the extremely high‐arm density generated by the phosphazene repeat unit, the second‐generation structures represent quite unique architectures for any type of polymer. Using two relativity straight forward iterative syntheses it is possible to prepare globular highly branched polymers with up to 30 000 functional end groups, while keeping relatively narrow polydispersities (1.2–1.6). Phosphine mediated polymerization of chlorophosphoranimine is first used to prepare three‐arm star polymers. Subsequent substitution with diphenylphosphine moieties gives poly(organo)phosphazenes to function as multifunctional macroinitiators for the growth of a second generation of polyphosphazene arms. Macrosubstitution with Jeffamine oligomers gives a series of large, water soluble branched macromolecules with high‐arm density and hydrodynamic diameters between 10 and 70 nm.

     

Reconstruction of the Chain Length Distribution for Vinyl‐Divinyl Copolymerization Using the Numerical Fractionation Technique

A proposed theory for evaluating the chain length distribution (CLD) using the numerical fractionation (NF) technique was extended to the vinyl‐divinyl (VDV) copolymerization. The CLD is reconstructed for this system, in which pendant double bond propagation leads to crosslinking and gel formation. The method was earlier developed for a non‐linear free‐radical polymerization scheme where chain transfer to polymer and termination by combination resulted in gel formation. The VDV study presented indicates that the proposed method of weighted summation (WS) accurately predicts the resulting CLDs evaluated using NF.

Comparison of the overall polymer NF and the direct solution CLDs near the gel point.     

A Fast Method to Compute the MSD and MWD of Polymer Populations Formed by Step‐Growth Polymerization of Polyfunctional Monomers With Identical Reactive Groups

A fast method is presented for the calculation of the MSD and the MWD of polymers obtained via step‐growth polymerization of polyfunctional monomers bearing identical reactive groups (i.e., systems of type “Afi”). Using this method, the complete distribution can be calculated rapidly, not just the statistical averages of the polymer population such as or . The computed MSD and MWD give more insight than these averages and can be compared to similar data measured on actual polymer systems. The low‐ and intermediate molecular size/weight part of the distribution curves are calculated using a recurrence scheme, while the high‐molecular tail (large and very large polymers) of the distributions is derived from an asymptotic approximation of the associated generating functions.

     

A Fast Method to Compute the MSD and MWD of Polymer Populations Formed by Step‐Growth Polymerization of Polyfunctional Monomers Bearing A and B Coreactive Groups

General step‐growth polymerization systems of order 2 are considered, i.e., systems of type “AfiBgi”, and a fast algorithmic method is presented to compute, at a given degree of conversion, the MSD and the MWD. The complete distribution is calculated; not just statistical averages of the polymer population such as or . For the computation of the low‐ and intermediate size/weight parts of the distribution curves, a set of recurrence relations is used. The high‐molecular size/weight parts of the curves (right tails) are computed using an accurate approximation derived from generating functions. In a previous paper, we applied our method to general order‐1 systems, i.e., systems of type “Afi”.

     

A Probability Model of Star‐Branched Polymers Formed by Connecting Polydispersed Primary Chains Onto a Multifunctional Coupling Agent

Summary: A probability model, based on the “in‐out” recursive analysis, is developed for obtaining the average molecular weights of star polymers formed by connecting polydispersed primary chains onto a multifunctional coupling agent. The average properties and the polydispersity index of the formed star polymers can be described as a function of the reaction conversion and the average properties of the polydispersed primary chains without the knowledge of the whole distribution. The results indicate that, although PI of the resulting star polymers might increase at the intermediate conversion for the higher functionalities of the core molecules, the resulting star polymers generally have narrower molecular weight distributions at the complete conversion compared to the initial polydispersed polymer chains.

A schematic illustration of the star polymer formation.     

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.     

Role of sodium dodecylbenzenesulfonate in 2,2,6,6‐tetramethy‐1‐piperidinyloxy‐mediated styrene miniemulsion polymerization

In studying 2,2,6,6‐tetramethy‐1‐piperidinyloxy (TEMPO)‐mediated styrene miniemulsions, we have observed that the surfactant sodium dodecylbenzenesulfonate (SDBS) not only provides colloidal stability but also influences the rate of polymerization. Increasing the SDBS concentration results in higher polymerization rates, although the molecular weight distribution and particle size distribution are not significantly impacted. We have also examined another common sulfonate surfactant, DOWFAX 8390. In contrast to SDBS, DOWFAX 8390 does not affect the polymerization rate. Furthermore, DOWFAX‐stabilized polymerizations are slower than SDBS‐stabilized polymerizations. TEMPO‐mediated bulk styrene polymerizations are also accelerated significantly in the presence of SDBS. Although the mechanism for the rate acceleration is unknown, the experimental evidence suggests that SDBS is participating in the generation of radicals capable of propagating, thereby reducing the TEMPO concentration within the particles. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 5974–5986, 2006     

The First Example of Main‐Chain Cyclic Azobenzene Polymers

A novel main‐chain azobenzene cyclic polymer, cyclic‐PEHPA, has been successfully synthesized by ‘click’ cyclization of the α‐alkyne‐ω‐azido hetero‐difunctional linear precursors (linear‐PEHPA), which is synthesized by a step‐growth polymerization of the 3′‐ethynylphenyl[4‐hexyl‐(2‐azido‐2‐methyl‐ propionate) phenyl] azobenzene (EHPA). Gel permeation chromatography, and ¹H NMR and FT‐IR spectra confirmed the complete transformation of linear‐PEHPA into cyclic‐PEHPA. With the same molecular weights, the cyclic‐PEHPAs are found to have higher glass transition temperatures than the linear‐PEHPAs, but almost the same decomposition temperatures. In addition, the obtained cyclic azobenzene polymer with lower molar mass shows a slightly better trans–cis–trans photoisomerization ability than the corresponding linear‐PEHPA.

     

Controlled Bimodal Molecular‐Weight‐Distribution Polymers: Facile Synthesis by RAFT Polymerization

The RAFT agents RAFT‐1 and RAFT‐2 were used for RAFT polymerization to synthesize well‐defined bimodal molecular‐weight‐distribution (MWD) polymers. The system showed excellent controllability and “living” characteristics toward both the higher‐ and lower‐molecular‐weight fractions. It is important that bimodal higher‐molecular‐weight (HMW) polymers and block copolymers with both well‐controlled molecular weight (MW) and MWD could be prepared easily due to the “living” features of RAFT polymerization. The strategy realized a mixture of higher/lower‐molecular‐weight polymers at the molecular level but also preserved the features of living radical polymerization (LRP) of the RAFT polymerization.