Polymer Chain Length Distribution Research Articles

A general model for the ideal chain length distributions of polymers made with reversible deactivation

A general model is developed for the distribution of polymers made with reversible deactivation. The model is applied to a range of experimental systems including RAFT, cationic and ATRP.

Random Branching of Polymer Chains with Schulz–Zimm Distribution. 2. Radius of Gyration and Maximum Span Length

AbstractMean‐square radius of gyration Rg2 and the maximum span length LMS of each polymer molecule are investigated for the random branching of primary chains that follow the Schulz–Zimm distribution, by using the Monte Carlo simulation method. It is found that the expected g‐ratio of Rg2 of the branched molecule to that of a linear molecule for a given number of branch points k does not change with the branching density. The expected g‐ratio becomes larger for the primary chains with broader distribution. An approximate formula for the relationship between g‐ratio and k that accounts for the distribution breadth is proposed. The relationship between the weight fraction of the maximum span chain LMS/(degree of polymerization) and k shows the analogous behavior with that for g‐ratio and k. The magnitude of Rg2 is proportional to LMS, with Rg2 = 0.18 LMS, irrespective of the breadth of the primary polymer chain length distribution.

Interplay between synthetic conditions and micromorphology in poly(3,4-ethylenedioxythiophene):tosylate (PEDOT:Tos): an atomistic investigation.

Micromorphology of conjugated polymers is expected to play a crucial role in both heat and charge transport properties. In this perspective, the details of the polymerization mechanism acquire a fundamental relevance, providing the link between the basic chemical reaction paths and the resulting molecular structure and arrangement. For PEDOT, the role played by the Brønsted bases (proton scavengers) and their impact on the distribution of polymer chain lengths are still a matter of debate. In the present work, we have systematically analyzed several reaction paths leading to PEDOT polymerization. By means of atomistic simulations, we identified the thermodynamically preferred reaction path, proving that tosylate anions rule proton scavenging. PEDOT chain length was computed to be ∼12-13 monomeric units. We could also demonstrate how the proton scavengers set at once the chain lengths and the sample crystallinity. Furthermore, we found that tosylate gives rise to a sharper multimodal distribution of chain length, a feature that supports hypotheses regarding the occurrence of a percolative transport regime mediated by tie chains bridging paracrystalline regions.

A Polymer Blend Approach for Creation of Effective Conjugated Polymer Charge Transport Pathways.

Understanding the role of the distribution of polymer chain lengths on process-structure-property relationships in semiconducting organic electronics has remained elusive due to challenges in synthesizing targeted molecular weights ( Mw) and polydispersity indices. Here, a facile blending approach of various poly(3-hexylthiophene) (P3HT) molecular weights is used to investigate the impact of the distribution of polymer chain lengths on self-assembly into aggregates and associated charge transport properties. Low and high Mw samples were blended to form a highly polydisperse sample which was compared to a similar, medium Mw control. Self-assembly was induced by preprocessing the polymer solution with UV irradiation and subsequent solution aging before deposition via blade-coating. Superior charge carrier (hole) mobilities were observed for the blend and control samples. Furthermore, their solution lifetimes exceeded 14 days. UV-vis spectral analysis suggests that low Mw P3HT lacks the mesoscale crystallinity required for percolative charge transport. In contrast, when the Mw is too high, the polymer rapidly aggregates, leading to paracrystalline disorder and structural inhomogeneity that interrupts charge-transfer pathways. The role of grain boundaries, fibrillar order, and macroscale alignment is characterized via grazing-incidence wide-angle X-ray scattering, atomic force microscopic, and optical microscopic techniques. The results presented here provide additional guidance on the interplay between polymer solubility, self-assembly, network interconnectivity, and charge transport to enable robust polymer ink formulations with reliable and reproducible performance attributes.

Comparison of chain-growth polymerization in solution versus on surface using reactive coarse-grained simulations

We developed a reactive coarse-grained model for chain-growth polymerization to study the chain length distribution of polymers grafted from a surface as compared to those grown in solution. The surface-initiated polymer chains exhibit a markedly broader distribution of chain lengths (larger dispersity) and slower chain growth kinetics as compared to the solution phase process. Two key factors that cause deviation from ideal grafting are identified: (1) the formation of chain “loops” with both termini attached to the substrate via recombination and (2) the “starvation” effect, in which the live chain ends of short polymers are sterically shielded from monomers by the presence of longer neighboring chains. Both effects are markedly amplified with increasing initiator surface density, thus limiting the density of high molecular weight chains grafted from the surface. Whereas the “loop” effect is absent in systems that are very strictly “living” (no termination), the starvation effect is likely operative in most real systems. Taken together, these results suggest a theoretical limit on the density of polymer brushes that can form on surfaces by a “grafting-from” technique, based on the kinetics of the chain-growth polymerization process.

A rubber elasticity and softening model based on chain length statistics

The classical statistical theory of polymerization predicts a random distribution of polymer chain lengths. This distribution has long ago been known in the polymerization theory but, to the best of our knowledge, has not so far been utilized in mechanics of polymers. In the present paper, we incorporate this chain length statistics into full network rubber models which are based on continuous directional distributions of polymer chains. The free energy of the full network results as an integral of single chain energies over the unit sphere. In the case of an initially isotropic spatial arrangement of chains and ideally elastic behavior an analytical solution in terms of micro-structural parameters of the network is obtained. Introducing a softening criterion formulated in terms of the minimal number of chain segments available in the distribution we can describe not only elastic behavior but also inelastic phenomena especially pronounced in filled rubbers. These are, for example, the Mullins effect, permanent set and strain induced anisotropy. In this case, numerical integration over the unit sphere is applied. Predictions of the model demonstrate good agreement with experimental data with respect to the above mentioned phenomena.

Chain length distribution and kinetic characteristics of an enzymatically produced polymer

Abstract Non-processive enzymatic polymerization leads to a distribution of polymer chain lengths. A polymerization model was developed to investigate the relation between the extent of this distribution on one hand, and the polymerization start conditions and reaction kinetics on the other hand. The model describes changes in concentration of chains of length n as the result of two elongation reactions: elongation by monomer addition to length n-1 and elongation by monomer addition to length n. Polymerization reactions were assumed to be zero order in monomer concentration and to obey Michaelis-Menten kinetics with respect to polymer concentrations. In addition, the amount of enzyme available for each individual reaction (n → n+1) is assumed to be proportional to the concentration of polymer substrate of length n. The development of the shape of the chain length distribution was found to be independent of the value of the overall reaction rate constant; only the rate at which these shapes developed was influenced by the 1st-order rate constant. The value of the Michaelis parameter did affect the form of the chain length distribution curve since it affects the reaction order. An increase in reaction order was found to promote widening of the chain length distribution. Differences in kinetic parameters between the subsequent polymerization reactions, if any, were also found to have a large effect on the development of the chain length distribution. An increase in rate constants with chain length entailed a wider distribution; a more narrow distribution would require a decrease in rate constants with chain length.

Identifying the Ideal Characteristics of the Grafted Polymer Chain Length Distribution for Maximizing Dispersion of Polymer Grafted Nanoparticles in a Polymer Matrix

We investigate, using theory and simulations, the role of the short and long grafted chains in polydisperse polymer grafted nanoparticles in stabilizing particle dispersion in a chemically similar polymer matrix in the presence of particle–particle attractions. The effect of the short and long chains in a polydisperse or bidisperse graft length distribution on the potential of mean force between the polymer grafted nanoparticles is coupled and distinct from their role in the corresponding deconstructed short and long monodisperse distributions. At high grafting density, the increased monomer crowding near the particle surface from both short and long chains maximizes shielding of particle–particle attraction, while the length and crowding of long chains away from the particle surface determine the location, range, and strength of the steric repulsion and midrange attraction. We find that to maximize grafted nanoparticle dispersion, it is best to synthesize grafted particles at high grafting density with p...

Three-dimensional analysis of membrane formation via thermally induced phase separation by dissipative particle dynamics simulation

The effects of polymer molecular weight (polymer chain length), polymer molecular weight distribution (polymer chain length distribution) and diluent structure on the membrane formation of polymer–diluent system via the thermally induced phase separation (TIPS) method were analyzed on the basis of a dissipative particle dynamics (DPD) simulation platform. The simulation platform was extended into three-dimensions (3-D) to make sure that the results were more comprehensive. The simulation results proved that both the polymer chain length and polymer chain length distribution had significant influences on the phase separation process. As the polymer chain length increased, phase separation rate first increased when the polymer chain was relatively short and then decreased when the polymer chain length was larger. In addition, as the chain length distribution got wider, the phase separation rate decreased gradually at the middle and last stages of the phase separation process. It was also demonstrated that the interaction between polymer and diluent could be regulated by changing the interaction of the polymer and diluent local part by the local modification on the diluent molecular structure, which was beneficial to the membrane preparation. The information obtained from this study is consistent with the experimental results in literatures, proving that the simulation method could be widely applied in dynamics analysis of the membrane formation process.

Role of diffusion in scaling of polymer chain aggregates found in vapor deposition polymerization

Polymer chain aggregates grown by (1+1)-dimensional Monte Carlo simulations of vapor deposition polymerization (VDP) were studied. The dynamic scaling behavior of polymer chain length distribution n(s)(t) was studied as a function of chain length (s), deposition time (t), and the ratio G=D/F of deposition rate (F) and free monomer diffusion (D). The dynamic scaling approach was employed to highlight the dependence of n(s)(t) on t, s, and G. With an increase in t, we found a power law increase in n(s)(t) and total number of polymer chains N(total)(t), given by N(total)(t)~t(ω) and n(s)(t)~t(ω) with exponent ω=1.01(2) that was invariant for a range of G=10 to 10^{4}. For small s and t=10^{3}, 5×10^{3}, and 10^{4}, n_{s}(t) decreased according to n(s)(t)~s(-τ) with τ=0.58(2). As G was increased from 10 to 10(4), we observed a systematic influence of G on the rescaled n(s)(t) data that prevented the manifestation of unique scaling function for polymer chain aggregates. The dependence of scaling functions of n(s)(t) on G elucidates the sensitivity of polymer chain aggregates to G and is thought to be a characteristic of VDP.

Kinetic Modeling as a Tool to Understand and Improve the Nitroxide Mediated Polymerization of Styrene

AbstractKinetic modeling is used to obtain insight in the complex interplay between reaction rates and obtained polymer properties in the SG1 and the TEMPO mediated bulk polymerization of styrene at 396 K. The increase of the viscosity during NMP is accounted for. At higher targeted chain lengths, chain transfer to dimer and transfer from nitroxide to dimer are shown to cause the experimentally observed reduced control over the average polymer properties and to result in a clear fronting of the polymer chain length distribution. The potential of kinetic modeling to design tailor‐made synthesis strategies is illustrated. Simulations indicate that careful control of the polymerization conditions allows to obtain an important improvement of the polymer properties. The approach is also applicable for NMP mediated by other alkoxyamines/nitroxides and allows to expand the application range of NMP for styrene polymerization in particular to synthesize complex polymer architectures by assembly of functionalized polymers. magnified image

The effects of energetic disorder and polydispersity in conjugation length on the efficiency of polymer-based light-emitting diodes

Energetic disorder plays a critical role in governing the performance of organic light-emitting diodes (OLEDs). This is particularly true for polymer-based OLEDs in which disorder can be controlled but not eliminated. Contrary to the common assumption that energetic disorder has only a negative effect on performance, we provide experimental evidence of improved efficiency when the energetic disorder is increased upon deliberate addition of traps via an increase in polymer chain length distribution. We use numerical calculations to extract the mechanisms responsible for the observed improvements and discuss the general conditions under which traps can be introduced to increase device efficiency.

A comprehensive Monte Carlo simulation of styrene atom transfer radical polymerization

An optimized and high-performance Monte Carlo simulation is developed to take thorough account of four different cases of termination in styrene ATRP. According to the simulation results, the bimolecular termination rate constant sharply drops throughout the polymerization when either chain-length dependency of termination rate constant, gel effect, or both together is applied to the simulation. In addition, as expected, the initiator is quickly decomposed at the early stages of the polymerization. The concentration of the catalyst in lower oxidation state decreases at first and then plateaus at higher conversion; furthermore, the steady concentration of M t Y/L in the polymerization is the highest when the chain-lengthdependent diffusion-controlled termination rate constant is employed in the simulation. The rates of deactivation and chain end degradation reactions are also smaller in this case. Therefore, the fraction of dormant chains is higher throughout the reaction and consequently the portion of dead polymers decreases. Besides, molecular weight increases linearly with conversion; however, when neither gel effect nor chain-length dependency of termination rate constant is considered, the molecular weight deviates from linearity at the end of the reaction. The peak of chain length distribution shifts toward higher molecular weight too during the reaction. Finally, the molecular weight distribution broadens at higher conversion; however, the chain length distribution of polymers produced under conditions of applying chain-length-dependent diffusion-controlled termination rate constant is narrower.

In vitro synthesis of heparosan using recombinant Pasteurella multocida heparosan synthase PmHS2

In vertebrates and bacteria, heparosan the precursor of heparin is synthesized by glycosyltransferases via the stepwise addition of UDP-N-acetylglucosamine and UDP-glucuronic acid. As heparin-like molecules represent a great interest in the pharmaceutical area, the cryptic Pasteurella multocida heparosan synthase PmHS2 found to catalyze heparosan synthesis using substrate analogs has been studied. In this paper, we report an efficient way to purify PmHS2 and to maintain its activity stable during 6 months storage at −80 °C using His-tag purification and a desalting step. In the presence of 1 mM of each nucleotide sugar, purified PmHS2 synthesized polymers up to an average molecular weight of 130 kDa. With 5 mM of UDP-GlcUA and 5 mM of UDP-GlcNAc, an optimal specific activity, from 3 to 6 h of incubation, was found to be about 0.145 nmol/μg/min, and polymers up to an average of 102 kDa were synthesized in 24 h. In this study, we show that the chain length distribution of heparosan polymers can be controlled by change of the initial nucleotide sugar concentration. It was observed that low substrate concentration favors the formation of high molecular weight heparosan polymer with a low polydispersity while high substrate concentration did the opposite. Similarities in the polymerization mechanism between PmHS2, PmHS1, and PmHAS are discussed.

Prediction of Bivariate Molecular Property Distributions in Free-Radical Polymerization Systems Using Monte Carlo and Sectional Grid Methods

An efficient Monte Carlo (MC) algorithm and a two-dimensional fixed pivot technique (2-D FPT) are described for the calculation of the average and distributed molecular properties of polymers in batch free-radical polymerization and copolymerization reactors. Simulations are carried out, under different reactor conditions, to calculate the individual monomer conversions, the leading moments of the 'live' and 'dead' polymer chain length distributions as well as the dynamic evolution of the distributed molecular properties (i.e., molecular weight distribution (MWD), long chain branching distribution (LCBD), copolymer composition distribution (CCD), joint LCB-MW and CC-MW distributions, etc.). The validity of the numerical calculations is examined via a direct comparison of the simulation results, obtained by the two numerical methods, with experimental data on the styrene-methyl methacrylate and vinyl acetate batch free-radical polymerization systems. Additional comparisons between the MC and the 2-D FPT methods are carried out for both polymerization systems, under different polymerization conditions. It is clearly shown that both numerical methods are capable of predicting the distributed molecular, branching and copolymer properties, with high accuracy, up to very high monomer conversions.

Dynamic prediction of the bivariate molecular weight–copolymer composition distribution using sectional-grid and stochastic numerical methods

In the present study, a two-dimensional fixed pivot technique (2-D FPT) and an efficient Monte Carlo (MC) algorithm are described for the calculation of the bivariate molecular weight–copolymer composition (MW–CC) distribution in batch free-radical copolymerization reactors. A comprehensive free-volume model is employed to describe the variation of termination and propagation rate constants as well as the variation of the initiator efficiency with respect to the monomer conversion. Simulations are carried out, under different reactor conditions, to calculate the individual monomer conversions, the leading moments of the ‘live’ and ‘dead’ polymer chain length distributions as well as the dynamic evolution of the distributed molecular properties (i.e., molecular weight distribution (MWD), copolymer composition distribution (CCD) and joint MW–CC distribution). The validity of the numerical calculations is examined via a direct comparison of the simulation results, obtained by the two numerical methods, with experimental data on the styrene–methyl methacrylate batch free-radical copolymerization. Additional comparisons between the 2-D FPT and the MC methods are carried out for different polymerization conditions. It is clearly shown that both numerical methods are capable of predicting the distributed molecular and copolymer properties, with high accuracy, up to very high monomer conversions. It is also shown that the proposed dynamic MC algorithm is less computationally demanding than the 2-D FPT.

Chain Length Distributions of Polyolefins Made in Stopped‐Flow Reactors for Non‐Instantaneous Site Activation

AbstractWe developed an analytical solution to describe how the chain length distribution (CLD) of polymers made with coordination polymerization catalysts vary as a function of time for very short polymerizations considering non‐instantaneous site activation. This solution is an extension of our previous analytical expression for instantaneous site activation. We validated the analytical solution with dynamic Monte Carlo simulation and obtained excellent agreement. Simulation results indicate that, unless the catalyst activation rate is much lower than the propagation rate, it will have only a minor effect on the initial shape of the CLD of polymers made in stopped‐flow reactors (SFR). We also show how incorrect polymerization kinetic parameters may be estimated when assuming instantaneous site activation when this hypothesis is not applicable to the polymerization data under investigation.magnified image

Kinetic Modeling of Non‐Linear Polymerization

AbstractRecent developments of a method based upon population balances of generating functions of polymer chain length distributions (CLD) are presented. The calculation of the CLD and how to take into account chain length dependent reactivity are discussed. Prediction of polymer properties is also possible but only easily done for the average molecular radius of gyration; some results are presented for a radical polymerization including transfer to polymer and propagation with terminal double bonds.

Effects of polymer polydispersity on the phase behaviour of colloid–polymer mixtures

We study the equilibrium behaviour of a mixture of monodispersehard sphere colloids and polydisperse non-adsorbing polymers at theirθ-point, using the Asakura–Oosawa model treated within the free-volume approximation.Our focus is the experimentally relevant scenario where the distribution of polymerchain lengths across the system is fixed. Phase diagrams are calculated usingthe moment free energy method, and we show that the mean polymer sizeξc at which gas–liquid phase separation first occurs decreases with increasing polymer polydispersityδ. Correspondingly, at fixed mean polymer size, polydispersity favours gas–liquid coexistence butdelays the onset of fluid–solid separation. On the other hand, we find that systems with differentδ values but the same mass-averaged polymer chain length have nearly polydispersity-independentphase diagrams. We conclude with a comparison to previous calculations for asemi-grand-canonical scenario, where the polymer chemical potentials are imposed; there itwas found that fluid–solid coexistence was favoured over gas–liquid in some areas of thephase diagram. Our results show that this somewhat counter-intuitive result arose becausethe actual polymer size distribution in the system is shifted to smaller sizes relative to thepolymer reservoir distribution.

Modeling of Solid-State Polymerization of Bisphenol A Polycarbonate

A mathematical model is presented to analyze the kinetics of solid-state polymerization of bisphenol A polycarbonate. The reaction kinetic model incorporated in the solid-state polymerization model is a hybrid of a molecular species model and a functional group model derived from the original model developed by Flory. The solid-state polymerization model allows for the calculation of time evolution of complete polymer chain length distributions in a polymer particle at different radial positions under various reaction conditions. The model simulations show that large polymer particles exhibit strong intraparticle diffusion resistance to phenol, and as a result, the chain length distribution varies significantly from position to position in the polymer particle. Also, some polymer chains have very large chain lengths, several times larger than the average chain length, and some polymer chains have less than 10 repeating units even with a very large average molecular weight. The model simulations also show that one of the most important kinetic parameters that affect the performance of solid-state polymerization is the end group mole ratio in the prepolymer used for solid-state polymerization. The more the end group mole ratio deviates from its stoichiometric ratio in a prepolymer, the longer the reaction time needed to obtain high molecular weight polymer in the solid state. Also, the stoichiometric imbalance in the prepolymer severely limits the maximum obtainable molecular weight. The model simulation shows that an increasing polymer crystallinity during the solid-state polymerization has an effect of increasing the concentrations of catalyst and reactive end groups and hence the polymerization rate.