Molecular weight distribution in free radical polymerization with long-chain branching

A new theory to predict the molecular weight distribution in free radical polymerization that includes chain transfer to polymer is proposed. This theory is based on the branching density distribution of the primary polymer molecules. The branching density distribution provides the information on how each chain is connected to other chains, and therefore, a full molecular weight distribution can be calculated by application of the Monte Carlo simulation. The present theory accounts for the history of the generated branched structure and can be applied to various reaction systems that involve branching and crosslinking regardless of the reactor types used. The present simulation confirmed the validity of the method of moments in a batch polymerization proposed earlier. It was shown clearly why gelation never occurs by chain transfer to polymer without the assistance of other interlinking reaction such as bimolecular termination by combination. © 1993 John Wiley & Sons, Inc.     

A simulation model for long-chain branching in vinyl acetate polymerization: 1. Batch polymerization

A new simulation model for the kinetics of long-chain branching formed via chain transfer to polymer and terminal double-bond polymerization is proposed. This model is based on the branching density distribution of the primary polymer molecules. The theory of branching density distribution is that each primary polymer molecule experiences a different history of branching and provides information on how each primary polymer molecule is connected with other chains that are formed at different conversions, therefore making possible a detailed analysis on the kinetics of the branched structure formation. This model is solved by applying the Monte Carlo method and a computer-generated simulated algorithm is proposed. The present model is applied to a batch polymerization of vinyl acetate, and various interesting structural changes occurring during polymerization (i.e., molecular weight distribution, distribution of branch points, and branching density of the largest polymer molecule) are calculated. The present method gives a direct solution for the Bethe lattice formed under nonequilibrium conditions; therefore, it can be used to examine earlier theories of the branched structure formation. It was found that the method of moments that has been applied successfully to predict various average properties would be considered a good approximation at least for the calculation of not greater than the second-order moment in a batch polymerization. © 1994 John Wiley & Sons, Inc.     

Markovian approach to nonlinear polymer formation: Free-radical crosslinking copolymerization

A Markovian model is used to extend the Flory/Stockmayer gelation theory to nonequilibrium reaction systems, by taking free-radical crosslinking copolymerization of vinyl and divinyl monomer as an example. Free-radical polymerizations are kinetically controlled; therefore, each primary polymer molecule experiences a different history of crosslinked structure formation. By assuming that the primary chains with identical birth time conform to the same chain connection probabilities, the nonlinear structural development can be viewed as a system in which the primary chains formed at different birth times are combined into nonlinear polymers in accordance with the first-order Markov chain statistics. According to the present Markovian model, the weight-average chain length, P̄_w is given by a matrix formula, P̄_w = W _p( E — Q )⁻¹ l where W _p is the row vector that concerns the weight contribution of a primary chain, E is a unit matrix, Q is the transition matrix representing the chain connection statistics, and I is a column vector whose elements are all unity. For an equilibrium system, W _p = P̄_wp (weight-average chain length of the primary chains), E = 1, Q = ρP̄_wp (ρ is the crosslinking density), and I = 1; therefore, the present formula reduces to the Flory/Stockmayer equation, P̄_w = P̄_wp/(1 − ρP̄_wp). The criterion for the onset of gelation is simply stated as a point at which the largest eigenvalue of the transition matrix Q reaches unity, i.e., det( E − Q ) = 0. The present Markovian approach elucidates important characteristics of the kinetically controlled network formation, and provides greater insight into nonequilibrium gelling systems.     

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.     

Crosslinking Index,Molecular Weight Distribution and Rubber Equilibrium Shear Modulus During Polyfunctional Crosslinking of Existing Polymer, 6. Primary Polymer with a Schulz–Zimm Distribution of Molecular Weights

A network model for the crosslinking of already existing polymer molecules with a so‐called Schulz–Zimm distribution of their molecular weights is presented. It is an extension of previously developed statistical network models applied to the crosslinking of primary polymers with several other molecular weight distributions and with crosslinks of any functionality. The model results in the possibility to obtain more insight into the structure of polymers, especially those with narrow distributions of the molecular weight. In more detail, the model can give a perspective on structural network parameters such as the weight fractions of ideal network, of dangling polymer ends, and of those molecules not connected to the network, i. e., the sol fraction, the number of crosslinks in which a polymer molecule is bound, the functionality of the crosslinks, or the average molar mass of the polymer molecules in between the crosslinks M̄_c. Results of calculations are shown for a hypothetical crosslinking process of polymers with various molecular weight distributions. Moreover, the dependency of the network parameters on the polydispersity index and the type of molecular weight distribution is shown. Finally the increase of the functionality of the crosslinks during the ageing process of a 9.9% poly(vinyl chloride) gel as a function of the polydispersity index of the molecular weight distribution is presented.     

Kinetics of polymeric network synthesis via free-radical mechanisms - polymerization and polymer modification

The kinetics of polymeric network formation via free radical mechanisms is an attractive research area because there are many phenomena which are not well understood and in addition, the commercial potential for crosslinked systems is great. Recently, a large research/development program was initiated at the McMaster Institute for Polymer Production Technology (MIPPT) to investigate the fundamentals and applications of polymeric network, in particular, the kinetics of synthesis via free-radical mechanisms and network characterization. The research on crosslinking involved both theoretical developments and experimentation. Herein is provided a comprehensive summary of this work. In the experimental polymerization, two comonomers, methyl methacrylate (MMA) / ethylene glycol dimethacrylate (EGDMA) and acrylamide (AAm) / N,N-methylene bisacrylamide (Bis), as model systems were studied in considerable detail. Measurements included: monomer conversions, radical concentrations, sol/gel fractions, crosslink densities (equilibrium swelling and swollen-state ¹³C-NMR) over the entire range of divinyl monomer levels as a function of polymerization time. In the polymer modification, high density polyethylenes were crosslinked using peroxides and γ-radiation. For this system, crosslinking and chain scission occur simultaneously. In the theoretical studies, it was shown that in general, network formation by free-radical mechanisms is highly irreversible requiring that the classical equilibrium gelation theories after Flory/Stockmayer be generalized. The general model which was developed using the pseudo-kinetic rate constant method predicts the existence of a crosslink density distribution (crosslink density of a primary polymer chain depends on its birth time) with a variance which can vary widely depending on network synthesis conditions.     

Elastic properties of crosslinked poly(vinyl alcohol) gels: Network topology

Current network theory exhibits inconsistencies which show up particularly clearly in deformation of networks prepared by crosslinking a polymer in solution. A check of theory can be obtained if one knows precisely the number of crosslinks in the network and if a range of deformations is applied to the network. In an effort to explore this problem we have examined the relation of shear modulus to crosslink density, primary molecular weight, and polymer concentration for a series of poly(vinyl alcohol) gels at low to intermediate concentrations. Aqueous poly(vinyl alcohol) solutions were crosslinked to form infinite networks using terephthalaldehyde. We find a large discrepancy with these poly(vinyl alcohol) gels between measured shear modulus and that calculated from classical elasticity theory assuming quantitative reaction of crosslinking. The ratio of measured to calculated modulus is independent of crosslink density for a given primary molecular weight and concentration. It shows linear dependence on polymer concentration prior to crosslinking and extrapolates to a critical concentration which is consistent with the effective sizes of the polymer molecules.     

Specific features of the formation of polymer-dye systems based on nanostructured polymer matrices prepared by solvent crazing

Specific features of the formation of polymer-dye systems based on various nanostructured polymer matrices prepared by the method of solvent crazing are discussed. In the general case, the formation of polymer-dye composites includes four main stages: sorption of dye molecules by the highly disperse fibrillar material of crazes, shrinkage of the polymer composite due to the removal of the solvent, migration of dye molecules from their localized sites on the surface of fibrils, and healing of the structure of crazes (internal interfacial boundaries) under thermal treatment. Analysis of the migration of dye molecules in the polymer matrix includes the following assumptions: first, a metastable (nonequilibrium) state of the system after solvent crazing and introduction of dye molecules into the fibrillar craze material and, second, the statement according to which both the depth and direction of the above migration processes are controlled by the free energy of mixing of components. For amorphous glassy systems (PVC, PS, PC), healing of the fibrillar craze material (after shrinkage and removal of the solvent) is observed. In the case of semicrystalline polymers (PP, vinylidene fluoride-trifluoroethylene copolymer) and amorphous crystallizable polymer matrices (PET), the intensity of healing upon thermal treatment decreases due to the presence of crystalline regions, which slow down the motion of macromolecules.     

Molecular weight distribution formed through chain-length-dependent crosslinking reactions

The molecular weight distribution formed through chain-length-dependent crosslinking reactions in free-radical vinyl/divinyl copolymerization is investigated by using Monte Carlo simulations. When the crosslinking reaction rates for larger polymer molecules are reduced, the high molecular weight tails cannot extend smoothly, resulting in distorted, sometimes bimodal distribution profiles, which exhibit qualitative similarity with those reported experimentally. Although the reduced crosslinking reaction rates between larger polymer molecules may not affect the average crosslinking density levels significantly, they can delay the developments of the weight-average molecular weight significantly. Because the wastage of pendant double bonds through cyclization would result in a similar tendency, one cannot distinguish these two types of nonidealities through the measurements of the pendant double bond consumption and the average molecular weight development.     

Markovian approach to nonlinear polymer formation: Free-radical polymerization with chain transfer to polymer

A Markovian model is proposed for nonrandom branching reactions, by using free-radical polymerization that involves chain transfer to polymer as an example. Free-radical polymerizations are kinetically controlled; therefore, each primary polymer molecule experiences different history of branched structure formation. By assuming that the primary chains with the identical birth time conform to the same chain connection probabilities, the nonlinear structural development can be viewed as a system in which the primary chains formed at different birth times are combined into nonlinear polymers in accordance with the first-order Markov chain statistics. An explicit formula for the weight-average chain length is derived in a matrix form. The onset of gelation is simply stated as a point at which the largest eigenvalue of the transition matrix X reaches unity, i.e., det(X − I) = 0. This criterion for the onset of gelation can be considered as an extension of the Flory/Stockmayer theory to a nonequilibrium reaction system. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36 : 357–371, 1998     

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.     

Effect of water contact on the density distributions of thin supported polymer films investigated by an X-ray reflectivity method

The diffusion processes of water molecules into polymer films (PMMA/PS homopolymers and random copolymers) in contact with liquid water were investigated using gravimetric methods and X-ray reflectivity (XRR) analysis. Methods of water contact and XRR measurement were designed for studying the systems in the nonequilibrium state of diffusion. Gravimetric measurements confirmed the Fickian diffusion behavior of films in contact with water. Vertical density distributions in PMMA and methylmethacrylate-rich copolymer films demonstrate the existence of a water-rich layer at the interface. However, with further absorption of water into the film, the overall density increased throughout the film. The results suggest that the diffusion of water into the polymer film occurs to recover density uniformity with a high concentration of water molecules at the surface. Some XRR data for the PS- and styrene-rich copolymer films could not be fit and converted to a vertical density distribution because of their huge diffusion coefficients. However, the reflectivity curves for these films and the vertical density distribution after sufficient water contact suggested that the surfaces of these films were commonly diffused after water contact. Atomic force microscopy (AFM) analysis demonstrated that the surface roughness of these films actually increased with water content.     

Microgel formation in emulsion copolymerization: II. Seeded polymerization

Microgel formation in seeded emulsion copolymerization of methyl methacrylate and ethylene glycol dimethacrylate is investigated both experimentally and theoretically. By introducing seed latex, the network structure development can be changed significantly. Even when the crosslinking density development takes a similar pattern as the crosslinking copolymerization in homogeneous media, the molecular weight development shows both types of behavior that is characteristic of emulsion polymerization without seed latex and of homogeneous polymerization, depending on the primary polymer chain length and the mole fraction of the divinyl monomer used. Once the microgels are formed, the weight-average molecular weight increases just linearly with conversion due to a very small locus of polymerization. The present investigation reveals important characteristics of gelation phenomena in a limited space. © 1996 John Wiley & Sons, Inc.     

Size exclusion chromatography of branched polymers: Star and comb polymers

Monte Carlo simulations were conducted to estimate the elution curve of size exclusion chromatography (SEC). The present simulation can be applied to various types of branched polymers, as long as the kinetic mechanism of nonlinear polymer formation is given. We considered two types of detector systems, (1) a detector that measures the polymer concentration in the elution volume to determine the calibrated molecular weights, such as by using the differential refractive index detector (RI), and (2) a detector that determines the weight‐average molecular weight of polymers within the elution volume directly, such as a light scattering photometer (LS). For polydisperse star polymers, both detector systems tend to give a reasonable estimate of the true molecular weight distribution (MWD). On the other hand, for comb‐branched polymers, the RI detector underestimates the molecular weight of branched polymers significantly. The LS detector system improves the measured MWD, but still is not exact. The present simulation technique promises to establish various types of complicated reaction mechanisms for nonlinear polymer formation by using the SEC data quantitatively. In addition, the present technique could be used to reinvestigate a large amount of SEC data obtained up to the present to estimate the true MWD.     

Light-fueled dynamic covalent crosslinking of single polymer chains in non-equilibrium states

While polymer synthesis proceeds predominantly towards the thermodynamic minimum, living systems operate on the reverse principle – consuming fuel to maintain a non-equilibrium state. Herein, we report the controlled formation of 3D macromolecular architectures based on light-fueled covalent non-equilibrium chemistry. In the presence of green light (525 nm) and a bivalent triazolinedione (TAD) crosslinker, naphthalene-containing polymers can be folded into single chain nanoparticles (SCNPs). At ambient temperature, the cycloaddition product of TAD with naphthalene reverts and the SCNP unfolds into its linear parent polymer. The reported SCNP is the first example of a reversible light triggered folding of single polymer chains and can readily be repeated for several cycles. The folded state of the SCNP can either be preserved through a constant supply of light fuel, kinetic trapping or through a chemical modification that makes the folded state thermodynamically favored. Whereas small molecule bivalent TAD/naphthalene cycloaddition products largely degraded after 3 days in solution, even in the presence of fuel, the SCNP entities were found to remain intact, thereby indicating the light-fueled stabilization of the SCNP to be an inherent feature of the confined macromolecular environment.

Synthetic polymers consume green light as fuel for intramolecular crosslinking, yielding non-equilibrium single chain nanoparticles that can be light-stabilised, kinetically and chemically trapped, or else unfold in the absence of light fuel.     

Cluster aggregation and fragmentation kinetics model for gelation

Gelation can occur in polymer, hydrogel, and colloid systems that undergo reversible aggregation-fragmentation (crosslinking accompanied by breakage). Gelation, characterized by rapid divergence of weight-average molecular weight and viscosity due to initial network formation, can be reversed if conditions change. In this paper, reversible aggregation and fragmentation in the pre-gelation time period are modeled with distribution kinetics. Moment equations are obtained from the population balance equation, and solved for eight different rate kernels. We identify the cases for which gelation is possible and obtain the critical values for the rate constants that allow gelation. The model provides a good simulation of published experimental data for aggregation and degradation of plasticized wheat gluten during thermo-mechanical treatments. We also evaluate two closure approximations based on Gamma and log-normal distributions, and conclude that log-normal closure predicts all five possible steady states, in agreement with the Vigil-Ziff criterion, and Gamma closure predicts only three. However, Gamma closure approximates the steady state either closely or exactly, whereas log-normal closure only poorly approximates the steady-state distribution.     

Statistical mechanical theory for steady state systems. V. Nonequilibrium probability density

The phase space probability density for steady heat flow is given. This generalizes the Boltzmann distribution to a nonequilibrium system. The expression includes the nonequilibrium partition function, which is a generating function for statistical averages and which can be related to a nonequilibrium free energy. The probability density is shown to give the Green-Kubo formula in the linear regime. A Monte Carlo algorithm is developed based upon a Metropolis sampling of the probability distribution using an umbrella weight. The nonequilibrium simulation scheme is shown to be much more efficient for the thermal conductivity of a Lennard-Jones fluid than the Green-Kubo equilibrium fluctuation method. The theory for heat flow is generalized to give the generic nonequilibrium probability densities for hydrodynamic transport, for time-dependent mechanical work, and for nonequilibrium quantum statistical mechanics.     

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.     

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.