Monte Carlo simulation of size exclusion chromatography for randomly branched and crosslinked polymers

The elution curves of size exclusion chromatography for nonlinear polymers formed through random branching and crosslinking of long polymer chains were simulated with a Monte Carlo method. We considered two types of measured molecular weight distributions (MWDs): (1) the MWD calibrated relative to standard linear polymers and (2) the MWD obtained with a light scattering (LS) photometer in which the weight‐average molecular weight of polymers within the elution volume is determined directly. The calibrated MWDs clearly underestimate the molecular weights for both randomly branched and crosslinked polymers, and this technique can be used to assess the degree of deviation from the true MWD. When the primary chains conform to the most probable distribution, the calibrated MWD can be estimated reasonably well with the Zimm–Stockmayer equation for the g factor with the help of the relationship between the average number of branch points per molecule and the degree of polymerization. However, the LS method gives good estimates of the true MWD for both randomly branched and crosslinked polymers, although the agreement is better for the branched ones. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2009–2018, 2000     

Stimulated echo method for investigation of structural and dynamic characteristics of branched polymers

The theory of stimulated echo (SE) without magnetic field gradient is proposed. The theory made it possible for the first time to establish a relationship between the stimulated echo signal and correlation function of molecular mobility. Signals of free induction decay (FID) and SE in polymer networks and branched polymers were modeled. The type of the correlation function at different average chain lengths N ₀ between knots and different distribution functions of the knots was determined. A strong influence of the molecular weight distribution (MWD) on the type of the correlation function in polymer melts was shown. Two methods were proposed for the numerical determination of the correlation function from the observed FID and SE signals. These methods gave an information about the molecular mobility and topological structure in the samples of branched polymethyl methacrylates of different structures and various molecular weights.

     

Long-chain branching in free-radical polymerization due to chain transfer to polymer

The Monte Carlo sampling technique is used to investigate the branched structure formation during free-radical polymerization that involves chain transfer to polymer. This method accounts for the history of the generated branched structure and can provide virtually any structural information, because one can observe each polymer molecule directly. In this paper, we investigate the whole molecular weight distribution (MWD) for both pre- and postgelation periods, the MWDs for polymer molecules containing 0, 1, 2, 3, … branch points, the branching density of polymer molecules as functions of both size and the number of branch points, the spatial distribution of the branched chains at the theta state, etc. Contrary to the term ‘long-chain’ branching, many branch chains are relatively small, and the branched structures formed are significantly different from those usually depicted to introduce ‘branched polymers’ in many introductory textbooks. The radii of gyration at the theta state can be approximated by the Zimm-Stockmayer equation for random branching, in spite of various violations against the assumptions used in deriving the equation © 1995 John Wiley & Sons, Inc.     

Branched structure formation in free radical polymerization of vinyl acetate

The branched structure formation during free radical polymerization of vinyl acetate is investigated in detail by application of the computer simulations on the basis of the Monte Carlo sampling technique. Simulations are made for the whole molecular weight distribution (MWD), the MWDs for polymer molecules containing 0, 1, 2, 3, etc., branch points, the branching density as functions of both size and the number of branch points, the spatial distribution of the branched chains, etc. It was found that the effect of polyradicals on the formed MWD could be neglected for batch polymerizations of the present reaction system. A large number of relatively small branch chains are formed due to both chain transfer to polymer (CTP) and the terminal double-bond polymerization (TDBP). The radius of gyration at a Θ state is found to agree satisfactorily with the Zimm-Stockmayer equation for random branching in spite of the heterogeneous branched structure formed in the polymerization. The present investigation reveals important characteristics of the complex molecular structure formation during free radical polymerization that involves both CTP and TDBP. © 1996 John Wiley & Sons, Inc.     

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.     

Dimensions of branched polymers formed in simultaneous long‐chain branching and random scission

In free‐radical olefin polymerizations, the polymer‐transfer reactions could lead to chain scission as well as the formation of long‐chain branches. The Monte Carlo simulation for free‐radical polymerization that involves simultaneous long‐chain branching and random scission is used to investigate detailed branched structure. The relationship between the mean‐square radius of gyration 〈s²〉 and degree of polymerization P as well as that between the branching density and P is the same for both with and without random scission reactions—at least for smaller frequencies of scission reactions. The 〈s²〉 values were larger than those calculated from the Zimm–Stockmayer (Z‐S) equation in which random distribution of branch points is assumed, and therefore, the Z‐S equation may not be applied for low‐density polyethylenes. The elution curves of size exclusion chromatography were also simulated. The molecular weight distribution (MWD) calibrated relative to standard linear polymers is much narrower than the true MWD, and high molecular weight tails are clearly underestimated. A simplified method to estimate the true MWD from the calibrated MWD data is proposed. The MWD obtained with a light scattering photometer in which the absolute weight‐average molecular weight of polymers at each retention volume is determined directly is considered a reasonable estimate of the true MWD. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2960–2968, 2001     

A comparative study on branched polymers formed by T‐shaped junctions and by H‐shaped junctions

In this paper, a statistical model is developed to determine the number‐average and weight‐average molecular weights of a branched polymer system formed through random branching of polymer chains. The average properties of the branched structures formed by T‐shaped junctions and by H‐shaped junctions through random branching of polymer chains are compared in detail. The model indicates that the H‐shaped chain connections can form a gel molecule while the T‐shaped chain connections alone can not cause gelation. Because only the randomly branched polymers for an equilibrium system are considered (under Flory's simplifying assumptions), the present results would not be a rigorous proof that the H‐shaped junctions are required to form a gel molecule in any type of reaction system. However, the present paper shows that the Macosko‐Miller model can be applied to this type of problem in a straightforward manner. The gel point for the branched structures formed by H‐shaped junctions can also be determined by the model.     

A basis for the determination of branching in polymers by use of gel-permeation chromatography

The technique for determining branching in polymers by using a combination of GPC and intrinsic viscosity data has been extended beyond current methods. Equations used in these analyses are presented. The derivations are based upon the assumption that branching is present only when there is a measurable reduction in the intrinsic viscosity. Techniques for calculating the functionality of the star branch point in starbranched polymers are given. Three random-branching parameters are calculated from a knowledge of the average branching density, \documentclass{article}\pagestyle{empty}\begin{document}$ \bar \lambda $\end{document}: (a) the lowest molecular weight branched polymer that can be measured, M?^*; (b) the average molecular weight between branch points, M?_bp; (c) the weight percentage of polymer that is branched. The applicability of this technique is demonstrated by using an analysis of published data on characterized fractions of a randomly branched polystyrene.     

Rheological properties of HyperMacs—long‐chain branched analogues of hyperbranched polymers

HyperMacs are long chain branched analogues of hyperbranched polymers, differing only in the sense that they have polymer chains, rather than monomers between branch points. Although the building blocks for HyperMacs and AB₂ macromonomers can be well defined in terms of molecular weight and polydispersity, the nature of the coupling strategy adopted for the synthesis of the HyperMacs results in branched polymers with a distribution of molecular weights and architectures. Melt rheology showed polystyrene HyperMacs to be thermorheologically simple, obeying William–Landel–Ferry behavior. Zero shear viscosities of the polymers were shown to increase with average molecular weight and the melts display shear‐thinning behavior. HyperMacs showed little evidence for relaxation by reptation and the rheological behavior agreed well with the Cayley tree model for hierarchical relaxation in tube models of branched polymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2762–2769, 2007     

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.     

Monte Carlo simulation of size exclusion chromatography for branched polymers formed through free‐radical polymerization with chain transfer to polymer

The elution curves of size exclusion chromatography (SEC) for branched polymers formed through free‐radical polymerization that involves chain transfer to polymer were theoretically investigated by using a Monte Carlo method. We considered two types of measured molecular weight distribution (MWD), (1) the calibrated MWD relative to standard linear polymers, and (2) the MWD obtained by using a light scattering photometer (LS) in which the weight‐average molecular weight of polymers within the elution volume is determined directly. It was found that the calibrated MWD clearly underestimates the high molecular weight tail, and the measured distributions are narrower than the true MWD. On the other hand, the present simulation results showed that the LS method gives reasonable estimates of the true MWDs. The mean square radius of gyration of the polymer molecules having the same molecular weight was also investigated. The radii of gyration showed clear deviation from the Zimm‐Stockmayer equation^[1] because of the non‐random nature of branched structure and the difference in the primary chain length distribution.     

Molecular weight distribution of a living polymer generated from a bifunctional initiator with non-equal initiation rate constants

This work deals with the kinetics of the living polymerization initiated by a bifunctional initiator with different rate constants of the first and the second step of initiation. The expressions of the molecular weight distribution (MWD) function, the number- and weight-average degrees of polymerization and the polydispersity index were derived rigorously. Numerical results show that the nonequal initiation rate constants often lead to a bimodal MWD except the ratio of k_i to k_i′ being very large (>500, for instance) or relatively small (<1, for example). The MWD of the resultant polymer is rather narrow even in the presence of double peaks. With decreasing initiator concentration, the MWD at full conversion becomes narrower, and the shoulder peak of the MWD curve declines.     

Randomly branched bisphenol A polycarbonates. I. Molecular weight distribution modeling,interfacial synthesis,and characterization

Randomly branched bisphenol A polycarbonates (PCs) were prepared by interfacial polymerization methods to explore the limits of gel‐free compositions available by the adjustment of various composition and process variables. A molecular weight distribution (MWD) model was devised to predict the MWD, G, and weight‐average molecular weight per arm (M_w /arm) values based on the composition variables. The amounts of the monomer, branching agent, and chain terminator must be adjusted such that the weight‐average functionality of the phenolic monomers (F_OH ) was less than 2 to preclude gel formation in both the long‐ and short‐chain branched (SCB) PCs. Several series of SCB and long‐chain branched PCs were prepared, and those lacking gels showed molecular weights measured by gel permeation chromatography–UV and gel permeation chromatography–LS consistent with model calculations. In SCB PCs, the minimum M_w /arm that could be realized without gel formation depended on both composition (molecular weight, terminator type) and process (terminator addition point, coupling catalyst) variables. The minimum M_w /arm achieved in the low molecular weight series studied ranged from ∼3300 to ∼1000. The use of long chain alkyl phenol terminators gave branched PCs with lower glass‐transition temperatures but a higher gel‐free minimum M_w /arm. SCB PCs where M_w /arm was less than ∼M_c spontaneously cracked after compression molding, a result attributed to their lack of polymer chain entanglements. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 560–570, 2000     

Molecular weight distribution in branched polymers

The distribution of molecular weights in realistic free-radical polymerizations with branching was analyzed theoretically. Series solutions were obtained for the fraction of molecules with r repeating units and the number of branch points contained in molecules with r repeating units. Branching by transfer processes was found to increase the proportion of both high and low molecular weight components in the system. The apportioning of branch points among r-mer molecules was shown to be somewhat narrower than a Poisson distribution. The major difference between the calculated distributions and previous model distributions for branched systems was the absence of discontinuities in the moments at all levels of branching.     

Architectural control of polystyrene physical properties using branched anionic polymerization initiated at ambient temperature

Ambient temperature-initiated anionic polymerization has generated branched polystyrenes of varying molecular weights and architectures by inclusion of a distyryl branching comonomer into a conventional sec-Butylithium-initiated polymerization of styrene. Primary chain length control within the branched polymers, and restriction of the branching points to varying segments of the primary chains, led to variations of glass transition temperature with no direct correlation to the branched polymer molecular weight but a strong relationship to the length of individual chains comprising the branched macromolecules.     

Controlling molecular weight distributions of polyethylene by combining soluble metallocene/MAO catalysts

A critical look at the possibility of controlling the molecular weight distribution (MWD) of polyolefins by combining metallocene/methylalumoxane (MAO) catalysts is offered. Catalysts investigated were bis(cyclopentadienyl)zirconium dichloride (Cp₂ZrCl₂), its titanium and hafnium analogues (Cp₂TiCl₂ and Cp₂HfCl₂), as well as rac-ethylenebis(indenyl)zirconium dichloride (Et(Ind)₂ZrCl₂). As observed by other researchers, the MWD of polyethylene can be manipulated by combining soluble catalysts, which on their own produce polymer with narrow MWD but with different average molecular weights. Combined in slurry polymerization reactors, the catalysts in consideration produce ethylene homopolymer just as they would independently. Unimodal or bimodal MWDs can be obtained. This effect can be mimicked by blending polymers produced by the individual catalysts. We demonstrate how a variability in catalyst activity translates into a variability in MWD when mixing soluble catalysts in polymerization. Such a variability in MWD must be considered when setting goals for MWD control. We introduce a more quantitative approach to controlling the MWD using this method. © 1998 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 36: 831–840, 1998     

Molecular weight distribution in living polymerization proceeding with reshuffling of polymer segments due to chain transfer to polymer with chain scission, 1. Determination of kp/ktr ratio from DPw/DPn data. Ideal reproduction of polymer chain activities

The molecular weight distribution (MWD) curves for polymerization systems with chain transfer to polymer leading to reshuffling of polymer segments (and broadening of the MWD), but not changing chain functionalities, were simulated by the Monte Carlo method. The bimodality observed in some distributions was explained by different distribution functions of chains which did not undergo reshuffling and of those which underwent the chain transfer reaction. Using this observation, a numerical integration method for computing DP _w/DP _n (and the MWD curves) in the systems under consideration was devised. Plots relating DP _w/DP _n to monomer conversion and k_tr/k_p are presented and a method of determination of k_tr/k_p from the DP _w/DP _n data is proposed.     

A Conditional Monte Carlo Method to Determine the Architectures of Metallocene Catalyzed Polyethylene

Two polymer molecules of the same length (n) and the same number of branch points (N) can have different properties, since they may possess distinct architectures. In this paper we present a conditional Monte Carlo algorithm for the virtual synthesis of metallocene‐catalyzed polyethylene (PE) in a continuous stirred tank reactor (CSTR). The condition for the Monte Carlo method consists of a fixed chain length distribution (CLD) and a degree of branching distribution (DBD). These distributions are calculated with a Galerkin finite element method. The synthesis method is a recursive algorithm that subsequently creates insertions of sub‐structures containing numbers of branch points according to a certain probability density function. This provides an adjacency matrix describing the connectivity between the branch points, while separately a vector containing the length of segments between branch points and terminal segments is generated. Characterization of the architectures proceeds by rheological features, seniorities and priorities, and molecular properties like the radius of gyration. Comparing the radii of gyration of metallocene polyethylene and low density PE (ldPE) shows the former to possess a more comb‐like structure on average. This is confirmed by the rheological characterization. The found bivariate seniority/priority distribution agrees well to the results of an analytical study of the same chemical system.

Constituting elements of the algorithm.