Molecular weight distribution formed during free‐radical polymerization in the presence of polyfunctional chain transfer agents

Free‐radical polymerization of styrene was carried out in the presence of chain transfer agents (CTAs) with functionality, f = 1–4. The size exclusion chromatography (SEC) with an ultraviolet absorption detector (UV) was used to measure the molecular weight distribution (MWD). A Monte Carlo simulation method proposed earlier was used to investigate the experimental results. In this simulation method, one can observe the structure of each polymer molecule directly, and very detailed information can be obtained in a straightforward manner, including the elution curve of SEC. It was found that up to the functionality f = 3, the equal reactivity model that assumes the reactivity of all functional groups in a CTA is equal agrees reasonably well with the experimental results. However, with f = 4, the reactivity of the fourth functional group seems to decrease and the substitution effects may need to be accounted for to fine control the formed branched structure. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 1267–1275, 1999     

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.     

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     

Simulation of gel permeation chromatography measurement for long chain branched metallocene polyethylene

The research about the polymerization reaction mechanism of long chain branched polymer provides a method to simulate the generation of LCB mPE (long chain branched metallocene polyethylene).^[1-3] In this work, after simulating the generation of one million LCB mPE molecules, we calculate the sizes (i.e. radii of gyration) of molecules in good solvents to obtain the molecular size distributions. Then we simulate the fractionation in GPC (gel permeation chromatography) measurement and the different GPC detector responses to obtain simulated GPC MWDs (molecular weight distributions). The simulated MWDs are compared to the real GPC results provided by the Dow Chemical Company.     

Monte Carlo simulation of diepoxides and monoepoxides cured with amines

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

Simultaneous long‐chain branching and random scission: I. Monte Carlo simulation

In free‐radical olefin polymerizations, the polymer transfer reactions could lead to chain scission as well as forming long‐chain branches. For the random scission of branched polymers, it is virtually impossible to apply usual differential population balance equations because the number of possible scission points is dependent on the complex molecular architecture. On the other hand, the present problem can be solved on the basis of the probability theory by considering the history of each primary polymer molecule in a straightforward manner. The random sampling technique is used to solve this problem and a Monte Carlo simulation method is proposed. In this simulation method, one can observe the structure of each polymer molecule formed in this complex reaction system, and virtually any structural information can be obtained. In the illustrative calculations, the full molecular weight distribution development, the gel point determination, and examples of two‐ and three‐dimensional polymer structure are shown. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 391–403, 2001     

Analysis of branched polymers by high resolution multidetector size exclusion chromatography: Separation of the effects of branching and molecular weight distribution

This article presents the SEC analysis of branched polyisobutylene PIB and polystyrene PS with high molecular weight and broad multimodal molecular weight distribution. Both polymers were synthesized using an inimer technique, which results in long‐chain branched polymers with statistical branching and broad multimodal distributions. Using high resolution multidetector Size Exclusion Chromatography SEC the polymers were analyzed based on three branching factors: g = (R_z,br/R_z,lin)_Mw; h = (〈R_h〉_z,br/〈R_h〉_z,lin)_{Mw ;} and ρ = (R ^1/2/〈R_h〉_z). It is generally accepted that for monodisperse branched polymers g and h < 1. In the case of our polydisperse PIB and PS, it was seen that g and h > 1, and ρ increases with molar mass and the number of chain ends as predicted earlier. The multidetector SEC system allowed for the separation of branching and polydispersity, reported here for the first time experimentally. The g parameter as a function of DP_i was compared to the theory developed by Zimm and Stockmayer. The plots followed a similar trend, but were shifted by a factor related to the average chain length between branching points. The ρ parameter decreased with increasing DP_i, as predicted theoretically by Kajiwara. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Off-lattice Monte Carlo studies on size and shape of isolated polymer chains with isotactic and syndiotactic attached side groups

Chains containing side groups were generated with up to 102 bonds in the backbone for both iso- and syndiotactic arrangements. The mean-square end-to-end distance <h ²>, the mean-square radius of gyration <r ²> and the ratios <l _i ² >/<l ₁ ² > of the axes of inertia were investigated. It could be shown that the mean size and shape of such chains are not influenced by the kind of stereoregularity.Dedicated to Prof. Dr. K. Ueberreiter on the occasion of his 70th birthday.     

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.     

Advanced Monte Carlo Modeling Using Weight‐Based Selection of Arborescent Polyisobutylene Molecules in a Batch Reactor

An advanced Monte Carlo (MC) method is developed, using weight‐based selection of polymer chains, to predict the molecular weight distribution (MWD) and branching level for arborescent polyisobutylene (arbPIB) at the end of a batch reaction. This new weight‐based MC method uses differential equations and random numbers to determine the detailed structure of arbPIB molecules. Results agree with those from an advanced number‐based MC method. The proposed weight‐based algorithm requires approximately twice the computation time of the number‐based method, but produces more accurate results in the high‐molecular‐weight portion of the MWD when the same number of polymer chains is assembled.

     

体积排阻色谱法测定低分子量肝素抗凝血因子Xa的活性

发展了一种基于体积排阻色谱测定低分子量肝素(LMWH)抗凝血活性的方法。利用肝素与抗凝血酶Ⅲ(ATⅢ)结合后可增强ATⅢ对凝血因子Xa(FXa)抑制作用的原理,通过测定加入LMWH后FXa水解其生色底物产生对硝基苯胺(pNA)这一反应的抑制程度确定LMWH的活性。首先将含有一定浓度LMWH的缓冲溶液与ATⅢ溶液混合,然后依次加入FXa和生色底物,分别孵育一段时间。底物被FXa水解,产生游离的pNA。体积排阻色谱可将小分子产物pNA与其他大分子分离开,因而可以在pNA的最大吸收波长下得到高灵敏度的测定,并且不再受其他成分的干扰。该方法重复性好,灵敏度高,极大地减少了样品的消耗量,降低了成本,并且还可进行各种复杂样品(如血浆)中LMWH抗FXa活性的监测。     

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.     

Error introduced into determination of pore size distribution by size exclusion chromatography

Simultaneous use of large standard molecules and small particles of the product examined gives rise to errors in pore size determination by size exclusion chromatography. This error is calculated for packings of spherical particles, thus making corrections possible.     

低分子量硫酸化多糖的体积排阻色谱法分离及其组成定量分析

建立了用于分离并定量测定低分子量硫酸化多糖中不同糖链数的各个组分分布比例的体积排阻色谱方法。系统考察了流动相的组成、离子强度和pH值、流速、柱温等因素对分离的影响。最佳分离条件: 两支TSK-GEL G2000 SWxl色谱柱(300 mm×7.8 mm)串联,流动相100 mmol/L Na2HPO4-NaH2PO4 (pH 7.0),流速0.5 mL/min,柱温35 ℃,进样量5 μL,样品质量浓度10 g/L。在最佳的分离条件下,可以将低分子量硫酸化多糖样品中不同糖链数的各个组分分离并对各个组分的分布进行了定量分析。用该方法对美国药典标准品(USP)、商品和实验室制备的低分子量硫酸化多糖糖链数分布进行了定量化比较,证明该方法可用于低分子量硫酸化多糖类药物的组成成分的质量控制。     

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.     

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.     

Effect of interchange reactions on the molecular weight distribution of poly(ethylene terephthalate): A Monte Carlo simulation

The effect of the interchange reactions of poly(ethylene terephthalate) (PET) on its molecular weight distribution (MWD) was analyzed using a Monte Carlo simulation method. Three kinds of motions, which correspond to the direct ester(SINGLEBOND)ester interchange reaction, alcoholysis, and internal alcoholysis in polyester, were performed in this simulation: bond flip, end attack, and backbite. Two systems with two different types of nonequilibrium distribution (monodisperse and bimodal distribution) were initially prepared. The initial biases from equilibrium MWD are rapidly relaxed to an equilibrium MWD as the reaction progresses. The MWD at equilibrium is well described by the most probable MWD proposed by Flory. From the polydispersity data, it is concluded that about 0.3 interchanges per segment are sufficient to equilibrate the nonequilibrium system. For the validity of the simulation, the variation of MWD of the mixtures of two PETs having different molecular weights were monitored using gel permeation chromatography. The agreement between simulation and experiment is remarkably good. © 1996 John Wiley & Sons, Inc.