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.

     

Mathematical Modeling of Arborescent Polyisobutylene Production in Batch Reactors

A novel model describes copolymerization of isobutylene and inimer (initiator‐monomer) via living carbocationic polymerization. Six different propagation rate constants and two types of equilibrium reactions are considered. Simplifying assumptions are made to enable implementation in PREDICI, so that the molecular weight distribution (MWD) could be predicted for molecules with different branching levels. Four apparent rate constants were estimated from experimental data with <5 branches per molecule. Model predictions provide a good fit to data, and simulation results show that polymers with high‐branching levels and ≥15 inimer units contribute significantly to the MWD, even though their concentrations are very low.

     

Dynamic Monte Carlo Simulation of ATRP in a Batch Reactor

A dynamic MC model was developed to simulate the polymerization kinetics and the detailed microstructure of copolymers made with ATRP in a batch reactor. The model was used to predict monomer conversion, average molecular weight, polydispersity index, and copolymer composition as a function of polymerization time. The model can also predict the distribution of molecular weight, chemical composition, and comonomer sequence length at any polymerization time or comonomer conversion. The simulation was used to explore the effects of rate constants and reactant stoichiometry on the microstructure of chains. Two copolymerization systems were chosen to demonstrate the effect of reactivity ratios and comonomer feed compositions on the final chemical composition distribution.

     

电解质模型流体的MonteCarlo分子模拟

采用正则系综Monte Carlo分子模拟方法，模拟得到了一系列状态条件下1：1 价和2：2价对称电解质模型流体的径向分布函数、构型能和压缩因子的“机器实验 数据”，这些数据对构筑电解质溶液的新参考流体具有重要意义。模拟过程中，电 解质溶液被简化为硬球离子和硬球点偶扳子的混合物，离子—离子、离子—偶极和 偶极—偶极长程位能计算采用了Ewald求和方法。最后，对离子—偶极子混合物的 结构和热力学性质与体系温度、密度和浓度的关系进行了分析和讨论。     

Monte Carlo Simulation of Controlled/Living Radical Polymerization in Emulsified Systems

A new MC simulation method is proposed for the controlled/living radical polymerization in a dispersed medium, assuming an ideal miniemulsion system. This tool is used to consider the effects of particle size on the polymerization rates and the molecular weight distributions. For NMP, the polymerization kinetics are basically governed by two conflicting factors, (i) the confined space effect that promotes the coupling reaction between a radical and a trapping agent and (ii) the isolation effect of radicals into different particles that suppresses the overall frequency of bimolecular termination. For RAFT polymerization, a significant rate enhancement by reducing the particle size could be observed only for the systems with fast fragmentation of adduct radicals.

     

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     

Simulation model for the molecular weight distribution in emulsion polymerization

A Monte Carlo simulation model for the kinetics of emulsion polymerization is proposed. In the present model, the formation of each polymer molecule is simulated by the use of only a couple of probability functions; therefore, the calculation can be handled well even on personal computers. It is straightforward to account for virtually any kinetic event, such as the desorption of oligomeric radicals and chain length dependence of kinetic parameters, and as a consequence very detailed information such as the full distributions of the dead polymer molecular weights and the macroradicals among various polymer particles can be obtained. When bimolecular terminations are the dominant chain stoppage mechanism, the instantaneous molecular weight distribution (produced in a very small time interval) becomes broader than that for homogeneous polymerizations due to a higher possibility that short and long polymer radicals react with each other if bimolecular reactions are fast enough. The increase in the polydispersity of the MWD is fairly large, especially when bimolecular termination by disproportionation is significant; however, the gel permeation chromatography (GPC) may not be a suitable analytical technique to detect such broadening since oligomeric peaks may not be observed in the elution curve. The present simulation method provides greater insight into the complicated phenomena of emulsion polymerizations. © 1995 John Wiley & Sons, Inc.     

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     

Chain Length Distributions of Polyolefins Made with Coordination Catalysts at Very Short Polymerization Times – Analytical Solution and Monte Carlo Simulation

We developed an analytical solution to describe how the CLD of polymers made with coordination polymerization catalysts vary as a function of time for very short polymerization times before the CLD becomes completely developed. We compared the analytical solution with a dynamic Monte Carlo model for validation, obtaining excellent agreement. Our analytical solution can be used to determine when the steady‐state hypothesis, commonly used in polymerization models, becomes valid as a function of polymer chain length. We also extended our model to describe polymerization with multiple‐site‐type catalysts. Depending on the polymerization kinetic parameters of the different site types on the catalyst, the fully developed CLD is reached through very different intermediate CLDs. This modeling approach, although rather simplified, can be used to interpret results from short polymerization time experiments such as the ones done in stopped‐flow reactors.

     

RAFT种子乳液聚合反应初期的Monte Carlo模拟

自由基聚合是制备聚合物材料最为重要的技术 .但由于自由基极易进行双基终止 ,一般很难对其结构进行精确的控制 ,所得产物分子量宽 ,组成分布不易控制 ,很难制备嵌段共聚物 . 2 0世纪 90年代出现的活性自由基聚合技术 (RAFT)克服了上述缺点 ,成为高分子化学研究的热点[1] .RAFT聚合以其适用单体广、聚合条件温和以及活性高而成为最具前途的活性自由基聚合技术之一 .迄今为止 ,RAFT的研究大多集中在溶液和本体等均相聚合体系 [2～ 5] .乳液聚合有聚合速率快、环境友好、体系粘度低等优点 ,是活性自由基聚合工业化首选工艺 ,因而近年来活…     

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     

分批投料模式下加核自缩合乙烯基体系的Monte Carlo模拟

通过Monte Carlo 模拟方法对加核自缩合乙烯基体系在分批投料模式下的聚合行为进行研究, 考察了引发核的加入时间及分批投料方式对超支化高分子的重均分子量及多分散指数的影响. 研究结果表明, 不同的分批投料方式对超支化高分子的重均分子量和多分散性具有显著的调控作用. 通过调整投料方式、 投料次数、 引发核官能度和转化率等因素, 可以得到分子量较高且多分散性较好的超支化高分子.     

Prediction of Chain Length Distribution of Polystyrene Made in Batch Reactors with Bifunctional Free‐Radical Initiators Using Dynamic Monte Carlo Simulation

The objective of this paper is to present a dynamic Monte Carlo model that is able to simulate the polymerization of styrene with bifunctional free‐radical initiators in a batch reactor. The model can predict the dynamic evolution of the chain length distribution of polystyrene in the reactor. The model includes all relevant polymerization mechanistic steps, including chemical and thermal radical generation, and diffusion‐controlled termination. The model was applied to styrene polymerization and the Monte Carlo estimates for chain length averages were compared to those obtained with the method of moments. Excellent agreement was obtained between the two methods. Although styrene polymerization was used as a case study, the proposed methodology can be easily extended to any other polymer type made by free‐radical polymerization.

     

Wavelet-accelerated Monte Carlo sampling of polymer chains

We introduce a new method for coarse-graining polymer chains, based on the wavelet transform, a multiresolution data analysis technique. This method, which assigns a cluster of particles to a coarse-grained bead located at the center of mass of the cluster, reduces the complexity of the problem significantly by dividing the simulation into several stages, each with a small fraction of the number of beads in the overall chain. At each stage, we compute the distributions of coarse-grained internal coordinates as well as potential functions required for subsequent simulation stages. We show that, with this wavelet-accelerated Monte Carlo method, coarse-grained Gaussian and self-avoiding random walks can reproduce results obtained from atomistic simulations to a high degree of accuracy in orders of magnitude less time. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 897–910, 2005     

Bimodal molecular weight distribution formed in the emulsion polymerization of ethylene

It is known that the molecular weight distribution (MWD) formed in an emulsion polymerization of ethylene can be bimodal. However, the origin of the bimodality has not been elucidated. In this article, a Monte Carlo simulation is conducted, mostly with parameters reported in the literature. The simulated MWDs are bimodal because of the limited volume effect; that is, the high molecular weight profiles are distorted by the small particle size, which is comparable to the size of the largest branched polymer molecule in a particle. The simulated MWDs agree reasonably well with the experimentally obtained MWDs. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 3426–3433, 2002     

Monte Carlo Simulations of Adsorbed Solutions in Heterogeneous Porous Materials

We present results of a Monte Carlo simulation study of binary mixtures of ethane and methane in silica gel. The molecular model treats the adsorbent as a matrix of silica microspheres. The adsorption isotherms, adsorption selectivities and isosteric heats of adsorption have been determined for these systems. The results are compared with predictions from the ideal adsorbed solution (IAS) theory and with experiment. The heats of adsorption are accurately described by the IAS theory. The adsorption isotherms are accurately described by the IAS theory at low bulk pressure but the IAS theory overpredicts the density at high bulk pressure. This latter effect is opposite to that observed in bulk mixtures of this type where nonidealities generally lead to a density increase on mixing. The pressure dependence of the selectivity does not exhibit a maximum at low pressure. We discuss this effect in terms of the adsorbent microstructure.     

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.