Monte Carlo Simulation of Ethylene Copolymers: A Look into the Monomer Sequence Distribution

A comprehensive mathematical model was developed using Monte Carlo simulation to describe the mechanism of ethylene and α-olefin copolymerization. The model studies the polymerization mechanism using coordination catalysts and is able to predict molecular weight and detailed chemical composition distributions. This work is considered to be a useful tool that enables us to understand and described the monomer sequence distribution as a function of chain length in semi-batch polymerization reactors.     

A Monte Carlo Simulation of Homogeneous Crystallization in Confined Spaces: Effect of Crystallization Kinetics on the Avrami Exponent

Homogeneous nucleation in polymers and subsequent crystallization in spaces confined relative to spherulitic dimensions in one direction has been simulated for a wide range of nucleation and growth rates. An empirical equation based on simulation results relating an Avrami exponent of crystallization to growth and nucleation rates along with the characteristic dimension of crystallizing volume has been proposed. It was found that for ideal homogeneous crystallization, the Avrami exponent can be significantly lower than the anticipated values for free unconstrained growth; therefore, all crystallization conditions and parameters must be taken into account in order to make a conclusion on the character of the observed crystallization.

     

Monte Carlo simulation of diffusion effects on chain‐extension reactions

The diffusion effects on chain‐extension reactions using carboxyl‐terminated polyamide‐12 as a model reactant with bisoxazolines were investigated by the stochastic Monte Carlo method. Thus, complicated direct modeling and numerical calculations were avoided. The chain‐length dependence and detailed diffusive behavior were discussed in depth. The diffusion effects retarded the progress of chain‐extension reactions and led to lower coupling efficiency. The simulated results indicated that the diffusion effects could make the final molecular weight distributions wider. In the presence of diffusion and with the progress of the coupling efficiency, peaks in the evolution curves of the weight‐average molecular weight and valleys in the evolution curves of the polydispersity index were observed, respectively, when the coupling efficiency was low enough. These phenomena were different from those without diffusion effects and were analyzed in detail. The critical entanglement chain length had strong effects on the simulated results of the diffusion effects, especially when its value was near the average chain length. The results also showed that the effects of the reactant degradation made the molecular weight distribution of the reaction system wider and weakened the diffusion effects on the coupling reaction. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2902–2911, 2006     

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.

     

Modeling of Low Concentrations of Reactive Species in Kinetic Monte Carlo Simulations

A new method for determination of contributions of species present in low concentration in kinetic Monte Carlo (MC) simulations of chemical processes is proposed. It allows, for instance, to calculate of exact concentrations when species being of interest appear in MC experiments only for short time intervals. The method is based on monitoring of the cumulative life‐time of the investigated species under consideration as a function of the reaction time and subsequent differentiation of the resulting curve. It is especially useful for determination of distributions of low‐concentration active propagation species in radical polymerizations, both free and controlled ones. The method enables to reduce significantly a simulation scale required for determination of the concentrations of investigated species.

     

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     

Bayesian Modeling and Markov Chain Monte Carlo Simulations for a Kinetic Study of Homo‐ and Co‐ Polymerization Systems

A Bayesian modeling and Markov Chain Monte Carlo simulation was developed for a kinetic study of homopolymerization and copolymerization systems at the molecular scale. Two copolymerization models – the terminal unit model and the penultimate unit model – were considered. Prior estimates of the kinetic parameters were obtained by L₁‐norm robust statistics. Using the structure of experimental data through a likelihood function, Bayesian modeling was employed to update the prior estimates. The joint posterior probability regions and shimmer bands were calculated for updated reactivity ratios. A method for assessing the power of experimental data in discrimination between copolymerization models is presented. This method was validated for free radical polymerization in binary systems. The evolution of species and radical populations during the course of polymerization were determined. The computational time was considerably decreased by calculating the propagation step from lifetime of the polymer chain and local monomer concentration. To avoid inaccuracies in the results caused by poor choice or false computation of the time step, the time step between successive Monte Carlo events was adapted to the time scale of the fastest reaction. The simulation algorithm is exact, in the sense that it takes full account of the fluctuations and correlations.

     

Kinetic analysis of the cross reaction between dithioester and alkoxyamine by a Monte Carlo simulation

A model reaction of dithioester and alkoxyamine is proposed to probe the reversible addition–fragmentation chain transfer (RAFT) process. The kinetics of the model reaction is analyzed and compared with that of pure alkoxyamine homolysis with a Monte Carlo simulation. Although the pure alkoxyamine obeys the law of persistent radical effect, the model reaction results in higher concentration of the persistent radical during the main period of the reaction. However, for a very fast RAFT process or a very low addition rate constant, the time dependence of the persistent radical concentration is quite close to that of pure alkoxyamine. Furthermore, the cross termination between the intermediate and alkyl radicals causes a retardation effect for the model reaction when the intermediate is relatively long‐lived. The Monte Carlo simulation indicates that it is feasible to measure the individual rate constants of the RAFT process, such as the rate constant of addition, with a large excess of alkoxyamine. In addition, the special feature of the system with different leaving groups in the alkoxyamine and dithioester is also discussed. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 374–387, 2007     

Carboxyl Terminated Polymer Chain Extension Using a Bisoxazoline Coupling Agent: Monte Carlo Simulation

Summary: The kinetics of the chain extension reaction of a carboxyl‐terminated polymer using a bisoxazoline coupling agent was simulated by the Monte Carlo method based on the master equation. The effects of temperature and stepwise addition of the chain extender was examined. A comparison between simulated results and those calculated by an amended kinetic model was made. The results show that the highest coupling efficiency and the highest can be obtained when the initial concentrations of carboxyl and oxazoline groups are equal, which is in good agreement with the experiments. It is found that a higher reaction temperature could lead to a bigger coupling efficiency, a higher and narrower MWDs. The stepwise addition of the chain extender can only postpone the chain extension reaction, but cannot affect the final coupling efficiency and the MWDs when the concentration of the oxazoline group is lower than that of the carboxyl group. However, stepwise addition of the chain extender favors bigger coupling efficiency and narrower MWDs when the concentration of the oxazoline group exceeds that of the carboxyl group.

The relationship among three group of polymers, r and ratios of k₂ to k₁. The dots are the data simulated by MC and solid lines are relative data calculated by the kinetic model.     

Experimentally Calibrated Random Walk of Branched Polymers: A Pragmatic Approach

In order to control the branching behavior of polymers, the comparison of experimental and simulated data is important. The utilization of a nonlattice, self‐avoiding necklace‐bead random walk simulator is reported, which allows for the calculation of radii of gyration r _g of polymer molecules with branched structures. The focus is on sensitivity toward short‐chain branches, long‐chain branches (LCBs), and the copolymer composition. Using only two parameters—the size of monomer beads and the minimum angle between three subsequent beads—a fast and reliable parameter fit procedure based on experimental data is described. The procedure is exemplarily shown for copolymers of vinylidene fluoride and hexafluoropropene (HFP) with HFP contents in the copolymer of at most 0.3 and is easily transferable to other polymers that may be analyzed by size‐exclusion chromatography/multiangle laser light scattering close to θ conditions. Applying the Zimm–Stockmayer equation to simulated r _g data allows for comparing the “effective” number of LCBs with the number of LCBs given by kinetic simulations. A tool for better estimation of rate coefficients associated with the formation of short‐ and long‐chain branches is provided.

     

Theoretical Consideration of the Grafting of Polymer Chains onto Fullerene C60

Summary: For the first time, the kinetics and statistics of the formation of polymer stars by grafting linear macromolecules onto the fullerene C₆₀ are considered. Two kinetic models are put forward providing an explanation for some important experimentally observed peculiarities of these macromolecular reactions. The comparison of the results found from the solution of the kinetic equations and those obtained by Monte Carlo simulations demonstrates their good agreement. Due to the minimal number of the parameters involved, the theoretical approach proposed is rather convenient for the treatment of experimental data.

     

Monte Carlo Computer Simulation of a Single Semi-Flexible Macromolecule at a Plane Surface

Summary: The properties of a single semiflexible mushroom chain at a plane surface with a long-ranged attracting potential are studied by means of lattice Monte Carlo computer simulation using the bond fluctuation model, configurational bias algorithm for chain re-growing and the Wang-Landau sampling technique. We present the diagram of states in variables temperature T vs. strength of the adsorption potential, ε_w, for a quite short semiflexible chain consisting of N = 64 monomer units. The diagram of states consists of the regions of a coil, liquid globule, solid isotropic globule, adsorbed coil and cylinder-like liquid-crystalline globule. At low values of the adsorption strength ε_w the coil–globule and the subsequent liquid–solid globule transitions are observed upon decreasing temperature below the adsorption transition point. At high values of ε_w these two transitions change into a single transition from an adsorbed coil to a cylinder-like liquid-crystalline solid globule. We conclude that for a semiflexible chain the presence of a plane attracting surface favors the formation of a globule with internal liquid-crystalline ordering of bonds.     

Random End‐Switching Configurational Bias Monte Carlo for Long Chain Molecules

Summary: The exponential attrition of configurational bias Monte Carlo for long chains can be reduced to almost quadratic by a simple modification of the basic move. Each trial move begins on the side of the remaining sub chain opposite to the cut location. This type of move is akin to large‐scale reptation and in the limit of one‐segment cut is reptation. The extension is shown to require the same Rosenbluth weighting scheme as the original algorithm. Several examples are used to demonstrate the reliability and the improved performance of the proposed method.

     

基于格子链的缩聚反应的动态Monte Carlo模拟

采用描述自回避格子链的键长涨落模型, 以动态Monte Carlo方法对AB型单体的线型缩聚反应动力学过程进行了模拟. 通过该方法可以得到反应过程中链的瞬时构象, 还可以得到反应程度、聚合度、分子量分布及其随时间的演化. 模拟得到了合理的结果, 同时验证了无规线团尺寸与平均链长的标度关系, 表明该方法用于研究逐步聚合反应过程是可行的, 并且与一般的研究聚合反应的Monte Carlo方法相比, 还能够同时得到构象等空间信息. 还比较了不同大小的模拟体系所得到的分子量和多分散系数的异同, 讨论了有限元胞效应.     

钛电极苯酚降解反应动力学的Monte Carlo模拟

应用Monte Carlo方法,结合化学反应随机过程理论,模拟钛电极上苯酚电催化氧化反应机理,研究苯酚降解时间、初始浓度、电极的催化性能及反应中间产物对降解过程的影响.结果表明:苯酚浓度随时间变化逐渐降低并渐趋稳定,苯酚初始浓度较高时降解速率快,具有较高催化性能的电极对降解第1步反应的影响较大,可有效减少中间产物停留时间,但控制第1步反应则对降解过程尤其是第2步反应的产物有明显影响,而控制第2步反应对降解过程影响很小.     

端点附壁的高分子链形状的Monte Carlo模拟

分别基于简立方格点和四面体格点模型对一端吸附在无限大平面的高分子链（平面接枝高分子链）的形状进行了Ｍｏｎｔｅ Ｃａｒｌｏ模拟,结果表明,接枝高分子链的形状更偏离球形,〈Ｌ＾２１〉：〈Ｌ＾２２〉：〈Ｌ＾２３〉的极限值约为１：２．７５：１２．５,其中〈Ｌ＾２１〉,〈Ｌ＾２２〉和〈Ｌ＾２３〉分别为回转半径张量的本征值Ｌ＾２１,Ｌ＾２２和Ｌ＾２３（Ｌ＾２１〈Ｌ＾２２〈Ｌ＾２３）的统计平均;链长相同时,接枝