An application of Monte Carlo method to simulate disorders in polymer crystals

The Monte Carlo method is applied to polymer crystals of idealized linear chain molecules of 30 carbon atoms, and the unharmonic, large-amplitude, oscillations and the subsequent conformational disorders of the chains are investigated. A crystalline field that confines the chain is treated by the molecular field approximation, and assumed to be cylindrical in this work. A production type simulation is adopted taking into account rigorous statistical weights for each sample conformation. Both the rotational isomeric model and the continuous rotation model of chain conformation are considered. By averaging over 10⁴–8 × 10⁴ chains, mean-square end-to-end distance, fractions of gauche and trans states and a detailed distribution of internal rotation angle are obtained. The effects of temperature and pressure on the conformation of the chain in the crystals are also simulated.     

A multistep method for steady-state Monte Carlo simulations of polymerization processes

Identifying the microscopic information of polymers is of great significance for polymerization processes. Monte Carlo (MC) simulation is a powerful tool to predict the microscopic structure of polymers. Currently, most MC methods are designed for dynamic polymerization processes based on time evolution. The study on MC simulation for steady-state processes is scarce and current approaches face challenges in addressing complex mechanisms. In this work, a multistep method is proposed for the steady-state MC simulation. By introducing the “buffer pool” concept, the proposed method is computationally efficient and flexible to derive accurate predictions for processes with various polymerization mechanisms. Three applications with increasing complexity in the kinetic mechanisms, including both linear and branching polymerizations, are presented to demonstrate the applicability of the proposed method.     

Metamers for assessing the quality of CIE D50 simulators

Simulators of CIE Illuminants D65 and D75 are widely used for visually judging colors, but D50 simulators are used in photography and color printing. The CIE method of assessing the quality of daylight simulators provides sets of virtual metamers for assessing D55, D65, and D75 simulators, but does not provide them for D50. Data specifying sets of metamers for assessing the quality of D50 simulators are presented. They are based in part on data in a Japanese standard and in part on newly derived data. © 1996 John Wiley & Sons, Inc.     

A three-dimensional Monte Carlo model for electrically conductive polymer matrix composites filled with curved fibers

A three-dimensional (3-D) Monte Carlo model is developed for predicting electrical conductivity of polymer matrix composites filled with conductive curved fibers. The conductive fillers are modeled as a 3-D network of finite sites that are randomly positioned. The percolation behavior of the network is studied using the Monte Carlo method, which leads to the determination of the critical fiber volume fraction (or the percolation threshold). The effect of fiber curliness on the percolation behavior is incorporated in the current model by using 3-D arm-shaped fibers, each of which needs five independent geometrical parameters (i.e., three coordinates for its vertex and two orientation angles) for its identification. There are three controlling parameters for such fibers, namely the fiber arm length, the fiber aspect ratio, and the fiber arm angle. The new model also considers the sample size and scaling effects. The simulation results reveal an exponential relationship between the fiber aspect ratio and the percolation threshold: the higher the aspect ratio, the lower the threshold. It is also found that the curliness largely influences the percolation threshold: the more curved the fiber, the higher the threshold. However, the effect of curliness diminishes with the increase of the fiber aspect ratio. With the percolation threshold obtained from the Monte Carlo model, the effective electrical conductivity of the composite is then determined by applying the theory of percolation. The numerical results indicate that the composite conductivity decreases as the fibers become more curved and as the fiber aspect ratio decreases. These predicted trends of the percolation threshold and composite conductivity are in good agreement with existing experimental and simulation results.     

Two methods for sensitivity analysis of coagulation processes in population balances by a Monte Carlo method

We consider two stochastic simulation algorithms for the calculation of parametric derivatives of solutions of a population balance equation, namely, forward and adjoint sensitivity methods. The dispersed system is approximated by an N-particle stochastic weighted ensemble. The infinitesimal deviations of the solution are accounted for through infinitesimal deviation of the statistical weights that are recalculated at each coagulation. In the forward method these deviations of the statistical weights immediately give parametric derivatives of the solution. In the second method the deviations of the statistical weights are used to calculate a finite-mode approximation of the linearized version of the population balance equation. The linearized equation allows for the calculation of the eigenmodes and eigenvalues of the process, while the parametric derivatives of the solution are given by a Lagrange formalism.     

Monte Carlo法跟踪随机运动颗粒中吸附动力学的快速计算方法

ＭｏｎｔｅＣａｒｌｏ法模拟颗粒随机运动体系的吸附过程仿真性强，但机时消耗严重。本文提出了一种在相同精度下比六点隐式差分格式数值求解吸附动力学方程的常规方法快５０００～２００００倍的吸附动力学快速计算新方法，并对实例做了计算比较。     

A kinetic Monte Carlo approach to diffusion in disordered nanoporous carbons

The efficiency of different technological processes where nanoporous carbons are used depends on their storage capacity and an appropriate gas transport. Different experimental and theoretical works that relate storage to global structural parameters of the solid such as specific surface area (SSA) or total pore volume (V_t) can be found in literature. The structure–transport relationships have been less studied. The combined use of the truncated pore network model (TPNM) and the kinetic Monte Carlo (KMC) method is proposed in this work to find hydrogen and methane effective self-diffusivities (D_eff) and to delve into those relationships. It was found that for Knudsen and free molecular diffusion in the simulated materials, the D_eff/V_t vs. SSA graphic nearly follows a power law. The KMC/TPNM approach was also used to predict the self-diffusion coefficients of hydrogen in Vulcan XC-72 and of methane in a carbon aerogel. The obtained values are within the expected range. KCM/TPNM is computationally fast and it allows a study of the diffusion synchronously and globally in the network, avoiding thus its fractionation in single pores and the use of just one geometric model to describe the porous spaces.     

A conditional Monte Carlo method to predict branched architectures from molecular weight and degree of branching distribution of polyethylene for single and mixed systems with a constrained geometry metallocene catalyst in continuous reactors

Any 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 conditional Monte Carlo algorithms for the virtual synthesis of polyethylene (PE) catalyzed by one branch forming constrained geometry metallocene catalyst (CGC, 1C-system) or by a mixture of CGC and linear metallocene catalyst (2C-system) in a continuous stirred tank reactor. The condition for the Monte Carlo method consists of a fixed chain length distribution and degree of branching distribution. To calculate these distributions explicit algebraic relations are available Macromolecules 36 (2003) 10037. The synthesis method is a recursive algorithm that subsequently creates insertions of sub-structures containing numbers of monomer units and branch points according to one- or two-dimensional probability density functions. We developed a general 2C-algorithm and a simplified 1C-algorithm applicable for 1C-systems only. They provide an adjacency matrix describing the connectivity between the branch points, while separately (1C-method Macromol Theor Simul 12 (2003) 484) or simultaneously (2C-method) a vector containing the length of segments between branch points and terminal segments is generated Macromol Theor Simul 10 (2001) 855. Characterization of the architectures proceeds by calculating the topological variables seniority and priority as well as the radius of gyration Macromol Theor Simul 10 (2001) 870. As regards seniority and priority good agreement is found with results from the analytical approach Macromolecules 36 (2003) 10037 is found. 2C-architectures turn out to possess more comb-like structures than 1C-architectures. Concerning molecular sizes an interesting effect of the segment length distribution in 2C-molecules is observed.     

Conditional Monte Carlo sampling to find branching architectures of polymers from radical polymerizations with transfer to polymer and recombination termination

A model is developed that predicts branching architectures of polymers from radical polymerization with transfer to polymer and termination by disproportionation and recombination, in a continuously stirred tank reactor (CSTR). It is a so-called conditional Monte Carlo (MC) method generating architectures of molecules of specified dimensions. The relevant dimensions in the present case are the number of branch points, n_p, and the number of combined parts a molecule consists of, n_c. These branch points and combination points together are decisive for the connectivity inside molecules. The modeling strategy is based on backtracking of the molecular growth history in terms of the chemical events determining connectivity, transfer to polymer and recombination termination. The recombination termination mechanism requires the model to develop parts of the architecture following several paths back to the initial primary polymers that form the starting points for the molecules. The algorithm requires the construction of probability density functions being evaluated using a fast Galerkin-FEM method. The architectures generated by the conditional Monte Carlo method are compared to those from a full MC method using several qualifiers. One of these is the number of initial primary polymers in a molecule as well as their lengths, another is the radius of gyration contraction factor. Perfect agreement is found between the architectures found by the conditional and full MC methods.     

Eulerian Monte Carlo method for the joint velocity and mass-fraction probability density function in turbulent reactive gas flows

A new Eulerian (field) Monte Carlo method for solving an equation that describes the one-time one-point probability density of species mass fractions in turbulent reactive gas flows has been proposed in a previous article. In the present paper, this method is extended to an equation for the joint velocity and mass-fraction probability density function. The method is based on passing from Lagrangian variables used in Lagrangian Monte Carlo methods to Eulerian variables. In this manner, stochastic ordinary differential equations for the Lagrangian trajectories of fluid particles are transformed to partial stochastic equations. As compared to the classical hydrodynamics, the stochastic velocity field satisfies only the mean continuity constraint and not the instantaneous one. As a consequence, one has to introduce a stochastic density, which differs from the physical density but has the same mean value. The case of the mass-fraction probability density is revised. The equations differ from those derived previously: they can be written in divergent form. Both formulations, however, are statistically equivalent. __________ Translated from Fizika Goreniya i Vzryva, Vol. 42, No. 6, pp. 144–155, November–December, 2006.     

Roughening of catalytic surface due to reversible surface reconstruction coupled with oscillatory dynamics: a Monte Carlo study

Pattern formation processes on single-crystal catalytic surfaces involve transitions between alternative surface phases coupled with oscillatory reaction dynamics. A qualitative Monte Carlo model has been developed in order to describe dynamics of Pt(1 1 0) surface reconstruction under reactive conditions on atomic scale. We use kinetics of a phenomenological time-dependent Ginzburg–Landau (TDGL) type, which is known to give a correct qualitative picture of oscillations. However, the lattice dynamics model involves realistic effects of nearest neighbor interactions dependent on the surface coverage. The surface phase distribution oscillates chaotically together with adsorbate coverage, and, in addition, undergoes a slow roughening process, resulting in the growth of island size and an increase of the oscillation period.     

A graph theory assisted Monte Carlo algorithm for simulating the topology structure of LDPE and connection with molecular dynamics simulation

Precise simulation of the topology structure of low-density polyethylene (LDPE) chain is significant for product properties research. However, the chain structure symbolization methods in literature all rely on specially designed methods or temporary tricks, which are hard to understand and may induce some deviations in branch distribution prediction. To overcome this problem, a graph theory assisted Monte Carlo algorithm is developed for chain topology structure simulation of LDPE. The new symbolization method provides better accuracy and is much easy for code implementation and integration in other software. In detail, the average long chain branching predicted by this method is always smaller than that predicted by traditional methods, and much closer to the experimental results. Furthermore, the predicted topology structure information of LDPE chains can be provided as input data for molecular dynamics simulation to study the crystal process, and the predicted crystallinity and density also show good agreement with experiments.     

Application of the Monte Carlo simulation method to the investigation of peculiar free‐radical copolymerization reactions: Systems with both reactivity ratios greater than unity (rA > 1 and rB > 1)

With a Monte Carlo simulationmethod, copolymer properties have been thoroughly studied, and the influence of the reactivity ratios and feed composition has been taken into consideration. Instantaneous alterations of the copolymer composition and copolymer heterogeneity, which is also called a randomness parameter, have been examined with data obtained from the simulation at each stage of the copolymerization reaction. The results prove the azeotropic behavior of copolymerization reactions in which both reactivity ratios are greater than unity, although some special reactivity ratio combinations ignore the azeotropic behavior. The copolymer composition reaches an azeotrope point at the end of the copolymerization reaction when the copolymerization is an azeotropic reaction. In addition, the randomness parameter takes its maximum value at the azeotrope point when reactivity ratio r_A is equal to reactivity ratio r_B. Finally, increasing the reactivity ratios causes no change in the trend of copolymer composition/feed composition curves when r_A is equal to r_B. However, the curves produced with larger r_A and r_B values show more fluctuations. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci, 2007     

基于蒙特卡罗方法的率值稳定性定量分析(Ⅲ)——原料计量设备误差对生料率值稳定性的影响

使用误差理论和方法对原料计量设备误差的性质、特点,及对出磨生料率值稳定性的影响规律和程度进行了分析;使用蒙特卡罗方法模拟计算了原料计量设备随机误差与出磨生料率值标准偏差之间的定量关系。结果表明,当4种原料计量设备准确度等级为1%时,KH、SM、IM的标准偏差增量分别为:0.0026、0.0027、0.0022。     

基于蒙特卡罗方法的率值稳定性定量分析(Ⅰ)——随机检验误差对生料率值稳定性的影响

使用蒙特卡罗方法对随机检验误差对生料率值稳定性的影响进行了模拟计算.结果表明,在生料化学成分SiO2、Al2O3、Fe2O3、CaO自身波动标准偏差分别为0.30%、0.10%、0.08%、0.45%的条件下,生料化学成分SiO2、Al2O3、Fe2O3、CaO检验标准偏差分别为0.050%、0.067%、0.050%、0.083%时,导致生料率值KH、SM、IM标准偏差分别增加0.000 3、0.007、0.0094.化学成分检验标准偏差按倍数增加时,生料率值标准偏差随之增加,但与化学成分检验标准偏差增加的倍数不同.得到了4种主要化学成分检验标准偏差单位变化量对应的生料率值标准偏差变化量.     

Part IV: Dynamic evolution of the particle size distribution in particulate processes. A comparative study between Monte Carlo and the generalized method of moments

The present work provides a comparative study on the numerical solution of the dynamic population balance equation (PBE) for batch particulate processes undergoing simultaneous particle aggregation, growth and nucleation. The general PBE was numerically solved using three different techniques namely, the Galerkin on finite elements method (GFEM), the generalized method of moments (GMOM) and the stochastic Monte Carlo (MC) method. Numerical simulations were carried out over a wide range of variation of particle aggregation and growth rate models. The performance of the selected techniques was assessed in terms of their numerical accuracy and computational requirements. The numerical results revealed that, in general, the GFEM provides more accurate predictions of the particle size distribution (PSD) than the other two methods, however, at the expense of more computational effort and time. On the other hand, the GMOM yields very accurate predictions of selected moments of the distribution and has minimal computational requirements. However, its main disadvantage is related to its inherent difficulty in reconstructing the original distribution using a finite set of calculated moments. Finally, stochastic MC simulations can provide very accurate predictions of both PSD and its corresponding moments while the MC computational requirements are, in general, lower than those required for the GFEM.     

A novel analysis of color component for top dyed melange yarn with support vector machine

In view of the difficulty of distinguishing the color component in top dyed melange yarn due to the spectral overlap of the component colors, a novel color component analysis method based on support vector machine is presented. With this method, spectra data can be distinguished more accurately and effectively than with the traditional method—human‐eye detection—and therefore, the method will be very helpful for accurate color matching. In our work, the core idea was to convert the overlapped spectra data into linearly separable ones in a high dimension space, followed by recognition and determination of the composition of melange yarn by trained support vector machine classifier. The effects of four kernel functions, i.e., linear, radial basis kernel, sigmoid, and polynomial, as well as five spectral preprocessing methods, including amplification, first derivative, second derivative, principal components analysis, and L*a*b* values were studied. The results demonstrated that with the amplification factor of 100 of reflectance spectra coupled with L*a*b* as input data, and using radial basis kernel as kernel function, the highest recognition rate was achieved, with an average recognition rate of eight colors of 96.5%, indicating that it was a better color component analysis method for top dyed melange yarn. © 2015 Wiley Periodicals, Inc. Col Res Appl, 41, 636–641, 2016     

基于蒙特卡罗方法的率值稳定性定量分析(Ⅱ)——原料化学成分波动对生料率值稳定性的影响

使用蒙特卡罗方法对原料化学成分波动与生料率值稳定性之间的定量关系进行了模拟计算。结果表明，一种原料单一波动，石灰石的SiO2、Al2O3、Fe2O3、CaO的标准偏差分别为0．10％、0．02％、0．05％、0．15％时，生料KH、SM、IM的标准偏差分别为0．006、0．028、0．031；砂岩的SiO2、Al2O3、Fe2O3、CaO的标准偏差分别为0．80％、0．20％、0．50％、0．10％时，生料KH、SM、IM的标准偏差分别为0．005、0．033、0．038。这一结果揭示，相对于石灰石，砂岩的预均化同样不容忽视。石灰石和砂岩化学成分同时按上述标准偏差波动，生料KH、SM、IM的标准偏差分别为0．008、0．043、0．049。     

A general framework to study the price‐color relationship in paintings with an application to Mark Rothko rectangular series

We propose a general framework to study the relationship between the price of a painting and its color‐related attributes. To this end we focus on four key aspects: dominant colors, features of the color palette, color harmony, and color emotions. We demonstrate the usefulness of this approach with an example based on Mark Rothko's post‐1950 paintings (the “rectangular” series), and auction data from the 1994 to 2018 period. We identify two distinct price‐color regimes in Rothko's market: (a) [1994‐2005], a period in which prices are explained mainly by the growing popularity of the artist regardless of the color attributes of the paintings sold; and (b) [2006‐2018], a period in which color‐related attributes explain most of the prices. Furthermore, we find that in this second period, the dominant colors and the diversity of the color palette, are by far the most relevant attributes that influence the price; color harmony and color emotions hold almost no explanatory power during this period. Finally, we propose a new metric based on the Herfindahl Index to describe color diversity; this metric seems to be promising at characterizing the effect of the color palette on the price of a painting.     

Combination of a Monte Carlo approach with the contact time distribution concept for the steady-state modeling of an isothermal heterogeneous coordinated anionic ring opening polymerization reactor

In this paper, the possibility of combining a molecular level Monte Carlo simulation with a chemical engineering model of an isothermal fixed-bed reactor is demonstrated. This approach is applied to the isothermal heterogeneous coordinated anionic ring opening polymerization of ε-caprolactone. The contact time distribution (CTD) concept as defined by Orcutt et al. (Chem. Eng. Prog. Symp. Ser. 38 (58) (1962) 1), Nauman and Collinge (Chem. Eng. Sci. 23 (1968a) 1309) and Shinnar et al. (Chem. Eng. Sci. 27 (1972) 1627) is used. The contact time is the time spent by a reactant within the pores of the pellets constituting the fixed-bed. The classical monodisperse model represents the hydrodynamic and mass transfer in the reactor. The CTD is calculated from this model according to Shinnar et al. theory. From that point, the reactor outlet monomer conversion can be calculated analytically, the ε-caprolactone consumption being a first-order process. Furthermore, the molecular size distribution (MSD) of the oligomers at the reactor outlet is computed on the basis of Monte Carlo simulations. Comparisons with experimental data are given.