Procedures and Guidelines for Inputting and Output Smoothening of Kinetic Monte Carlo Distributions

Abstract

Population balance modeling is very popular in several branches of modern science, including (bio)chemical engineering, geography, and demography. A growing trend is the use of stochastic solvers (e.g., kinetic Monte Carlo), but care should be taken upon inputting and outputting distributed data for the populations involved. In the present contribution, several procedures and guidelines are formulated facilitating such data treatment. The solutions are exemplified through polymerization and polymer modification case studies, taking the molar mass or chain length as core stochastic variable. It is shown that rediscretization is often necessary, as experimentally measured distributions are not directly interpretable at the input side of stochastic solvers. This rediscretization should take place both for the abscissa and ordinate values of the distributions. As stochastic solvers often display noisy data as model output, attention is paid to the smoothening of the output distributions as well. It is highlighted that the kernel‐smoothening method is very promising, as it can reliably tackle postprocessing of nonsymmetric distributions. Guidelines are formulated regarding the correct interpretation of the distribution tail both for the input and output modifications.

Kinetic Monte Carlo residence time distributions and kinetics in view of extrusion-based polymer modification and recycling

“…Unless mentioned otherwise, in the present work, a MTF MC value of 1 s −1 is adopted. Notably, if MTF MC is lower than 1 s −1 a rediscretization is carried out on the RTD, following the procedure developed by De Keer et al 106 Macro-scale convective mass transfer events between the CSTR's in this second phase can be seen as transfers of individual polymer chains which are selected randomly on a number basis but describing a desired physical transfer over all compartments, e.g. not allowing for a direct consecutive transfer (so backward transfer is still allowed) and acknowledging a preservation of compartment volume.…”

“…Unless mentioned otherwise, in the present work, a MTF MC value of 1 s −1 is adopted. Notably, if MTF MC is lower than 1 s −1 a rediscretization is carried out on the RTD, following the procedure developed by De Keer et al 106…”

On the Precise Determination of Molar Mass and Dispersity in Controlled Chain-Growth Polymerization: A Distribution Function-Based Strategy

“…Meanwhile, the lognormal distribution was employed to describe the number MMD instead of the number CLD, which is convenient for the users to directly convert the SEC-RI results into the true ones, omitting transformation from chain length averages to molar mass averages. The n ln ( M r ) plot obeys the normalization criterion: ∑ M r = M mon ∞ n ln ( M r ) × normalΔ M r = ∑ M r = M mon ∞ n ln ( M r ) × ( Δ r × M mon ) = 1 in which Δ M r is the abscissa value interval. To approximate the experimental SEC-RI trace, the number MMD was transformed into w (ln M r ), shown as eq .…”

“…As explained above, the as-proposed molar mass rectification strategy based on the lognormal distribution is feasible for the polymers obtained by the well-controlled chain-growth polymerizations that produce polymers with relatively narrow CLDs (e.g., Đ < 1.5), and the guidance for the users is illustrated in the Scheme . It is noteworthy that the results derived from the lognormal distribution framework in this work are also potentially useful in numerical simulation, for instance, as an initial distribution input code in the kinetic Monte Carlo ( k MC) model or as a predefined description for distribution output in the extended method of moments (MoM) . Also, the distribution function used in this method is alterable so that it can be extended to other polymerization techniques.…”

Toward an Automated Convergence Tool for Kinetic Monte Carlo Simulation of Conversion, Distributions, and Their Averages in Non-dispersed Phase Linear Chain-Growth Polymerization

“…In many situations, the selection of the appropriate control volume is made by visual thus human-based inspection of a set of average simulation targets (alongside the monomer conversion), and in certain more specialized studies, the resulting CLD of one or more than one of the products is inspected. Numerical strategies can be used such as distribution curve smoothening , to extract the core distribution information and minimize the signal-to-noise ratio. Notably, benchmarking studies with deterministic solvers have also been performed, − highlighting the overall consistency of k MC simulations.…”