Polymer Reaction Engineering

Saldívar‐Guerra, Enrique

doi:10.1002/0471440264.pst669

Cited by 5 publications

(6 citation statements)

References 158 publications

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“…This is undoubtedly the situation in polymer reaction engineering (PRE) with polymerization reactions in parallel and series and the additional complexity introduced by (chain length-dependent) diffusional limitations on the microscale (scale to define concentrations) and mixing/ temperature heterogeneities at the macroscale (scale of the reactor), 24 prohibiting the availability of an exact analytical solution. 22,25 For a general process, one thus needs to resort to numerical frameworks based on deterministic or stochastic problem formulations but maintain fundamental principles such as the consideration of Arrhenius laws and the mass law of action. 24,26−37 A deterministic modeling framework consists of a set of (coupled) differential equations, one for every species, representing mass balances in a given reaction locus.…”

Section: Introductionmentioning

confidence: 99%

“…For systems with elemental species, analytical or pseudoanalytical solutions (e.g., based on the quasi-steady-state approximation; QSSA) are often readily available. , However, to model the time evolution of processes involving distributed species, only in some cases, an exact analytical solution may exist. − In most cases, the likely occurrence of several competing phenomena complicates the feasibility of an analytical approach. This is undoubtedly the situation in polymer reaction engineering (PRE) with polymerization reactions in parallel and series and the additional complexity introduced by (chain length-dependent) diffusional limitations on the microscale (scale to define concentrations) and mixing/temperature heterogeneities at the macroscale (scale of the reactor), prohibiting the availability of an exact analytical solution. , For a general process, one thus needs to resort to numerical frameworks based on deterministic or stochastic problem formulations but maintain fundamental principles such as the consideration of Arrhenius laws and the mass law of action. ,− …”

Section: Introductionmentioning

confidence: 99%

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Gillespie-Driven kinetic Monte Carlo Algorithms to Model Events for Bulk or Solution (Bio)Chemical Systems Containing Elemental and Distributed Species

Trigilio

Marien

Steenberge

et al. 2020

Ind. Eng. Chem. Res.

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Stochastic modeling techniques have emerged as a powerful tool to study the time evolution of processes in many research fields including (bio)chemical engineering and biology. One of the most applied techniques is kinetic Monte Carlo (kMC) modeling according to the stochastic simulation algorithm (SSA) as pioneered by Gillespie, in which MC channels and time steps are discretely sampled from probability distributions. In the last decades, the SSA algorithm, as originally developed for systems with elemental species (e.g., A, B, C, etc.), has been further adapted (i) to also tackle systems with distributed species, therefore, populations and (ii) to enable faster algorithm execution. In the present contribution, we highlight the most important developments, taking bulk/solution polymerization as the reference distributed chemical process. We address SSA principles based on conventional array data structures, common acceleration methods (e.g., τ leaping and the scaling method), and the strength of tree- and matrix-based data structures for detailed storage of molecular information per distribution type and even individual population member. In addition, we report advancement regarding array programming and MC sampling methods complemented by the introduction of the use of higher-order trees and root-finding sampling tools. The contribution gives thus a detailed overview of the available main kMC algorithm steps to study kinetics, irrespective of the specific field of application due to their generic nature.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gillespie-Driven kinetic Monte Carlo Algorithms to Model Events for Bulk or Solution (Bio)Chemical Systems Containing Elemental and Distributed Species

Trigilio

Marien

Steenberge

et al. 2020

Ind. Eng. Chem. Res.

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show abstract

“…The corresponding term in the G ‐th moment of dead polymer can be generalized assuming some conventions as follows [ 39 ] :

\frac{d λ_{G}}{italicdt} = \sum_{n = 1}^{\infty} \frac{n^{G} {italicdD}_{n}}{italicdt} + = \frac{1}{2} k_{italictc} \sum_{n = 1}^{\infty} \sum_{m = 1}^{n - 1} n^{G} P_{m} P_{n - m}; G = 0, 1, 2 \dots

…”

Section: Additional Mechanistic Stepsmentioning

confidence: 99%

“…The corresponding term in the G-th moment of dead polymer can be generalized assuming some conventions as follows [39] :…”

Section: Termination By Combination (Alternative Derivation)mentioning

confidence: 99%

The method of moments used in polymerization reaction engineering for 70 years: An overview, tutorial, and minilibrary

Zapata‐González¹,

Saldívar‐Guerra²

2023

Can J Chem Eng

Self Cite

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In the last seven decades, the method of moments (MoM) has become an invaluable tool in the field of polymerization reaction engineering, due to the simplicity of translating a complex set of population balance equations (PBE) into a system with a limited number of equations. In this work, we offer an overview of the MoM, describing the derivation of the moment equations in a basic kinetic mechanism. Some tools and strategies for the derivation of the moment equations are reviewed and explained in detail, such as the binomial theorem, the method of series expansion and pattern identification (SEPI), extensively used by the community, and a graphical approach of summation inversion. The treatment for multivariate distributions is also exposed, taking advantage of the complete and partial moment techniques. The derivation of the MoM contribution by kinetic mechanisms beyond the basic ones or involving special difficulties, such as depropagation, long chain branching (LCB), random chain scission, LCB and β‐scission, short chain branching (SCB) and scission, internal double bond (IDB) (polymerization), termination by combination, reversible deactivation radical polymerization (RDRP), and intermolecular transesterification reactions (ITRs), are explained in a tutorial way. Additionally, the fundamentals of the MoM in copolymerization and the application of the pseudo‐homopolymerization approach are briefly described. An introduction of the MoM to emulsion polymerization is also presented. Finally, some advanced applications of MoM in recent works are exposed: MoM models with chain‐length dependent or diffusion‐controlled termination, and the extension of the MoM for the prediction of the molecular weight distribution (MWD).

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“…Alternatively, the full set of differential equations can be numerically solved using proper numerical approaches and/or simplifications. 15–17…”

Section: Introductionmentioning

confidence: 99%

Simulation time analysis of kinetic Monte Carlo algorithmic steps for basic radical (de)polymerization kinetics of linear polymers

et al. 2023

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Polymer Reaction Engineering

Cited by 5 publications

References 158 publications

Gillespie-Driven kinetic Monte Carlo Algorithms to Model Events for Bulk or Solution (Bio)Chemical Systems Containing Elemental and Distributed Species

Gillespie-Driven kinetic Monte Carlo Algorithms to Model Events for Bulk or Solution (Bio)Chemical Systems Containing Elemental and Distributed Species

The method of moments used in polymerization reaction engineering for 70 years: An overview, tutorial, and minilibrary

Simulation time analysis of kinetic Monte Carlo algorithmic steps for basic radical (de)polymerization kinetics of linear polymers

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