Numerical Techniques for the Solution of the Molecular Weight Distribution in Polymerization Mechanisms, State of the Art

Abstract

The molecular weight distribution (MWD) is possibly the most important characteristic of a polymer. Polymers derive many of their physical properties from their MWD. Therefore, since the origins of polymer science, the theory provides a link between the kinetic mechanism and the mathematical expression of the MWD, and there are analytical solutions for ideal cases. However, the MWD formed in real‐life polymerization processes is usually more complex; the solution of the mathematical models that describe them can be quite challenging and has been the focus of enormous research efforts. These models may consist of systems of very large dimension: thousands of differential equations, often stiff, which demand special numerical techniques for their solution. In this paper the numerical techniques that can be used to solve this challenging problem are reviewed and contrasted, including weighted residual methods, direct integration, numerical inversion of transformed equations, and lumping methods. Stochastic techniques are also surveyed.

Benchmarking Stochastic and Deterministic Kinetic Modeling of Bulk and Solution Radical Polymerization Processes by Including Six Types of Factors Two

“…[22][23][24] An alternative for event-driven kMC modeling methods is conditional Monte Carlo (MC) methods in which distributed features are sampled based on predetermined distributions. [25,26] The strength of MoM lies in its limited number of differential equations but this goes hand in hand with the calculation of only the evolution of the monomer conversion and averages of the molar mass distribution (MMD) or chain length distribution (CLD) as a function of t. [27][28][29] In turn, event-driven kMC modeling is free from computational stiffness issues so that more complex reaction schemes and the calculation of complete MMD/CLDs can be tackled. [30] A downside of kMC modeling is that it requires a sufficiently large ensemble of initial molecules thus large memory capacities, which became only truly accessible in the last decade thanks to progress in the field of computer science and informatics.…”

Cost‐efficient modeling of distributed molar mass and topological variations in graft copolymer synthesis by upgrading the method of moments

“…In this table, and are the ( i , j )th moment of living PS radicals, dead PS, ungrafted PB, living GC radicals, and dead GC with topology ( p , q ) according to the definitions in Figure 2. Note that one has here sometimes several indices and calls this type of MoM a topology‐based NF strategy 38,49–52 . NF represents a discrete mathematical treatment with regard to polymers with different molecular topology features rather than the overall polymer, for example, p and q , which are however connected through elementary reaction steps in the kinetic simulations.…”

“…Note that one has here sometimes several indices and calls this type of MoM a topology-based NF strategy. 38,[49][50][51][52] NF represents a discrete mathematical treatment with regard to polymers with different molecular topology features rather than the overall polymer, for example, p and q, which are however connected through elementary reaction steps in the kinetic simulations.…”

“…This is done by calculating only on a few leading statistical moments so that the actual chain length distributions (CLDs) are inevitably sacrificed, at least in the conventional MoM format. One feasible solution is to extend the average characteristics to distributions using some posteriori (and validated) functions 49–52 . Another MoM downside is the need of so‐called closure relations if nonlinear polymerizations involving monomer unit dependent reaction steps (e.g., chain transfer to polymer) are handled 38,51–54 .…”

“…One feasible solution is to extend the average characteristics to distributions using some posteriori (and validated) functions. [49][50][51][52] Another MoM downside is the need of so-called closure relations if nonlinear polymerizations involving monomer unit dependent reaction steps (e.g., chain transfer to polymer) are handled. 38,[51][52][53][54] This means that the MoM equations can be indeterminate without extra approximations, which is attributed to the inherent F I G U R E 1 Strong heterogeneity with respect to molar mass (MM), chemical composition (CC), and molecular topology in the synthesis product of free-radical-induced grafting (FRIG).…”