Previous results concerning the modeling of scission of linear homopolymer chains are reviewed an... more Previous results concerning the modeling of scission of linear homopolymer chains are reviewed and new approaches are proposed, extending to length distributions for chains with different numbers of scission points and to copolymer chains. Analytical solutions for random and parabolic scission of a uniform population of polymer chains are compared and extended with solutions from a numerical scheme using Chebyshev polynomials (PREDICI®), giving rise to interesting conclusions concerning the relative lengths of the 3 different types of chains. The outcomes were applied to a reaction system with recombination of chain fragments, yielding an analytical solution of the 2-dimensional population balance problem in chain length and numbers of combination points. As regards copolymer scission an exact equation and a simple approximation for the fragment length distribution has been obtained based on all possible configurations of sequences of unbroken bonds between two types of monomer units. Scission of weaklinks could also be successfully treated using the simple approximation based on a copolymer composition weighted average scission probability.
ABSTRACT This paper proposes a set of models to calculate contraction factor, which to maximum ex... more ABSTRACT This paper proposes a set of models to calculate contraction factor, which to maximum extent account for the kinetics of a radical polymerization with transfer to polymer and recombination termination. The models are alternatives to the Zimm and Stockmayer’s (1949) analytical expression of contraction factor for molecules with terminal branching. The results, being representative for molecules such as low-density Polyethylene (ldPE), show significantly stronger contraction than predicted by the model of Zimm and Stockmayer. In the case of termination by disproportionation only, molecular sizes turn out to be smaller by a factor of almost two. In presence of recombination termination molecules are less compact. It is shown that the interpretation of contraction factors as measured by the Size Exclusion – Multi-Angle Light Scattering to find the branchedness of ldPE, with the new model would lead to a considerably lower estimate of branching than by using the standard Zimm and Stockmayer’s model.
When a particle model simulates fluid behavior, the calculation of all particle interactions caus... more When a particle model simulates fluid behavior, the calculation of all particle interactions causes long computation times. Especially in mesoscale simulations, the bulk areas can be computationally demanding. To reduce the time spent on such regions, we propose a model that combines different length scales in one system. This is a particle analog to mesh refinement in, for instance, finite-element methods. To this end, we define particles of a coarse-grained scale within the framework of dissipative particle dynamics. These particles have a lower number density, but the same mass density, pressure, temperature, and viscosity as the original description. Furthermore, the coarse-grained particles can directly interact with the “normal” particles. The two length scales are combined in one system, coupled by an overlap region. At the edges of this region, particles transform into the other scale, through local refining or coarse graining. The resulting combined system adequately reproduces the properties and flow behavior of a normal system. When half the system is coarse grained, the computation time reduces by a factor of two. Thus, computational efficiency can be greatly increased for a variety of mesoscale applications.
ABSTRACT A population balance model for the prediction of molecular weight distribution (MWD) in ... more ABSTRACT A population balance model for the prediction of molecular weight distribution (MWD) in a continuous stirred tank reactor (CSTR) has been developed accounting for multiradicals and gel formation in the framework of Galerkin-FEM. In the absence of recombination, gel does not form, but accounting for multiradicals leads to a better prediction of the long MWD tail. Results of the multiradical model with topological scission are well in line with Monte Carlo (MC) simulations. For the case of recombination without scission the multiradical model leads to perfect agreement with MC simulations as regards prediction of the gel fraction and chain length distribution. The classical monoradical model fails to describe the gel regime. We account for gel fragmentation in systems with gelation and scission. Results for this case are in agreement with MC simulations. A nongel assuming variant allows properly detecting the gelpoint and the associated distribution. The scission model adopted, linear or topological scission, turns out to be of extreme importance for the gel regime prediction.
Previous results concerning the modeling of scission of linear homopolymer chains are reviewed an... more Previous results concerning the modeling of scission of linear homopolymer chains are reviewed and new approaches are proposed, extending to length distributions for chains with different numbers of scission points and to copolymer chains. Analytical solutions for random and parabolic scission of a uniform population of polymer chains are compared and extended with solutions from a numerical scheme using Chebyshev polynomials (PREDICI®), giving rise to interesting conclusions concerning the relative lengths of the 3 different types of chains. The outcomes were applied to a reaction system with recombination of chain fragments, yielding an analytical solution of the 2-dimensional population balance problem in chain length and numbers of combination points. As regards copolymer scission an exact equation and a simple approximation for the fragment length distribution has been obtained based on all possible configurations of sequences of unbroken bonds between two types of monomer units. Scission of weaklinks could also be successfully treated using the simple approximation based on a copolymer composition weighted average scission probability.
ABSTRACT This paper proposes a set of models to calculate contraction factor, which to maximum ex... more ABSTRACT This paper proposes a set of models to calculate contraction factor, which to maximum extent account for the kinetics of a radical polymerization with transfer to polymer and recombination termination. The models are alternatives to the Zimm and Stockmayer’s (1949) analytical expression of contraction factor for molecules with terminal branching. The results, being representative for molecules such as low-density Polyethylene (ldPE), show significantly stronger contraction than predicted by the model of Zimm and Stockmayer. In the case of termination by disproportionation only, molecular sizes turn out to be smaller by a factor of almost two. In presence of recombination termination molecules are less compact. It is shown that the interpretation of contraction factors as measured by the Size Exclusion – Multi-Angle Light Scattering to find the branchedness of ldPE, with the new model would lead to a considerably lower estimate of branching than by using the standard Zimm and Stockmayer’s model.
When a particle model simulates fluid behavior, the calculation of all particle interactions caus... more When a particle model simulates fluid behavior, the calculation of all particle interactions causes long computation times. Especially in mesoscale simulations, the bulk areas can be computationally demanding. To reduce the time spent on such regions, we propose a model that combines different length scales in one system. This is a particle analog to mesh refinement in, for instance, finite-element methods. To this end, we define particles of a coarse-grained scale within the framework of dissipative particle dynamics. These particles have a lower number density, but the same mass density, pressure, temperature, and viscosity as the original description. Furthermore, the coarse-grained particles can directly interact with the “normal” particles. The two length scales are combined in one system, coupled by an overlap region. At the edges of this region, particles transform into the other scale, through local refining or coarse graining. The resulting combined system adequately reproduces the properties and flow behavior of a normal system. When half the system is coarse grained, the computation time reduces by a factor of two. Thus, computational efficiency can be greatly increased for a variety of mesoscale applications.
ABSTRACT A population balance model for the prediction of molecular weight distribution (MWD) in ... more ABSTRACT A population balance model for the prediction of molecular weight distribution (MWD) in a continuous stirred tank reactor (CSTR) has been developed accounting for multiradicals and gel formation in the framework of Galerkin-FEM. In the absence of recombination, gel does not form, but accounting for multiradicals leads to a better prediction of the long MWD tail. Results of the multiradical model with topological scission are well in line with Monte Carlo (MC) simulations. For the case of recombination without scission the multiradical model leads to perfect agreement with MC simulations as regards prediction of the gel fraction and chain length distribution. The classical monoradical model fails to describe the gel regime. We account for gel fragmentation in systems with gelation and scission. Results for this case are in agreement with MC simulations. A nongel assuming variant allows properly detecting the gelpoint and the associated distribution. The scission model adopted, linear or topological scission, turns out to be of extreme importance for the gel regime prediction.
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Papers by Piet Iedema