Polymer length distributions for catalytic polymerization within mesoporous materials: Non-Markovian behavior associated with partial extrusion

Abstract

We analyze a model for polymerization at catalytic sites distributed within parallel linear pores of a mesoporous material. Polymerization occurs primarily by reaction of monomers diffusing into the pores with the ends of polymers near the pore openings. Monomers and polymers undergo single-file diffusion within the pores. Model behavior, including the polymer length distribution, is determined by kinetic Monte Carlo simulation of a suitable atomistic-level lattice model. While the polymers remain within the pore, their length distribution during growth can be described qualitatively by a Markovian rate equation treatment. However, once they become partially extruded, the distribution is shown to exhibit non-Markovian scaling behavior. This feature is attributed to the long-tail in the "return-time distribution" for the protruding end of the partially extruded polymer to return to the pore, such return being necessary for further reaction and growth. The detailed form of the scaled length distribution is elucidated by application of continuous-time random walk theory. KeywordsCatalytic polymerization, Continuous-time random walk theories, Kinetic Monte Carlo simulation, lattice models, length distributions, Markovian, polymer length, pore openings, rate equations, scaling behavior, single file diffusion, mesoporous materials, monomers, polymerization, silicon dioxide, algorithms We analyze a model for polymerization at catalytic sites distributed within parallel linear pores of a mesoporous material. Polymerization occurs primarily by reaction of monomers diffusing into the pores with the ends of polymers near the pore openings. Monomers and polymers undergo single-file diffusion within the pores. Model behavior, including the polymer length distribution, is determined by kinetic Monte Carlo simulation of a suitable atomistic-level lattice model. While the polymers remain within the pore, their length distribution during growth can be described qualitatively by a Markovian rate equation treatment. However, once they become partially extruded, the distribution is shown to exhibit non-Markovian scaling behavior. This feature is attributed to the long-tail in the "return-time distribution" for the protruding end of the partially extruded polymer to return to the pore, such return being necessary for further reaction and growth. The detailed form of the scaled length distribution is elucidated by application of continuous-time random walk theory.

Kinetic Monte Carlo Simulation of Statistical Mechanical Models and Coarse-Grained Mesoscale Descriptions of Catalytic Reaction–Diffusion Processes: 1D Nanoporous and 2D Surface Systems

“…One example is the oxidatively catalyzed formation of poly­(phenylene butadinylene) polymer (PPB) in a Cu 2+ -functionalized MCM-41 silica . Modeling reveals unusual spatial and kinetic features of polymerization in 1D nanoporous materials subject to SFD. , …”

“…In the above reaction–diffusion systems, nontrivial spatiotemporal behavior generally arises from the interplay between the typically nonlinear reaction kinetics and diffusive transport. In the 1D nanoporous catalytic systems, especially with inhibited transport within pores, net reactivity is often localized near pore openings either for catalytic conversion − or polymerization , reactions. This feature induces strong concentration variations within the nanopores typically on a length scale of tens of nanometers.…”

Langevin and Fokker-Planck Analyses of Inhibited Molecular Passing Processes Controlling Transport and Reactivity in Nanoporous Materials

“…Examples include Pd-catalyzed neopentane conversion in zeolites [11], and aldol condensation in amine-functionalized MSN [10]. Catalytic polymerization in narrow pores is also strongly impacted by inhibited passing [12].…”

Beyond the Markovian assumption: response–excitation probabilistic solution to random nonlinear differential equations in the long time

“…Such excitations play a significant role in many scientific and engineering disciplines, e.g. material sciences [12,13], medical physics [14,15], ecology [16], systems biology [17], neurosciences [18], communications [19], oceanography [20], earthquake modelling and engineering [21,22], wind engineering [23,24] and ship dynamics [25].…”