Overall rate constants for diffusion and incorporation in clusters of spheres

Abstract

Three numerical schemes and one approximate model are developed to compute the overall rate constants for diffusion and incorporation of small entities in clusters of spheres. These include the Brownian dynamic simulation, multipole expansion, boundary collocation, and a model linking diffusion-limited (DL) and nondiffusion-limited (NDL) data. The Brownian dynamic simulation is speeded up with a first-passage technique and is capable of taking the finite surface incorporation rate into account. The multipole expansion truncated at the dipole moment gives an excellent approximation while the second order boundary collocation is satisfactorily accurate. The DL to NDL model offers a quick and reasonably accurate estimate of the rate constant. Clusters of Euclidean dimensions, including 1D strings, 2D squares, and 3D cubes, are particularly investigated. The screening effect arising from the long range nature of the disturbance concentration field is found responsible for the variation in the overall rate constant due to structural variation in clusters, and becomes less pronounced as P increases. Here, P measures the relative dominance of surface incorporation over the diffusion. Also, the rate constants for the Euclidean clusters are found to obey the similar scaling laws as those confirmed by Tseng et al. [Phys. Rev. Lett. 86, 5494 (2001)] for the translational drag coefficient of clusters of spheres in the low Reynolds number flow regime.

Shadowing Effect in Catalyst Activity: Experimental Observation

“…A more clear visualization of this trend can be seen in Figure B, which displays the evolution of CO conversion values as a function of the reactor configurations under Knudsen conditions. The experimental trends presented in Figure are also consistent with numerical simulations in chemical physics and TAP theory that found catalyst particles to be less active in spatial arrangements with closer proximities. − ,− On the other hand, the obtained results directly challenge some of the theories in chemical physics that predict a proportional increase of the rate coefficients with the density of catalyst particles. − …”

“…Such low concentrations tend to subsist, within the passing of the gas pulse, because the shadowed spaces are not steadily replenished with reactants on account of the random motion of the diffusing molecules. Numerical simulations have previously detected such concentration-deprived zones in the vicinities of closely neighboring particles exhibiting an overall reduced apparent activity. − …”

“…Numerous theories have been proposed throughout the years with the objective of describing competitive interactions taking place within an ensemble of catalyst particles in terms of their intrinsic activity, that is, not affected by transport limitations, and the mass transport of reactants. − However, not all of these theories agree on whether the apparent rate coefficients increase or decrease proportionally with the density of active particles . Some numerical simulations and model calculations of particle clusters in different spatial configurations suggest a screening effect between closely neighboring catalyst particles that is detrimental to their apparent activity. − As they are in closer proximity, their mutual screening “shadows” the latter by limiting their exposure to the diffusing reactants. Nevertheless, systematic experimental data are required to rigorously validate the occurrence of shadowing as suggested by these theoretical trends and assess its implications to heterogeneous catalysis and chemical technology.…”

Selectivity for patch‐distributed reactive spherical surfaces

“…In this study, the accelerated Brownian dynamic scheme, developed by Lu9 and colleagues19, 20 by incorporating the first‐passage technique, with the Brownian dynamics algorithm proposed by Zhou et al,21 is adopted to evaluate the selectivity of reactions occurring on the two distinct surfaces: the patches and the carrier. The details of the accelerated Brownian dynamic scheme can be found in Lu,9 Lu et al,20 and Zhou et al21 The basic idea to accelerate the Brownian dynamic simulation is to replace a large number of random‐walk steps with a big jump to the surface of a virtual homogeneous sphere, spending some statistically average time, so that the simulation time can be dramatically reduced 22…”

“…The trajectories are traced until the species either are reacted on the sphere surface or escape to infinity. Here, the judgment as to whether the species would escape to infinity is made based on a statistical argument as detailed in Lu et al20 By collecting the number of reactions occurring on both patch ( NR patch ) and carrier ( NR carrier ) surfaces, the selectivity of the patch over carrier surfaces, S , can be defined as Two different patch sizes, s / R values of 0.314 and 0.063, are considered in this study with patch numbers of 2 to 16 and 10 to 405, respectively, to demonstrate the size effect on the selectivity under the same patch coverage fraction. For the present work, results for selectivity are averages of 100 patch configurations, each probed with 10 6 test reactant species released one at a time.…”