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.