Patch size effect on diffusion and incorporation in dilute suspension of partially active spheres

The normalized overall rate constant kp/kf for diffuse-and-incorporate processes in dilute suspension of spheres partially covered with randomly distributed nonoverlapping active patches is studied with a sped-up Brownian dynamic simulation scheme. The normalized overall rate constant is found to increase with decreasing characteristic size of the active patch under the condition of fixed fcover, the surface area fraction covered by the active patches. A scaling relation is proposed as (kp/kf)NDL/[(1 + P) - (1 + P/fcover)(kp/kf)NDL] approximately Np(1/2) with Np being the number of active patches on the sphere, P a parameter characterizing the relative dominance of surface incorporation over diffusion, and NDL signifying the nondiffusion-limited condition. This scaling relation is verified with rate constant data from the sped-up Brownian dynamic simulation. From this scaling relation, the maximum achievable kp/kf is derived to be (1 + P)/(1 + P/fcover). This result implies that kp/kf can approach unity under the diffusion-limited situation even for a partially active sphere by reducing the size of active patch, while it is not possible to achieve unit kp/kf for nondiffusion-limited systems.