Steady absorption of Brownian particles by a sphere

The problem of steady Brownian motion in the presence of an absorbing sphere is studied in the full Brownian particle position, velocity phase space. Approximate solutions are found for the Brownian particle distribution function by solving the Fokker–Planck equation subject to the boundary condition that the solution for emergent particles vanish on the surface of the sphere. Comparisons are made with results from the macroscopic theory based on the diffusion equation. We find that the macroscopic theory is not valid in a boundary layer of extent depending on the radius of the Brownian particle and absorber and the viscosity and temperature of the host fluid in a fairly complicated manner. Close to the boundary, Fick’s law only holds if the diffusion coefficient is taken to be space dependent. One significant consequence of our results is that the macroscopic theory of diffusion-controlled rate constants is shown not to be generally valid. The corrected values for the rate constant that we find for aerosol particles are in good agreement with the values found by Fuchs based on an ad hoc semiempirical theory.