Diffusional Nanoimpacts: The Stochastic Limit

The probability expressions for the average number of diffusional impact events on a surface are established using Pick's diffusion in the limit of a continuum flux. The number and the corresponding variance are calculated for the case of nanoparticles impacting on an electrode at which they are annihilated. The calculations show the dependency on concentration in the limit of non-continuous media and small electrode sizes for the cases of linear diffusion to a niacroelectrode and of convergent diffusion to a small sphere. Using random walk simulations, we confirm that the variance follows a Poisson distribution for ultradilute and dilute solutions. We also present an average "first passage time" for the ultradilute solutions expression that directly relates to the tower limit of detection in ultradilute solutions as a function of the electrode size. The analytical expressions provide a straightforward way to predict the stochastics of impacts in a "nanoimpact" experiment by using Fick's second law and assuming a continuum dilute flux. Therefore, the study's results are applicable to practical electrochemical systems where the number of particles is-very small but much larger than one. Moreover, the presented analytical expression for the variance can be utilized to identify effects of particle inhomogeneity in the solution and is of general interest in all studies of diffusion processes toward an absorbing wall in the stochastic limit.