Validity of the diffusion equation at the atomic scale investigated via numerical simulations

In the absence of theoretical predictions or experimental determinations, numerical simulations are used for establishing the region over which diffusive mass fluxes linearly relate to concentration gradients (Fick's law). Two different physical situations have been studied: (i) vacancy-mediated hopping diffusion of atoms in a rigid face-centered-cubic lattice and (ii) continuous atom diffusion in a model liquid. In the former, the random walk of the vacancy-inducing atom diffusion is simulated, whereas, in the latter, continuous atom motion is studied via molecular dynamics. The results show that, in both systems, Fick's law is valid and, thus, the diffusion equation applies, even in the presence of the strongest possible tracer concentration gradients, provided that the diffusion time exceeds the value for the spreading of the tracer to become larger than a couple of nearest-neighbor distances. These findings are discussed and are compared with results available in the literature.