Nonlinear diffusion in Cu-Au multilayer thin films

Multilayered films with an average concentration of Cu-16 at. % Au and a layering periodicity of 3.31 nm were produced by vapor deposition on sodium chloride substrates. Diffraction studies were carried out at Stanford Synchrotron Radiation Laboratory, thus allowing the observation of up to three orders of satellites, the evolutions of which were measured as a function of annealing time at 515 K. It was found that the first-order satellite intensity decayed nearly exponentially with time, whereas intensities of both second- and third-order satellites decreased very rapidly at first, then increased before decaying exponentially. These results could be explained by following the evolution of satellites during annealing of a one-dimensional modulated system governed by a nonlinear diffusion equation. The diffusivity in this system was assumed to be of the form D0+D1u+D2u2, with u=c−c0, where Di’s are the coefficients, c is the concentration, and c0 represents the average composition of the alloy. The nonlinear diffusion equation was solved numerically to obtain one-dimensional concentration profiles at successive annealing times. Satellite amplitudes, obtained by Fourier transforming the concentration profiles, were found to exhibit a remarkable phenomenon: The mth-order satellite amplitude changes its sign (m−1) times before decaying asymptotically to zero. By comparing the evolution of the square of the amplitude with that of the satellite intensities, a set of Di’s was determined that produced realistic concentration profiles and reproduced the experimentally evaluated satellite intensities reasonably well.