We investigate the dynamical behavior of spinodal decomposition in binary mixtures using a variation on the Rothman-Keller cellular automaton, in which particles of type A(B) move toward domains of greater concentration of A(B). Domain growth and system morphologies are determined via the pair correlation function, revealing that (i) the characteristic domain size, R, grows with time as t^1/3 and (ii) the configurations of the mixtures at different times are self-similar. These results did not change when different game rules were adopted, indicating that self-similarity and the 1/3 scaling law constitute fundamental properties of any diffusion-driven phase separation process. The same model also was applied to describe the mixing process, and, as expected, it was determined that the characteristic time was dependent on he square of a characteristic linear dimension of the system.