We study theoretically two types of kinetic models of a binary alloy at zero temperature. In the phase separation model, a nearest-neighbor interchange can occur if the fraction of AB bonds (where A and B denote distinct species of atoms in a binary alloy) is thereby decreased. The crystallization model is defined by the opposite evolution rule. We examine these models in one dimension and obtain exact analytical results for the densities of domain walls, defects, and for a number of other correlators. Nonergodic zero-temperature dynamics
leads to final states strongly dependent on initial conditions. For generalized models, in which nearest-neighbor interchange is also performed if the portion of AB bonds is not changed, a very rich kinetic behavior is observed.