During the chemical vapor deposition (CVD) growth of graphene on Cu foils, evaporation of Cu and changes in the dimensions of Cu grains in directions both parallel and perpendicular to the foils are induced by thermal effects. Such changes in the Cu foil could subsequently change the shape and distribution of individual graphene domains grown on the foil surface, and thus influence the domain structure and electrical properties of the resulting graphene films. Here, a slower cooling rate is used after the CVD process, and the graphene films are found to have an improved electrical performance, which is considered to be associated with the Cu surface evaporation and grain structure changes in the Cu substrate.