of impurity atoms is also approximately 0.25. In the representation to the right, however, the same number of impurity atoms are shown segregated on the surface, so that an observation of the surface composition would yield a much higher impurity fraction (in this case, about 0.69). In fact, in this example, were impurities to completely segregate to the surface, an impurity fraction of just 0.36 could completely cover the surface of the material. In an application where surface interactions are important, this result could be disastrous. While the intergranular failure problems noted above are sometimes severe, they are rarely the cause of major service failures (in structural steels, for example), as suitable safety margins are included in the designs. Perhaps the greater concern is that with the development of new technologies and materials with new and more extensive mechanical property requirements, and with the increasing impurity contents as a result of the increased recycling of materials, we may see intergranular failure in materials and situations not seen currently. Thus, a greater understanding of all of the mechanisms surrounding segregation might lead to being able to control these effects in the future. Modeling potentials, experimental work, and related theories are still being developed to explain these segregation mechanisms for increasingly complex systems. Theories of Segregation Several theories describe the equilibrium segregation activity in materials. The adsorption theories for the solid-solid interface and the solid-vacuum surface are direct analogues of theories well known in the field of gas adsorption on the free surfaces of solids. Langmuir–McLean theory for surface and grain boundary segregation in binary systems This is the earliest theory specifically for grain boundaries, in which McLean uses a model of P solute atoms distributed at random amongst N lattice sites and p solute atoms distributed at random amongst n independent grain boundary sites. The total free energy due to the solute atoms is then: where E and e are energies of the solute atom in the lattice and in the grain boundary, respectively and the kln term represents the configurational entropy of the arrangement of the solute atoms in the bulk and grain boundary. McLean used basic statistical mechanics to find the fractional monolayer of segregant, , at which the system energy was minimized (at the equilibrium state), differentiating G with respect to p, noting that the sum of p and P is constant. Here the grain boundary analogue of