allow for interactions between two co-segregating species in multicomponent systems. This modification is vital to explaining the segregation behavior that results in the intergranular failures of engineering materials. More complex theories are detailed in the work by Guttmann and McLean and Guttmann. The free energy of surface segregation in binary systems The Langmuir–McLean equation for segregation, when using the regular solution model for a binary system, is valid for surface segregation (although sometimes the equation will be written replacing with ). The free energy of surface segregation is . The enthalpy is given by where and are matrix surface energies without and with solute, is their heat of mixing, Z and are the coordination numbers in the matrix and at the surface, and is the coordination number for surface atoms to the layer below. The last term in this equation is the elastic strain energy , given above, and is governed by the mismatch between the solute and the matrix atoms. For solid metals, the surface energies scale with the melting points. The surface segregation enrichment ratio increases when the solute atom size is larger than the matrix atom size and when the melting point of the solute is lower than that of the matrix. A chemisorbed gaseous species on the surface can also have an effect on the surface composition of a binary alloy. In the presence of a coverage of a chemisorbed species theta, it is proposed that the Langmuir-McLean model is valid with the free energy of surface segregation given by , where and are the chemisorption energies of the gas on solute A and matrix B and is the fractional coverage. At high temperatures, evaporation from the surface can take place, causing a deviation from the McLean equation. At lower temperatures, both grain boundary and surface segregation can be limited by the diffusion of atoms from the bulk to the surface or interface. Kinetics of segregation In some situations where segregation is important, the segregant atoms do not have sufficient time to reach their equilibrium level as defined by the above adsorption theories. The kinetics of segregation become a limiting factor and must be analyzed as well. Most existing models of segregation kinetics follow the McLean approach. In the model for equilibrium monolayer segregation, the solute atoms are assumed to segregate to a grain boundary from two infinite half-crystals or to a surface from one infinite