text: Other models exist to model more complex binary systems. The above theories operate on the assumption that the segregated atoms are non-interacting. If, in a binary system, adjacent adsorbate atoms are allowed an interaction energy , such that they can attract (when  is negative) or repel (when  is positive) each other, the solid-state analogue of the Fowler adsorption theory is developed as
  When  is zero, this theory reduces to that of Langmuir and McLean. However, as  becomes more negative, the segregation shows progressively sharper rises as the temperature falls until eventually the rise in segregation is discontinuous at a certain temperature, as shown in the following figure.
  Guttman, in 1975, extended the Fowler theory to 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.