Langmuir adsorption at free surfaces becomes: Here, is the fraction of the grain boundary monolayer available for segregated atoms at saturation, is the actual fraction covered with segregant, is the bulk solute molar fraction, and is the free energy of segregation per mole of solute. Values of were estimated by McLean using the elastic strain energy, , released by the segregation of solute atoms. The solute atom is represented by an elastic sphere fitted into a spherical hole in an elastic matrix continuum. The elastic energy associated with the solute atom is given by: where is the solute bulk modulus, is the matrix shear modulus, and and are the atomic radii of the matrix and impurity atoms, respectively. This method gives values correct to within a factor of two (as compared with experimental data for grain boundary segregation), but a greater accuracy is obtained using the method of Seah and Hondros, described in the following section. Free energy of grain boundary segregation in binary systems Using truncated BET theory (the gas adsorption theory developed by Brunauer, Emmett, and Teller), Seah and Hondros write the solid-state analogue as: where is the solid solubility, which is known for many elements (and can be found in metallurgical handbooks). In the dilute limit, a slightly soluble substance has , so the above equation reduces to that found with the Langmuir-McLean theory. This equation is only valid for . If there is an excess of solute such that a second phase appears, the solute content is limited to and the equation becomes This theory for grain boundary segregation, derived from truncated BET theory, provides excellent agreement with experimental data obtained by Auger electron spectroscopy and other techniques. More complex systems 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