text: 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 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.