Theoretical Isotherms for Adsorption on a Square Lattice

A method which is not self‐consistent has been used by J. K. Roberts to calculate the adsorption isotherm of adatoms, which are larger than the spacing of the adsorption sites on the substrate so that each adatom excludes all others from neighboring sites. It consists in completely formulating the internal statistics of a sample of the film surface and allowing for the effect of the rest of the film by introducing one or more “environmental” parameters as undetermined functions. These are determined by the requirement that each site of the sample shall be an average site. The failure of this method is traceable to the fact that these functions, when determined, are found to vary with surface concentration in a manner inconsistent with the physical requirements. The same difficulty is found to be latent in Peierls’ treatment of adsorption with interaction between adatoms and in Bethe’s treatment of superlattices. The adsorption isotherms for two cases with no interaction between neighbor adatoms are here calculated by another method. The one case is that of Roberts, the other that in which 9 sites are excluded by each adatom. The method requires separate treatment of dilute (small θ) and concentrated (θ nearly unity) films. At each extreme it involves successive approximations. For the dilute film’ the number of free sites is determined as a function of the number of adatoms by enumerating the number of ways that adatoms occur (1) without overlapping of exclusion patterns, (2) with the exclusion patterns of only two adatoms overlapping, etc. This yields the adsorption isotherm. For the concentrated film, the interactions of the free sites themselves are treated in an analogous but necessarily somewhat different manner to give the adsorption isotherm near $\text{θ\hspace{0.17em}=\hspace{0.17em}1.}$ Some fairing of the relations to bridge the intermediate values of θ is necessary. The results are: for the 5‐site exclusion pattern $${\text{θ\hspace{0.17em}=\hspace{0.17em}2Ap[1−(5/2)θ+(3/2)θ}}^2{\text{+3/4θ}}^3\text{],\hspace{1em}0≦θ≦0.5}$$ $${\text{θ\hspace{0.17em}=\hspace{0.17em}Ap[1−θ−3(1−θ)}}^3{\text{+5(1−θ)}}^4\text{],\hspace{1em}0.5≦θ≦1}$$ and for the 9‐site exclusion pattern $${\text{θ\hspace{0.17em}=\hspace{0.17em}4AP[1−(9/4)θ+θ}}^2\mathrm{],\hspace{1em}\theta \hspace{0.17em}}\mathrm{near}\hspace{0.17em}\mathrm{zero}$$ $${\text{θ\hspace{0.17em}=\hspace{0.17em}2Ap(1−θ)}}^2\mathrm{,\hspace{1em}\theta \hspace{0.17em}}\mathrm{near}\hspace{0.17em}\mathrm{unity}$$ . These cases are compared with the limiting cases, first the 1‐site exclusion pattern and second the infinity‐site pattern which is the same case as adsorption on a siteless surface.