At the atomic level, metal surfaces are not always made of a regularly smooth array of atomic sites. They may have steps and terraces depending on how they are cut. A terrace has a finite number of atomic sites in one direction, and is rather long in the other. The study of adsorption on terraces is motivated by the fact that certain chemical reactions are enhanced on terraces with the substrate playing the role of a catalyst, and in many cases preferential adsorption occurs on the steps of terraces. The modeling of such terraces considered adsorbate-substrate interaction on edge-sites to be different from that on bulk-sites. The system made of the the surface exposed to an atomic or molecular gas is at thermodynamic equilibrium. The chemical potential energy of the gas particles depends on the pressure which is allowed to vary keeping the temperature relatively low. Adsorbate-adsorbate interactions have been limited to nearest and next nearest neighbors. The model is purely phenomenological and no a priori assumptions are made as to the relative strength of the interaction energies, and as to whether the forces are attractive or repulsive. A transfer matrix associated with the system is constructed recursively using a technique developed in 1993 (Article .pdf) and recently adapted to study adsorption on nanotubes as well (Monomer Adsorption on Terraces and Nanotubes). For a given gas pressure, one determines the coverage of the terrace, the numbers per site of first and second neighbor adsorbates, and the number per site of adsorbates on the edges, as well as the entropy per site. The calculations are carried out on the Cray XT3 (Big Ben) of the Pittsburgh Supercomputing Center and on a PC-cluster located in the Physics Department at Villanova. The determination of the eigenvalues of the transfer matrix, from which all the pertinent quantities mentioned above are determined are computed using ScaLAPACK a linear package software specially designed for multiparallel supercomputing to which we have added long double precision arithmetic. The computation also involved the numerical evaluation of the derivatives of the largest eigenvalue with respect to the adsorbate-adsorbate interaction energies. For numerical accuracy we developed for the Numerical Computation of Derivatives to Arbitrary Precision which appeared  in the International Journal of Theoretical Physics, Group Theory, and Nonlinear Optics, Vol. 10, pp.415-424 (2004). Phases and transitions between phases have been determined in all possible ranges of interaction energies for terraces made of a squares, rectangles, isosceles and, recently, equilateral triangles. Adsorption properties of zigzag single walled carbon nanotubes are currently being investigated with 10 carbon atoms in the cross-section of the tube. The model has been shown to provide an experimental method for determining the interaction energies from the knowledge of the phases encountered at increasing pressure, and from the conditions prevailing at the transitions between phases.

Combinatorics Function Technique

The combinatorics function technique (CFT) is a method developed in the late 70's and perfected in the early 80's (see publication list) which has been used with some success to solve a certain category of problems. The CFT provides an algorithm for obtaining the solution of linearly coupled, multidimensional partial difference equations with variable coefficients. One application of the CFT is in obtaining exact analytic solutions of the radial Schrodinger equation with power-type potentials. Of particular interest is the combined linear and Coulomb potential relevant to quark confinement.