The interaction between ions and the activation barrier of elementary events of crystal growth and evaporation

Abstract

The activation barrier of the elementary processes of the growth and evaporation of an NaCl-type ionic crystal in contact with its vapour such as surface diffusion, incorporation and detachment, are examined in terms of the sum of the pair potential energy between adions and lattice ions. Lattice-sum studies of the Coulombic part of the potential energy was evaluated exactly using integral transform which is equivalent to the solution of the Poisson equation derived earlier. The Coulombic potential-energy function experienced by adions decays rapidly to zero within a distance less than one interionic separation above the surface. Three forms of non-Coulombic potential energy were evaluated by direct lattice summation. Both the pair potential-energy function derived from ab initio quantum mechanics and the effective Born–Mayer potential-energy function are short-ranged. Moreover, at a binding site, the ab initio potential energy experienced by an adion is attractive and slightly larger than that of the Coulombic part, whilst the Born–Mayer potential function gives rise to a small but repulsive contribution to the total potential energy. The total potential energy of an adion was shown not to be proportional to the number of nearest neighbours, despite being effectively short-ranged. The non-Coulombic potential energy derived from a combination of the Mott–Littleton polarisation energy and overlap repulsion is long-ranged, and at an equilibrium adsorption site it is attractive and is very large compared with the Coulombic potential-energy function. All non-Coulombic potential energies are dependent on the type of charge of the adion. The activation barrier to incorporation, either directly from the vapour phase or after surface diffusion, is zero, whilst the barrier to detachment is determined by the total potential energy of the surface ion. The activation barrier for surface diffusion, as calculated from the variation of adion potential energy along a diffusion path, is shown to be contributed almost exclusively from the Coulombic part. Of the two possible diffusion paths, the lateral path along the unit cell (path 1) has a larger activation barrier than the preferred diffusion path diagonal to the unit cell (path 2) which has a barrier of almost half the size. The corresponding diffusion will be more restricted on the surface of bivalent metal sulphates because of the larger charges of lattice ions.