The Influence of the Shape and Size of the Nanostructures to the Atoms and Molecules Transport

The discrete Frenkel-Kontorova model for the movement of a foreign atom in a solid, obtained in the local chains approximation, led to the movement of the atom in the Frenkel-Kontorova (F-K) soliton form. This model made it possible to reveal the structure of the F-K soliton, to design the shapes of nanostructures and to take into account their influence on the F-K soliton. The transition to field variables leads the Frenkel-Kontorova equation to the sine-Gordon equation for the displacement field of an atom having solutions also in the form of a soliton. This equation and its solution (soliton) contains coefficients depending on the shape and size of nanostructures. The transition to the sine-Gordon equation allowed us to use the results of Theodorakopoulos 's works related to the consideration of the interaction of elastic vibration modes with a soliton. This made it possible to calculate the diffusion coefficient of the soliton and find the dependence of the diffusion coefficient on the shape and size of nanostructures and temperature.