Smeared Lattice Model as a Framework for Order to Disorder Transitions in 2D Systems

Abstract

Order to disorder transitions are important for two-dimensional (2D) objects such as oxide films with cellular porous structure, honeycomb, graphene, Bénard cells in liquid, and artificial systems consisting of colloid particles on a plane. For instance, solid films of porous alumina represent almost regular crystalline structure. We show that in this case, the radial distribution function is well described by the smeared hexagonal lattice of the two-dimensional ideal crystal by inserting some amount of defects into the lattice.Another example is a system of hard disks in a plane, which illustrates order to disorder transitions. It is shown that the coincidence with the distribution function obtained by the solution of the Percus–Yevick equation is achieved by the smoothing of the square lattice and injecting the defects of the vacancy type into it. However, better approximation is reached when the lattice is a result of a mixture of the smoothed square and hexagonal lattices. Impurity of the hexagonal lattice is considerable at short distances. Dependencies of the lattice constants, smoothing widths, and contributions of the different type of the lattices on the filling parameter are found. The transition to order looks to be an increase of the hexagonal lattice fraction in the superposition of hexagonal and square lattices and a decrease of their smearing.