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.
Highlights
It is well known that for particles on a plane interacting via a certain potential, the existence of a periodic crystal is impossible at nonzero temperature [1,2,3,4]
We propose to take into consideration two mechanisms in order to simulate the radial distribution function of 2D
Let us first analyze a real sample of porous aluminum oxide in order to understand the form of the radial distribution function that we should seek
Summary
It is well known that for particles on a plane interacting via a certain potential, the existence of a periodic crystal is impossible at nonzero temperature [1,2,3,4]. As was believed in [5], the experiment confirmed the validity the two competing theories of the order–disorder transition They involve, on the one hand, the theory of crystallites [6], according to which an amorphous substance incorporates crystal-like agglomerates linked by disordered regions, and, on the other hand, the theory of random networks (see review [7]), which assumes that the order is destroyed throughout the entire volume of a substance, so that the crystal lattice transforms into random chains of atoms due to the distortion and breakage of some bonds. We consider a slightly different method consisting in the random shifting of the crystal nodes
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