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

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## Abstract

**:**

## 1. Introduction

## 2. Radial Distribution Function for Porous Aluminum Oxide Layer

## 3. Hard Disks in a Plane

#### Comparison with Solutions of the Percus–Yevick Equation

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Photograph of the sample of porous aluminum oxide layer from [8]. Circles around pore centers are plotted. (

**b**) Array of pore centers obtained by a parallel translation of the pore centers in the initial array along the coordinate axes and the diagonals of the sample. The region containing pore centers in the starting sample is selected by a rectangle.

**Figure 3.**Radial distribution function of pores in porous aluminum oxide: experimental sample (solid curve) and our calculation (dashed curve).

**Figure 4.**Radial distribution function for filling $\eta =0.62$ (

**a**) in the range $\rho \in \{1,5\}$, and (

**b**) in the range $\rho \in \{4,9\}$ (“a tail of the distribution function”). Solid line is a solution of the Percus–Yevick equation; dashed line corresponds to the smeared lattice model.

**Figure 5.**Radial distribution functions for filling $\eta =0.3$ (

**a**) in the range $\rho \in \{1,3.5\}$, and (

**b**) in the range $\rho \in \{2.5,5.0\}$ (“a tail of the distribution function”). Solid line is a solution of the Percus–Yevick equation; dashed line corresponds to the smeared lattice model.

**Table 1.**Constants of lattices ${a}_{squ}$ and ${a}_{gex}$ for modeling of the radial distribution function at various packing fractions $\eta $, and all the other parameters in Equations (7)–(9).

$\mathit{\eta}$ | ${\mathit{a}}_{\mathit{s}\mathit{q}\mathit{u}}/(\mathbf{2}\mathit{R})$ | ${\mathit{a}}_{\mathit{g}\mathit{e}\mathit{x}}/(\mathbf{2}\mathit{R})$ | $\mathit{\gamma}/{(\mathbf{2}\mathit{R})}^{\mathbf{1}/\mathbf{2}}$ | $\mathbf{2}\mathit{\lambda}\mathit{R}$ | $\mathit{b}{(\mathbf{2}\mathit{R})}^{\mathbf{2}}$ |
---|---|---|---|---|---|

0.62 | 1.04 | 1.1 | 0.158 | 0.027 | 0.4 |

0.6 | 1.056 | 1.12 | 0.176 | 0.03 | 0.41 |

0.55 | 1.095 | 1.14 | 0.192 | 0.1 | 0.45 |

0.5 | 1.11 | 1.2 | 0.2 | 0.2 | 0.53 |

0.45 | 1.112 | 1.3 | 0.209 | 0.5 | 0.7 |

0.4 | 1.115 | 1.6 | 0.213 | 0.9 | 1.1 |

0.35 | 1.117 | 2.1 | 0.224 | 1.4 | 0.75 |

0.3 | 1.12 | 2.3 | 0.236 | 2 | 0.6 ^{1} |

^{1}All the quantities are dimensionless.

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**MDPI and ACS Style**

Cherkas, N.L.; Cherkas, S.L.
Smeared Lattice Model as a Framework for Order to Disorder Transitions in 2D Systems. *Crystals* **2018**, *8*, 290.
https://doi.org/10.3390/cryst8070290

**AMA Style**

Cherkas NL, Cherkas SL.
Smeared Lattice Model as a Framework for Order to Disorder Transitions in 2D Systems. *Crystals*. 2018; 8(7):290.
https://doi.org/10.3390/cryst8070290

**Chicago/Turabian Style**

Cherkas, Nadezhda L., and Sergey L. Cherkas.
2018. "Smeared Lattice Model as a Framework for Order to Disorder Transitions in 2D Systems" *Crystals* 8, no. 7: 290.
https://doi.org/10.3390/cryst8070290