# Short-Range Order Modeling in Alloys

## Abstract

**:**

## 1. Introduction

## 2. Theory of Inhomogeneous Short-Range Order

- –
- It achieves maximal value when atoms are distributed chaotically;
- –
- It is correctly normalized at this condition, providing the same value as Equation (5).

## 3. Quasichemical Theory

## 4. Improved Combinatorial Factor

- –
- Size of the cell k (number of atoms included into cell);
- –
- Geometrical factor $\gamma $, defining location of the critical point;
- –
- Dimensionless ratio of interchange energy ${U}_{\Delta}$ to the critical immiscibility temperature of alloy ${\theta}_{c}$ [9];
- –
- Dimensionless ratio of the energy per atom taken at critical temperature E to the interchange energy ${U}_{\Delta}$;
- –
- Dimensionless $S/{k}_{B}$ expression for entropy at critical immiscibility temperature.

**BW**contains critical values calculated using Bragg–Williams approximation.

**QT**contains critical values calculated using Quasichemical Theory.

**SAM**contains critical values calculated using Surrounded Atom Model. Even if the cell size used by SAM is larger than the size of the quasichemical cell (equal two), it is evident from the table, that SAM provides precision significantly worse than standard QT. This error introduced by standard Surrounded Atom Model will be explained and corrected in the Section 5.

**IQT**corresponds to the Improved Quasichemical Theory—i.e., Quasichemical Theory where parameter of model $\gamma $ is adjusted to reproduce the correct critical temperature, as explained above. Comparison of column

**QT**and

**IQT**shows that adjustment of critical temperature significantly improves quality of approximation.

**IQT2**column, which considers cells that better match the geometrical structure of the simulated grids. Accordingly, the following choices are made:

- –
- for a square $2D$ lattice, a regular square (k = 4) is a natural choice;
- –
- for a $2D$ triangular lattice, a regular triangle (k = 3) is the choice;
- –
- for a simple cubical lattice, a regular cube (k = 8) is the choice;
- –
- for a Body Centered Cubical (BCC) lattice, an irregular tetrahedron (k = 4) with two vertices in the main grid and two vertices from the sublattice formed by central atoms;
- –
- for a Face Centered Cubic (FCC) lattice, a regular tetrahedral (k = 4) is the choice.

## 5. Surrounded Atom Model

## 6. Associated Solution Model

## 7. Order–Disorder Transformations

_{3}Au, which appears in the Cu–Au system below a certain critical temperature and has an FCC lattice [2]. It is also well known that the Cu

_{3}Au superstructure can be represented as the union of two intertwined sublattices, one of which is preferable for Cu and has three times as many nodes as the second sublattice, preferable for Au. It is convenient for our consideration to distinguish these sublattices by an index (which, we assume, varies from one to two). Naturally, in our case, the selection of a tetrahedron as the basic cell results in identifying two sub-cells inside every cell. One of the sub-cells has three nodes and is preferable for Cu, while the second sub-cell has one node and is preferable for Au. We assign an index to every sub-cell to be identical with the index of the corresponding sublattice.

## 8. Multicomponent Systems

## 9. Results and Disscussion

- –
- A new formalism, equally applicable to several independent models;
- –
- Identification and correction of serious errors in ASM and SAM;
- –
- An extended version of the formalism that allows the recreation of the correct critical temperature and significantly improves the accuracy of calculations for QT, ASM, and SAM.

- –
- Three temperature-dependent parameters to describe a binary system;
- –
- Three additional temperature-dependent parameters to describe the comprising ternary system;
- –
- One additional temperature-dependent parameter to describe the containing quaternary system;
- –
- No additional parameters for the systems with five or more components.

## 10. Applications of TISR

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ASM | Association Solution Model |

BW | Bragg–Williams approximation |

CEF | Compound Energy Formalism |

CF | Combinatorial Factor |

CSA | Cluster Site Approximation |

CVM | Cluster Variation Method |

FCC | Face Centered Cubic lattice |

IQT | Improved Quasichemical Theory |

LRO | Long-Range Order |

QT | Quasichemical Theory |

SRO | Short-Range Order |

SAM | Surrounded Atom Model |

TISR | Theory of Inhomogeneous Short-Range Order |

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**Figure 1.**Splitting the grid into cells. Cells are represented by solid lines while bonds between cells are dashed. Some dotted lines are omitted so as not to clutter up the drawing.

**Figure 3.**Versions of Combinatorial Factor $g\left(\nu \right)$ for FCC Lattice: Quasichemical approximation—green line; Improved Quasichemical approximation with $k=4$—red line; Location of the critical point by [12]—large dot.

**Figure 4.**Molar free energy of alloy near the critical point as a function of the concentration of atoms a in the first sublattice: ${x}_{1}=0.25$ and $\gamma =4.88$.

**Figure 5.**System Ag-In (T = 1200 K): solid lines—[39]; dotted lines—TISR.

**Figure 6.**System Al-Ni (T = 1923 K): solid lines—[40]; dotted lines—TISR.

**Figure 7.**System Al-Si (T = 1575 K): solid lines—[41]; dotted lines—TISR.

**Figure 8.**System Ca-Ni (T = 1120 K): solid lines—[42]; dotted lines—TISR.

**Figure 9.**System Cu-Si (T = 1575 K): solid lines—[41]; dotted lines—TISR.

**Figure 10.**System La-Zr (T = 1800 K): solid lines—[43]; dotted lines—TISR.

**Figure 11.**System Al-Mn (T = 1520 K): solid lines—[44]; dotted lines—TISR.

**Figure 12.**System Al-Sc (T = 1873 K): solid lines—[45]; dotted lines—TISR.

**Figure 13.**System Ca-Zn (T = 1520 K): solid lines—[46]; dotted lines—TISR.

**Figure 14.**System Ce-Si (T = 1923 K): solid lines—[47]; dotted lines—TISR.

**Figure 15.**System Ce-Zn (T = 1873 K): solid lines—[48]; dotted lines—TISR.

**Figure 16.**System Cu-Fe (T = 1873 K): solid lines—[49]; dotted lines—TISR.

LATTICE | BW | SAM | QT | IQT | IQT2 | Domb | |
---|---|---|---|---|---|---|---|

2D square | k | 1 | 5 | 2 | 2 | 4 | |

$\gamma $ | 1 | 5 | 4 | 2.784 | 1.608 | ||

${U}_{\Delta}/{\theta}_{c}$ | 0.5 | 0.575 | 0.693 | 0.882 | 0.882 | 0.882 | |

$E/{U}_{\Delta}$ | 1.000 | 0.857 | 0.667 | 0.439 | 0.412 | 0.292 | |

$S/{k}_{B}$ | 0.693 | 0.652 | 0.580 | 0.461 | 0.430 | 0.306 | |

$2D$ triangular | k | 1 | 7 | 2 | 2 | 3 | |

$\gamma $ | 1 | 7 | 6 | 3.002 | 1.956 | ||

${U}_{\Delta}/{\theta}_{c}$ | 0.333 | 0.365 | 0.405 | 0.549 | 0.549 | 0.549 | |

$E/{U}_{\Delta}$ | 1.5 | 1.364 | 1.2 | 0.751 | 0.715 | 0.501 | |

$S/{k}_{B}$ | 0.693 | 0.668 | 0.633 | 0.497 | 0.471 | 0.330 | |

Simple cubical | k | 1 | 7 | 2 | 2 | 8 | |

$\gamma $ | 1 | 7 | 6 | 4.44 | 1.673 | ||

${U}_{\Delta}/{\theta}_{c}$ | 0.333 | 0.365 | 0.405 | 0.443 | 0.443 | 0.443 | |

$E/{U}_{\Delta}$ | 1.5 | 1.364 | 1.2 | 1.064 | 1.051 | 0.966 | |

$S/{k}_{B}$ | 0.693 | 0.668 | 0.633 | 0.598 | 0.588 | 0.541 | |

BCC | k | 1 | 9 | 2 | 2 | 4 | |

$\gamma $ | 1 | 9 | 8 | 5.234 | 2.729 | ||

${U}_{\Delta}/{\theta}_{c}$ | 0.250 | 0.267 | 0.288 | 0.315 | 0.315 | 0.315 | |

$E/{U}_{\Delta}$ | 2.000 | 1.867 | 1.714 | 1.528 | 1.524 | 1.452 | |

$S/{k}_{B}$ | 0.693 | 0.675 | 0.652 | 0.619 | 0.617 | 0.585 | |

FCC | k | 1 | 13 | 2 | 2 | 4 | |

$\gamma $ | 1 | 13 | 12 | 5.845 | 2.566 | ||

${U}_{\Delta}/{\theta}_{c}$ | 0.166 | 0.174 | 0.182 | 0.204 | 0.204 | 0.204 | |

$E/{U}_{\Delta}$ | 3.000 | 2.870 | 2.727 | 2.381 | 2.362 | 2.270 | |

$S/{k}_{B}$ | 0.693 | 0.682 | 0.668 | 0.63 | 0.623 | 0.597 |

**Table 2.**Notation correspondence between [34] and the current article.

This Text | [34] | Meaning |
---|---|---|

k | n | count of atoms in a cell |

m | p | count of interatomic bonds in a cell |

${y}_{1}^{\left(m\right)}$ | ${y}_{A}^{m}$ | concentration of atoms a in sublattice m |

${k}_{m}/k$ | ${f}_{m}$ | sublattice fraction |

$\gamma =zk/2m$ | $zn/2p$ | structural parameter of theory |

$ln{b}_{m}={\lambda}_{m}k/\theta \gamma {k}_{m}$ | ${\mu}_{A}^{m}$ | Lagrangian multiplier |

System | Figure | ${\mathit{W}}_{1}$ | $\mathit{\delta}{\mathit{W}}_{1}$ | ${\mathit{W}}_{2}$ | $\mathit{\delta}{\mathit{W}}_{2}$ | ${\mathit{W}}_{3}$ | $\mathit{\delta}{\mathit{W}}_{3}$ |
---|---|---|---|---|---|---|---|

Ag-In | Figure 5 | 5.722 | −0.271 | 10.747 | −4.883 | 11.364 | −6.433 |

Al-Ni | Figure 6 | 24.84 | −61.332 | 93.534 | −322.812 | 22.291 | −53.686 |

Al-Si | Figure 7 | 5.474 | −1.291 | 8.717 | −2.701 | 4.458 | −0.519 |

Ca-Ni | Figure 8 | 8.675 | −8.07 | 5.213 | −3.981 | 4.362 | −2.086 |

Cu-Si | Figure 9 | 5.511 | −1.724 | 8.52 | −2.49 | 7.394 | −13.796 |

La-Zr | Figure 10 | 2.136 | 1.692 | 2.673 | 3.423 | 2.626 | 1.405 |

Al-Mn | Figure 11 | 7.212 | −8.068 | 17.858 | −29.19 | 11.002 | −14.233 |

Al-Sc | Figure 12 | 17.534 | −25.934 | 63.219 | −165.46 | 26.227 | −44.62 |

Ca-Zn | Figure 13 | 34.888 | −71.432 | 39.524 | −79.745 | 15.304 | −19.907 |

Ce-Si | Figure 14 | 103.668 | −353.391 | 342.362 | −1750.719 | 30.137 | −95.847 |

Ce-Zn | Figure 15 | 22.379 | −47.129 | 17.701 | −35.039 | 8.535 | −9.154 |

Cu-Fe | Figure 16 | 2.13 | 1.607 | 3.302 | 2.191 | 2.033 | 1.647 |

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**MDPI and ACS Style**

Kremer, E.
Short-Range Order Modeling in Alloys. *Thermo* **2023**, *3*, 346-374.
https://doi.org/10.3390/thermo3030022

**AMA Style**

Kremer E.
Short-Range Order Modeling in Alloys. *Thermo*. 2023; 3(3):346-374.
https://doi.org/10.3390/thermo3030022

**Chicago/Turabian Style**

Kremer, Edward.
2023. "Short-Range Order Modeling in Alloys" *Thermo* 3, no. 3: 346-374.
https://doi.org/10.3390/thermo3030022