# Development of the RF-MEAM Interatomic Potential for the Fe-C System to Study the Temperature-Dependent Elastic Properties

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Ground State DFT Calculation

#### 2.2. Low-Convergence AIMD Calculation

#### 2.3. High-Convergence DFT Calculation

#### 2.4. Potential Parameter Fitting

#### 2.5. Potential Selection and Validation

## 3. Results and Discussion

#### 3.1. Parameters for AIMD Calculations

#### 3.2. Ground State Elastic Calculation

#### 3.3. Finite-Temperature Elasticity Calculation

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Fluctuations in regulated parameters over time in the NPT ensemble using $LANGEVIN\_$$GAMMA$ and $LANGEVIN\_GAMMA\_L$ equal to 0.5 and $PMASS$ equals 1: (

**a**) Pressure vs. time, (

**b**) Temperature vs. time, where $LANGEVIN\_GAMMA$ and $LANGEVIN\_GAMMA\_L$ equal to 50 and $PMASS$ equals to 1: (

**c**) Pressure vs. time, (

**d**) Temperature vs. time. The black (solid) line is the actual data at a certain time, whereas the red (dotted) line is the running average over every 20 datapoints.

**Figure 3.**Comparison of energy vs. volume curves for (

**a**) cementite, (

**b**) B1, and (

**c**) L12 with experimental and DFT results. Both DFT and experimental results used for the comparison are from Lalitha et al. [35]. These curves are shifted vertically to match the RF-MEAM calculated equilibrium energy.

**Figure 4.**Comparison of phonon dispersion curves calculated with the optimized potential (blue) with the ones calculated with DFT (red) for (

**a**) cementite and (

**b**) O-Fe${}_{7}$C${}_{3}$ structures. For a color version of this figure, please refer to the online version of this article.

**Figure 6.**Bulk moduli of O-Fe${}_{7}$C${}_{3}$ at various temperatures (up to 1200 K) in comparison with the DFT results of T. Chihi et al. [57].

**Table 1.**Description of various structures used for ground state calculations. The unit cells of each structure are detailed in supplementary information (Figure S1).

Fe-C Structure | k-Points (Gamma) | No. of Atoms | Supercell |
---|---|---|---|

B1 | 7 × 15 × 15 | 128 | 4 × 2 × 2 |

B2 | 9 × 12 × 16 | 120 | 5 × 4 × 3 |

B3 | 7 × 15 × 15 | 128 | 4 × 2 × 2 |

Cementite | 23 × 10 × 7 | 64 | 1 × 2 × 2 |

L12 | 10 × 15 × 15 | 48 | 3 × 2 × 2 |

Fe-C Structure | k-Points (Gamma) | No. of Atoms | Supercell |
---|---|---|---|

B1 | 15 × 15 × 15 | 64 | 2 × 2 × 2 |

B2 | 16 × 16 × 16 | 54 | 3 × 3 × 3 |

B3 | 15 × 15 × 15 | 64 | 2 × 2 × 2 |

Cementite | 23 × 10 × 7 | 64 | 1 × 2 × 2 |

L12 | 10 × 15 × 15 | 48 | 3 × 2 × 2 |

**Table 3.**Comparison of our results with the published DFT and experimental data for equilibrium volume, bulk modulus, and pressure derivative calculated by the Birch–Murnaghan equation of state. Experimental data are included in the brackets.

Structure | V${}_{0}$ (Å${}^{3}$) | B${}_{0}$ (GPa) | B${}_{0}$’ | |||
---|---|---|---|---|---|---|

RF-MEAM | Literature | RF-MEAM | Literature | RF-MEAM | Literature | |

B1 (FeC) | 65 | $64{\phantom{\rule{3.33333pt}{0ex}}}^{a,b}$ | 338 | $329{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 5.28 | $4.40{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ |

B2 (FeC) | 16 | $15{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 302 | $343{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 4.12 | $4.40{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ |

B3 (FeC) | 78 | $77{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 222 | $251{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 4.50 | $4.20{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ |

Cementite (Fe3C) | 155 | $154{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 252 | $234{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 4.05 | $4.00{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ |

O-Fe${}_{7}$C${}_{3}$ | 94 | $89{\phantom{\rule{3.33333pt}{0ex}}}^{b}$, $91{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ ($93{\phantom{\rule{3.33333pt}{0ex}}}^{d}$) | 288 | $263{\phantom{\rule{3.33333pt}{0ex}}}^{b}$, $262{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 3.10 | $5.04{\phantom{\rule{3.33333pt}{0ex}}}^{b}$, $3.70{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ |

**Table 4.**Single-crystal and polycrystalline elastic constants for different alloys calculated using the produced RF-MEAM potential. Elastic constants of other alloys (B2, B3, and L12) are included in the Supplementary Information’s Table S1.

Elastic Constants (GPa) | Fe3C-Cementite | FeC-B1 | O-Fe${}_{7}$C${}_{3}$ | ||||||
---|---|---|---|---|---|---|---|---|---|

This Study | Literature | This Study | Literature | This Study | Literature | ||||

RF-MEAM | DFT | DFT/MEAM | RF-MEAM | DFT | DFT/MEAM | RF-MEAM | DFT | DFT/MEAM | |

C11 | 399 | 296 | $388{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 585 | 584 | $566{\phantom{\rule{3.33333pt}{0ex}}}^{b}$ | 399 | 344 | $394{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

C22 | 459 | 392 | $345{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | - | - | - | 469 | 428 | 445c |

C33 | 364 | 330 | $322{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | - | - | - | 444 | 423 | $452{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

C12 | 236 | 137 | $156{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 214 | 205 | $213{\phantom{\rule{3.33333pt}{0ex}}}^{b}$ | 238 | 166 | $185{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

C13 | 210 | 181 | $164{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | - | - | - | 245 | 170 | $179{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

C23 | 204 | 155 | $162{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | - | - | - | 255 | 182 | $170{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

C44 | 116 | 131 | $134{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 125 | 82 | $145{\phantom{\rule{3.33333pt}{0ex}}}^{b}$ | 106 | 95 | $126{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

C55 | 40 | 19 | $15{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | - | - | - | 98 | 89 | $112{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

C66 | 119 | 135 | $134{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | - | - | - | 81 | 66 | $78{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

- | |||||||||

B | 280 | 218 | $224{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 338 | 331 | $331{\phantom{\rule{3.33333pt}{0ex}}}^{b}$ | 310 | 247 | $262{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

G | 93 | 93 | $95{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 149 | 125 | $158{\phantom{\rule{3.33333pt}{0ex}}}^{b}$ | 95 | 95 | $114{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

Y | 252 | 245 | $249{\phantom{\rule{3.33333pt}{0ex}}}^{a}$ | 390 | 334 | $408{\phantom{\rule{3.33333pt}{0ex}}}^{b}$ | 259 | 253 | $298{\phantom{\rule{3.33333pt}{0ex}}}^{c}$ |

T(K) | FeC-0.2% | FeC-0.4% | ||||
---|---|---|---|---|---|---|

B (GPa) | G (GPa) | E (GPa) | B (GPa) | G (GPa) | E (GPa) | |

73 | 215 | 81 | 217 | 214 | 80 | 214 |

144 | 211 | 79 | 211 | 210 | 79 | 210 |

200 | 206 | 77 | 206 | 204 | 78 | 207 |

294 | 202 | 76 | 203 | 202 | 76 | 203 |

422 | 195 | 74 | 196 | 194 | 72 | 192 |

533 | 186 | 70 | 187 | 185 | 70 | 186 |

589 | 186 | 69 | 184 | 183 | 67 | 179 |

644 | 179 | 67 | 179 | 180 | 65 | 173 |

700 | 173 | 64 | 172 | 174 | 60 | 162 |

811 | 167 | 53 | 144 | 169 | 52 | 141 |

866 | 165 | 49 | 134 | 162 | 50 | 135 |

922 | - | - | - | 154 | 42 | 116 |

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**MDPI and ACS Style**

Risal, S.; Singh, N.; Duff, A.I.; Yao, Y.; Sun, L.; Risal, S.; Zhu, W.
Development of the RF-MEAM Interatomic Potential for the Fe-C System to Study the Temperature-Dependent Elastic Properties. *Materials* **2023**, *16*, 3779.
https://doi.org/10.3390/ma16103779

**AMA Style**

Risal S, Singh N, Duff AI, Yao Y, Sun L, Risal S, Zhu W.
Development of the RF-MEAM Interatomic Potential for the Fe-C System to Study the Temperature-Dependent Elastic Properties. *Materials*. 2023; 16(10):3779.
https://doi.org/10.3390/ma16103779

**Chicago/Turabian Style**

Risal, Sandesh, Navdeep Singh, Andrew Ian Duff, Yan Yao, Li Sun, Samprash Risal, and Weihang Zhu.
2023. "Development of the RF-MEAM Interatomic Potential for the Fe-C System to Study the Temperature-Dependent Elastic Properties" *Materials* 16, no. 10: 3779.
https://doi.org/10.3390/ma16103779