# Fractal Nature of Advanced Ni-Based Superalloys Solidified on Board the International Space Station

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## Abstract

**:**

## 1. Introduction

#### 1.1. Some Previous Results of Fractal Technique Application on Ceramics Samples

#### Short Description of the Applied Technique for the Grain Cluster Shape Reconstruction

## 2. Materials and Methods

#### 2.1. Short Experimental Review on the Differences in Solidification of CMSX-10 in 1g and 0g

_{liq}= 1706 K, then the liquid was further overheated until a maximum temperature of about 1900 K. Subsequently, the sample was cooled freely. This way, the sample undercooled about 140 K below its equilibrium melting point. In comparison, a sample was solidified on ground, while placed on a water cooled copper mold. Due to heterogeneous nucleation on the contact area, this represents the case of minimal undercooling. Figure 9 shows the temperature-time diagram recorded during the relevant melt cycle performed on ISS in microgravity of the 6.5 mm sphere of CMSX-10.

#### 2.1.1. SEM Images of the Surface

- CMSX-10–solidified onboard the ISS, “0g-Sample”
- CMSX-10–solidified on top of a water-cooled copper block, on the ground, in the Arc-Melter, “1g-Sample”

#### 2.1.2. Images of Cross-Sections

- 3.
- CMSX-10–solidified onboard the ISS, “0g-Sample”.
- 4.
- CMSX-10–solidified by suction casting, on the ground, in the Arc-Melter, “1g-Sample”.

#### 2.1.3. Mathematical Fractal Analysis Technique

## 3. Results

#### 3.1. Comparison of the Surface Images

#### 3.2. Comparison of the Cross-Section Images

#### 3.3. Fractal Analysis of the Images Consolidated in Space

#### 3.4. Fractal Analysis of an Image Consolidated on Earth

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Siauwand, T.; Bayen, A.M. An Introduction to MATLAB Programming and Numerical Methods for Engineers; Elsevier: Oxford/London, UK; Waltham, MA, USA; San Diego, CA, USA; Amsterdam, The Netherlands, 2015. [Google Scholar]
- Serpa, C.; Buescu, J. Constructive Solutions for Systems of Iterative Functional Equations. Constr. Approx.
**2017**, 45, 273–299. [Google Scholar] [CrossRef] - Serpa, C.; Buescu, J. Explicitly defined fractal interpolation functions with variable parameters. Chaos Solitons Fractals
**2015**, 75, 76–83. [Google Scholar] [CrossRef] - Buescu, J.; Serpa, C. Fractal and Hausdorff dimensions for systems of iterative functional equations. J. Math. Anal. Appl.
**2019**, 480, 1–19. [Google Scholar] [CrossRef] - Barnsley, M.F. Fractal functions and interpolation. Constr. Approx.
**1986**, 2, 303–329. [Google Scholar] [CrossRef] - Kenney, J.F.; Keeping, E.S. Linear Regression and Correlation, Ch. 15. In Mathematics of Statistics, 3rd ed.; Van Nostrand: Princeton, NJ, USA, 1962; pp. 252–285. [Google Scholar]
- Terner, M.; Yoon, H.Y.; Hong, H.U.; Seo, S.M.; Gu, J.H.; Lee, J.H. Clear path to the directional solidification of Ni-based superalloy CMSX-10: A peritectic reaction. Mater. Charact.
**2015**, 105, 56–63. [Google Scholar] [CrossRef] - Wilson, B.C.; Cutler, E.R.; Fuchs, G.E. Effect of solidification parameters on the microstructures and properties of CMSX-10. Mater. Sci. Eng. A
**2008**, 479, 356–364. [Google Scholar] [CrossRef] - Mohr, M.; Furrer, D.; Fecht, H. Thermophysical Properties of Advanced Ni-Based Superalloys in the Liquid State Measured on Board the International Space Station. Adv. Eng. Mater.
**2020**, 22, 1901228. [Google Scholar] [CrossRef] - Mitić, V.; Goran, L.; Vesna, P.; Hwu, J.R.; Tsay, S.C.; Perng, T.P.; Sandra, V.; Branislav, V. Ceramic materials and energy-Extended Coble’s model and fractal nature. J. Eur. Ceram. Soc.
**2019**, 39, 3513–3525. [Google Scholar] [CrossRef] - Mandelbrot, B. The Fractal Geometry of Nature; W. H. Freeman: New York, NY, USA, 1977. [Google Scholar]
- Arnold, W.; Fischer, H.-H.; Knapmeyer, M.; Krüger, H. Surface Mechanical Properties of Comet 67P. Jpn. J. Appl. Phys.
**2019**, 58, SG0801. [Google Scholar] [CrossRef] - Mitić, V.; Kocić, L.J.M.; Mitrović, I.Z. Fractals in Ceramic Structure. In Proceedings of the IX World Round Table Conference on Sintering, Belgrade, Serbia, 1–4 September 1998; Advanced Science and Technology of Sintering. Stojanović, B.D., Skorokhod, V.V., Nikolić, M.V., Eds.; Kluwer Academic/Plenum Publishers: New York, NY, USA, 1999; pp. 397–402. [Google Scholar]
- Mitic, V.V.; Lazovic, G.; Mirjanic, D.; Fecht, H.; Vlahovic, B.; Arnold, W. The Fractal Nature as New Frontiers in Microstructure Characterization and Relativisation the Scale Sizes within the Space. Mod. Phys. Lett. B
**2020**, 34, 2050421. [Google Scholar] [CrossRef]

**Figure 3.**Affine Fractal Regression of the reconstruction of BaTiO

_{3}sample structure by using fractal method.

**Figure 4.**International Space Station (ISS). Picture taken by a crew member of the space shuttle Atlantis after undocking from the space station (Image source NASA/Crew of STS-132).

**Figure 11.**Series of 5 SEM images of the CMSX-10 surface of the 0-g sample (

**A**,

**C**,

**E**,

**G**,

**I**) and the 1-g sample (

**B**,

**D**,

**F**,

**H**,

**J**).

**Figure 12.**Series of 5 SEM images of the CMSX-10 surface of the 0-g sample (

**A**,

**C**,

**E**,

**G**,

**I**) and the 1-g sample (

**B**,

**D**,

**F**,

**H**,

**J**).

**Figure 13.**Series of 5 SEM images of the CMSX-10 surface of the 0-g sample (

**A**,

**C**,

**E**,

**G**,

**I**) and the 1-g sample (

**B**,

**D**,

**F**,

**H**,

**J**).

Composition in wt% | CMSX-10 |
---|---|

Ni | Bal. |

Al | 5.7 |

Cr | 2.0 |

Co | 3.0 |

Mo | 0.4 |

W | 5.0 |

Ti | 0.2 |

Re | 6.0 |

Ta | 8.0 |

Hf | 0.03 |

Nb | 0.1 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|---|

${a}_{j}$ | 0.018 | 0.011 | −0.046 | −0.175 | −0.229 | −0.073 | −0.044 | −0.006 | −0.038 | −0.051 |

${b}_{j}$ | −0.03 | 0.401 | −0.231 | −0.861 | 1.046 | 0.778 | 0.032 | 0.069 | 0.008 | 0.063 |

${c}_{j}$ | 0.967 | 1.043 | 1.47 | 1.437 | 0.46 | 0.824 | 1.494 | 1.41 | 1.566 | 1.699 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${a}_{j}$ | −0.14 | 0.09 | 0.059 | 0.055 | 0.097 | 0.268 | 0.025 | −0.098 | −0.012 | 0.043 | −0.05 | −0.015 |

${b}_{j}$ | −0.89 | 0.324 | 0.318 | 0.348 | 0.291 | 0.941 | −0.651 | −0.398 | −0.49 | −0.254 | −0.813 | −0.607 |

${c}_{j}$ | 5.487 | 3.788 | 4.03 | 4.103 | 4.072 | 3.009 | 4.062 | 3.987 | 3.462 | 2.513 | 2.513 | 1.702 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${a}_{j}$ | −0.051 | −0.105 | 0.01 | 0.141 | 0.272 | 0.042 | −0.066 | −0.018 | −0.158 | 0 | 0.133 | 0.028 | −0.026 |

${b}_{j}$ | 0.63 | 0.599 | 0.262 | 0.453 | 0.128 | 0.004 | −0.332 | 0.269 | −0.191 | 0.155 | −0.364 | −0.831 | −0.632 |

${c}_{j}$ | 1.873 | 2.685 | 3.155 | 2.75 | 2.862 | 3.257 | 3.587 | 3.226 | 3.683 | 0.955 | 2.713 | 2.447 | 1.835 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|---|

${a}_{j}$ | 0.051 | −0.025 | 0.092 | −0.07 | −0.017 | 0.037 | −0.001 | 0.025 | 0.192 | −0.026 |

${b}_{j}$ | −0.58 | 0.101 | 0.032 | −1.117 | 0.351 | 0.561 | −0.653 | 0.334 | 0.462 | −0.374 |

${c}_{j}$ | 3.588 | 2.859 | 2.474 | 2.899 | 2.111 | 3.057 | 1.407 | 0.818 | 0.544 | 0.363 |

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

${a}_{j}$ | −0.079 | −0.038 | 0.071 | 0.06 | 0.051 | 0.015 | −0.035 | −0.023 | 0.075 | −0.033 | −0.023 |

${b}_{j}$ | −0.488 | 0.733 | 1.136 | −0.534 | 0.087 | −0.904 | −0.419 | −0.095 | 0.843 | −0.619 | 0.1 |

${c}_{j}$ | 2.102 | 1.552 | 2.247 | 3.309 | 2.994 | 2.979 | 2.158 | 1.751 | 1.742 | 2.566 | 1.891 |

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**MDPI and ACS Style**

Mitić, V.; Serpa, C.; Ilić, I.; Mohr, M.; Fecht, H.-J.
Fractal Nature of Advanced Ni-Based Superalloys Solidified on Board the International Space Station. *Remote Sens.* **2021**, *13*, 1724.
https://doi.org/10.3390/rs13091724

**AMA Style**

Mitić V, Serpa C, Ilić I, Mohr M, Fecht H-J.
Fractal Nature of Advanced Ni-Based Superalloys Solidified on Board the International Space Station. *Remote Sensing*. 2021; 13(9):1724.
https://doi.org/10.3390/rs13091724

**Chicago/Turabian Style**

Mitić, Vojislav, Cristina Serpa, Ivana Ilić, Markus Mohr, and Hans-Jörg Fecht.
2021. "Fractal Nature of Advanced Ni-Based Superalloys Solidified on Board the International Space Station" *Remote Sensing* 13, no. 9: 1724.
https://doi.org/10.3390/rs13091724