*3.2. Comparison of the Cross-Section Images*

Partitioning of the elements in the 1g and 0g-Sample are relatively similar, only Hf and Re have a higher partitioning coefficient in the 1g-sample.

The 1g-Sample shows more small dendrites with finer secondary dendrite arm spacings. The arm spacings are also dependent on the distance to the sample surface.

Voids between the primary dendrite arms can be found in the sample center for the 0g-sample, while this effect is not visible in the 1g-sample.

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

A part of an image from the CMSX-10 sample is given in Figure 19, where red points are marked. After a few simulations on the program Fractal Real Finder, we find that with *p* = 10 and *L* = 2, a good fit is obtained (*p<sup>L</sup>* = 100 points). The captured result is plotted in Figure 20.

**Figure 19.** CMSX-10 marked part of the sample.

**Figure 20.** Affine Fractal Regression of the CMSX-10 reconstruction.

The statistical Kolmogorov-Smirnov test, as well as the the respective plot, confirms the good reconstruction of the original data. From the output of the program, the relevant fractal coefficients are −0.175388 and −0.229489. The list of estimated coefficients is presented in Table 2.

**Table 2.** Estimated coefficients of image from the CMSX-10\_ISS\_post-flight\_007.


With these estimated fractal coefficients, an upper estimative for the Hausdorff dimension was computed as the solution of the nonlinear equation:

$$8\left(\frac{1}{10}\right)^D + 0.175388^D + 0.229489^D = 1.\tag{8}$$

The calculated Hausdorff dimension is *D* = 1.0906.

We successfully performed the fractal reconstruction of the sample morphology. In the next result, the rebuilding of two samples consolidated in the space is analyzed. The image A is depicted in Figure 21 and the image B is in Figure 22. In each image, red dots have been placed to mark identified boundary lines.

**Figure 21.** Image A of marked superalloy part.

**Figure 22.** Image B of marked superalloy part.

With the software for numerical computation of the solution, Fractal Real Finder, we obtain the coefficients given in Tables 3 and 4. For the image A, we used 12 fractal periods (*p* = 12) and, for image B, 13 fractal periods (*p* = 13). Both simulations worked with two fractal levels (*L* = 2).


**Table 3.** Estimated coefficients from Image A.


**Table 4.** Estimated coefficients from Image B.

We performed the statistical Kolmogorov-Smirnov test for two samples, and it indicates the goodness of the reconstructions.

The plot of the original data and the estimated curves for the image A and the image B are given respectfully in Figures 23 and 24.

**Figure 23.** Fractal estimated curve from Image A.

**Figure 24.** Fractal estimated curve from Image B.

The relevant fractal coefficients, in the case of the image A, are *a*0, *a*1, *a*4, *a*<sup>5</sup> and *a*<sup>7</sup> and of image B, *a*1, *a*3, *a*4, *a*<sup>8</sup> and *a*10. From the Proposition above, the upper estimators of Hausdorff dimension are respectively *D* = 1.11368 and *D* = 1.16975.

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

A CMSX-10 spare arc-melter image (Figure 13J) was imported into a pptx file and inserted a grid on it. Then, equally spaced red points were marked on a portion of the contour and the following image was obtained (see Figure 25).

**Figure 25.** CMSX-10 spare\_arc-melter image, with red points on contour.

Then we defined a scale, and we imported the data into a tabulated file in the form of an ordered list of these red points. We tried some options for the number of fractal periods *p* and fractal levels *L* and simulated with the software Fractal Real Finder. The result fitted very nicely with *p* = 10, *L* = 2 and *p<sup>L</sup>* = 100 points. The estimated coefficients are given in Table 5.


**Table 5.** Estimated coefficients from the CMSX-10 spare-arc-melter image.

The statistical analysis in the PAST software the Kolmogorov−Smirnov test for the independent samples' comparison of the equality of distributions showed undoubtedly that these distributions could be considered equal. The test showed that the largest difference between corresponding values was 0.06 and from *p*-value = 0.9921 > 0.05 we concluded that the null hypothesis was not rejected, showing that there is no significant difference between the distribution for the two samples. Please see the respective plot in Figure 26 showing that the reconstruction of the sample morphology is with high accuracy acceptable.

**Figure 26.** Fractal estimated curve.

The relevant fractal coefficient is *a*<sup>8</sup> = 0.1915881, and, consequently, the upper of the Hausdorff dimension is *D* = 1.04007 (see Proposition above).

From the above results and discussion, it is obvious that fractals help to overbridge the complex structures and processes leading towards controlled disorder and finally the ordered structures [10,11].

Additionally, from the same image (Figure 13J) we selected a circular region and then applied the fractal regression to compare the Hausdorff dimensions of the same sample, but taking into consideration different contour and another fragment of the image. Figure 27 represents this selected region with a polar grid.

We considered the series of the radius (distance between the center point (in blue) and the corresponding red point). We performed the fractal regression, and we obtained the coefficients for the estimated fractal curve (see Table 6). The fit was done for 2 fractal levels and 11 fractal periods.

**Figure 27.** CMSX-10 spare\_arc-melter image, with red points on contour with a polar grid.

**Table 6.** Estimated coefficients from Image J 2000x.


None of the fractal coefficients have sufficient fractal relevance to estimating the Hausdorff dimension bigger than 1, i.e., there is no estimated coefficient above 0.09 = 1/11. Therefore, the estimated Hausdorff dimension is *D* = 1, which means the estimated fractal oscillations are soft. Reconverting the estimated values back to polar coordinates, we obtained the plot in Figure 28.

**Figure 28.** Fractal estimated curve from polar coordinates.

The use of the fractal principle is highly relevant for the evaluation and estimation of dimensional properties of irregular structures in nature. Undoubtedly, they help us in understanding the morphological organization of complex structures that appear in Space and Earth [12–14].

#### **4. Discussion**

At the end of this very complex research overview, we obtained different results based on the analysis which included different parameters from surface and cross-sections of the samples studied by SEM and optical microscopy methods. Also, we included some comparative results that inlight the phenomena at the space and land consolidation. From all of these points of view, the fractal analysis provides a better understanding of the consolidation conditions influence on the final morphology of structures. Only fine fractal microstructure analysis enables the comparative differences between the dendrites' sizes, their orientations and voids between the primary dendrite arms.

This is very important for the reason of similarities among the images of space and land samples. Consequently, we can notify a very strong influence of the gravity in the consolidation process, which makes certain differences at microstructures based on different consolidation effects. We obtained certain differences among the land and space Hausdorff dimensions of the samples, but possibly due to different starting conditions in the experiments. However, we must highlight that the comparison of space and land samples was not the target of this paper. The comparison of the basic sample data according to fractal reconstruction could be a direction for some future researches. The aim of this paper is the fractal reconstruction of different surface samples in order to achieve the precise mathematical characterization of their roughness and consequently to predict and design the desired microstructures.

In future research, we plan to analyze more deeply the observed differences between space and land structures and try to reveal potential influences of the internal forces within constituents of samples matter (cohesion and adhesion).

The more biomimetic similarity in our material structures related to nature is a necessity. In our future research, we can also include the questions and relations between entropy and fractals.

### **5. Conclusions**

The importance of these results lies in the fact that we can use these fractal dimensions' characterizations for an additional understanding and insights of microstructures. Since we achieved an improved possibility of getting reconstructed morphology of shapes, we established a completely new perspective and frontiers on the advanced structures' prediction. All this phenomenology is extremely important for the relativization of the scale sizes in the space through fractal nature. In addition, in our further study, we extended fractal analysis on the micro images based on the land-consolidated samples what could be potentially very attractive for the future research in this area.

Finally at the end, when analyzing the data review of all presented experiments we must underline the existence of the dominant internal forces in and between the dendrites. It is necessary to emphasize the real influence of the mentioned forces especially at the space conditions where there is no gravity. Besides, we can also mention the micro capillary and surface tension effects.

Further, it is crucially important to take into consideration that all of these phenomena have been included in the understanding and explanations of the processes in the Space bodies consolidations and even though the whole Space. This fact is not related to any of the Space consolidation theories caused neither by the explosion nor by the high pressure on the micro level. On the other hand, this focuses the roll of the gravity itself, when we have the land consolidation processes and this can potentially provide much thorough approach in the explanation of gravity effect even in the evolution.

**Author Contributions:** Conceptualization, the idea and supervision: V.M.; Formal analysis, software and writing-original draft preparation: V.M., C.S. and I.I.; Validation, resources data curation and supervision: M.M. and H.-J.F. All authors have read and agreed to the published version of the manuscript.

**Funding:** V. Mitic and I. Ilic are gratefully thankful to the Ministry for the Education, Science and Technology Development of Serbia regarding the basic science research funds. Cristina Serpa acknowledges partial support from National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project: UIDB/04561/2020. M. Mohr and H.J. Fecht like to thank DLR (50WM1759) and ESA (AO-099-022) for the generous financial support. M. Mohr and H.J. Fecht acknowledge the access to the ISS-EML, which is a joint undertaking of the European Space Agency (ESA) and the DLR Space Administration. The reported work was conducted in the framework of the ESA project ThermoProp (AO-099-022). We further acknowledge funding from the DLR Space Administration with funds provided by the Federal Ministry for Economic Affairs and Energy (BMWi) under Grant No. 50WM1759.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We would like to thank A. Minkow, R. Wunderlich and Y. Dong for their expert technical support and fruitful discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

