2.1.2. Images of Cross-Sections

We have done SEM and optical images on the polished and etched cross-sections of two samples (Figures 14–16):


**Figure 14.** (**A**) 0g sample, (**B**) 1g sample.

(**C**) (**D**)

**Figure 15.** Optical images, 0g sample (**A**,**C**) and 1g sample (**B**,**D**).

(**A**) (**B**)

**Figure 16.** *Cont*.

**Figure 16.** SEM images, 0g sample (**A**,**C**) and 1g sample (**B**,**D**).

On the Figures 17 and 18 the EDX measurements on the cross section of the 0g-sample are presented showing the partitioning coefficients.



**Figure 17.** EDX measurements on the cross section of the 0g-sample.

EDX measurements on the cross section of the 1g-sample show partitioning coefficients:



**Figure 18.** EDX measurements on the cross section of the 1g-sample.

#### 2.1.3. Mathematical Fractal Analysis Technique

In order to describe the precise surface geometry, we introduced the concept of fractals and applied a new method of fractal reconstruction. The fractal analysis of real images was performed by using the technique based on a new affine fractal regression model. This process exploits certain mathematical formulations [2–4] designated for obtaining the coefficients of the equations system that best fit the data. The modeled system is:

$$
\rho \left( \frac{\mathbf{x} + j}{p} \right) = a\_{\rangle} \mathbf{e}(\mathbf{x}) + b\_{\rangle} \mathbf{x} + c\_{j\prime} \tag{2}
$$

where *x* ∈ [0, 1), 0 ≤ *j* ≤ *p* − 1, and *aj*, *bj*, *cj* are the parameters (real numbers) to estimate, with 0 < *aj* <sup>&</sup>lt; 1. The default domain is [0, 1).

The solution of this system is a function *<sup>ϕ</sup>* : [0, 1) <sup>→</sup> <sup>R</sup> and is called a fractal function [2]. In fact, it is proved [4] that such functions have a mathematical fractal structure, meaning that the plot of their graph is a fractal curve. Theoretical mathematical properties and explicit solutions are provided [2,3]. This model is originated by the system constructed by [5].

The fractal regression method consists of estimating the parameters *aj*, *bj*, *cj* such that they fit the real data. Hence, we consider the form of the explicit solution of the problem that depends on the *p*-expansion of numbers in [0, 1). For *L* = 2, this solution is:

$$\varphi(0) = \frac{c\_0}{1 - a\_0}'\tag{3}$$

$$\varphi\left(\frac{\mathfrak{z}\_1}{p}\right) = a\_{\mathfrak{z}\_1}\frac{c\_0}{1 - a\_0} + c\_{\mathfrak{z}\_1}\,\,\, \mathfrak{z}\_1 \neq 0,\tag{4}$$

$$\log\left(\frac{\mathfrak{J}\_1}{p} + \frac{\mathfrak{J}\_2}{p^2}\right) = a\_{\mathfrak{J}\_1} \left(a\_{\mathfrak{J}\_2}\frac{c\_0}{1 - a\_0} + c\_{\mathfrak{J}\_2}\right) + b\_{\mathfrak{J}\_1}\frac{\mathfrak{J}\_2}{p} + c\_{\mathfrak{J}\_1}, \ \mathfrak{J}\_2 \neq 0. \tag{5}$$

In order to obtain the best coefficients, the theoretical approach computes the SSR sum of square residuals in between the formal definition and the real values. Next, it uses the partial derivatives of this SSR and equals to zero, for minimizing this error. The best solution of the problem is given when:

$$\frac{\partial SSR}{\partial a\_j} = 0, \frac{\partial SSR}{\partial b\_j} = 0, \frac{\partial SSR}{\partial c\_j} = 0,\tag{6}$$

for all *j* = 0, 1, 2, ... , *p* − 1. This is a problem with 3*p* parameters to estimate where the equations to solve are nonlinear. The regression method is widely used as linear regression, a much simpler model in data analysis science. For detailed information on this subject see [6].

Parameters *aj* are the fractal coefficients and *bj* are the directional coefficients. Bigger fractal coefficients mean strong fractal oscillations. Parameter *p* is the fractal period and *L* is the fractal level of a curve defined by the system.

The mathematical analytical solution of this partial derivative system (for the fractal regression) is not possible to compute and a numerical approach is needed. By applying the newly available software Fractal Real Finder designed for the numerical computation of the solution, we processed the given samples and estimated the curves and Hausdorff dimension D. With an input of the real data, the program executes simulations and gives an output with a fractal curve as modeled above. With the estimated fractal curves, we may upper estimate the Hausdorff dimension.

Proposition. The Hausdorff dimension of the graph of the function ϕ solution of the above system is upper bounded by the solution of:

$$\sum\_{j=0}^{p-1} \beta\_j^{\;D} = 1,\tag{7}$$

where *<sup>β</sup><sup>j</sup>* <sup>=</sup> *max*#<sup>1</sup> *p* , *aj* \$ , 0 ≤ *p* ≤ *p* − 1.

The coefficients with fractal relevance are those *aj* such that *aj* <sup>&</sup>gt; 1/*p*.

All of these additional mathematical calculations with novelty solutions are highly important for better understanding the fractal nature analysis applications in material sciences and specifically for the research data in this paper. The fractal approach is based on the self-similarity of surfaces at different scales. Its superiority is that it is insensitive to the structural details and the structure is specified by a single descriptor, the fractal (noninteger) dimension D. So, the fractal surface analysis was used to describe, by a single parameter, surface roughness over many orders of magnitude. The increasing value of D represents an increasing surface roughness. It gives information on the measure of complexity of different surface topographies. That way, fractal dimension becomes highly suitable for the characterization of various topographies. Particularly, it is crucial in the cases when the land or space microstructures have lots of irregular peaks and valleys that cannot be easily defined and evaluated. Conclusively, fractal dimensions that we obtained from SEM micrographs of surfaces indeed do give us a very good description of the overall topography of the surface, due to the self-similarity.

#### **3. Results**

We implement the fractal structures that are originally coming from nature and beunivocally corresponding to chaotic structures in the matter. We considered the microstructures of different images samples. All samples have been processed onboard the International Space Station. In this particular case, the fractal analysis is implemented as the most effective. We applied several aspects of fractal analyses and obtained certain results regarding the Hausdorff dimensions related to the surface and structural characteristics of the CMSX-10 samples.

Several authors in the literature have investigated the mechanisms during the solidification of CMSX-10. It is generally accepted, that a primary solidification takes place, where the gamma-phase dendrites are formed. Subsequently, the interdendritic regions are solidified, together with the precipitation of the gamma'-phase. The solidification sequence of the interdendritic region is a subject of significant research and the assumed solidification paths by different authors do vary [7]. The dendritic formation is common in alloys where solute partitions between the solid and liquid phases. During the growth of the crystal in the melt, solute and heat can accumulate ahead of the growth interface and can lead to an unstable interface and dendritic solidification.

It is commonly observed in Ni-based superalloys, that the heavy elements, such as W and Re are segregated to the gamma-dendrites, the lighter elements, Ti, Ta, and Al are enriched in the remaining liquid [7–9].
