# Small-Angle Scattering from Fractional Brownian Surfaces

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## Abstract

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## 1. Introduction

## 2. Theoretical Background

#### 2.1. Small-Angle Scattering Technique

#### 2.2. Small-Angle Scattering from Fractal Surfaces

#### 2.3. Fractional Brownian Surfaces

## 3. Methodology for Generating the Fractional Brownian Surfaces and for Calculating the Pair Distance Distribution Function

- Class I fBss (CI): distances between points are kept unchanged; thus, $x,y$ and z are of the same orders of magnitude. This corresponds to the classical structure of fBss, as shown in Figure 1, with a globular-like shape.
- Class II fBss (CII): distances between points are stretched by the same amount along x and y directions by a factor of b; thus, $x=y\ll z$. This gives rise to fBss with rectangular, planar-like shapes.
- Class III fBss (CIII): distances between points are stretched along a single direction by a factor of b; thus, x or $y\ll z$. This gives rise to fBss with rod-like shapes.

## 4. Results and Discussion

#### 4.1. Pair-Distance Distribution Functions

#### 4.2. Scattering Intensities

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic representation of the three main classes of fractals that can be described in a SAS experiment. (

**Left**) Surface fractal (${D}_{\mathrm{m}}={D}_{\mathrm{p}}=d$ and ${D}_{\mathrm{s}}<d$). (

**Middle**) Mass fractal (${D}_{\mathrm{s}}={D}_{\mathrm{m}}<d$ and ${D}_{\mathrm{p}}=d$). (

**Right**) Pore/volume fractal (${D}_{\mathrm{s}}={D}_{\mathrm{p}}<d$ and ${D}_{\mathrm{m}}=d$). Here, $d=2$, and it represents the Euclidean dimension of the embedding space. See main text for details.

**Figure 2.**Fractional Brownian surfaces on a square grid with dimensions $x=y$ at various values of Hurst exponent H. (

**a**–

**c**) 3D representation. (

**d**–

**f**) Density plot. (

**a**,

**d**,

**g**) $H=0.9$. (

**b**,

**e**,

**h**) $H=0.6$. (

**c**,

**f**,

**i**) $H=0.3$. The peaks and bottoms are represented by light and dark regions along Oz-axis, respectively. (

**g**–

**i**) are the same as (

**d**–

**f**) but are represented in a single color for better visualization of the variation of density plot roughness. (

**a**) $x\vee y=2.31z$. (

**b**) $x\vee y=1.78z$. (

**c**) $x\vee y=1.19z$.

**Figure 3.**Schematic representation of fBS and the associated grid (6 × 6 points) at $H=0.9$. (

**Left**) fBs. Red—the highest points (at 1.5); Blue—the lowest ones (at −1.5). (

**Middle**) The corresponding grid used. The gray plane is at $z=0$ and stands as the reference level for the heights of the grid points. Red points are above the plane, and blue ones are below it. (

**Right**) Projection of the grid on the 2D $xy$ plane. ${l}_{\mathrm{min}}$ is the minimum distance between the points in the grid, and a is the length of the grid in either x or y direction.

**Figure 4.**Pddfs from fBss at various geometries. (

**a**) Class CI: symmetric bell-like curves reveal the globular-like shape. (

**b**) Class CII with stretching factor $b=10$: symmetric bell-like curves reveal planar-like structures, since one dimension is kept fixed while the other two are stretched, by a factor of $b=10$. (

**c**) Class CIII with stretching factor $b=10$: curves with long linear domains reveal elongated structures.

**Figure 5.**SAS from fBs at various grid geometries. (

**a**) Three-dimensional. Fractal regions follow immediately the Guinier region (i.e., the region where $I\left(q\right)\propto {q}^{0}$) and, thus, are completely visible. (

**b**) Two-dimensional. Fractal regions are expected to follow the region where $I\left(q\right)\propto {q}^{-2}$. (

**c**) One-Dimensional. Fractal regions follow the region where $I\left(q\right)\propto {q}^{-1}$ and are partially visible.

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**MDPI and ACS Style**

Anitas, E.M.
Small-Angle Scattering from Fractional Brownian Surfaces. *Symmetry* **2021**, *13*, 2042.
https://doi.org/10.3390/sym13112042

**AMA Style**

Anitas EM.
Small-Angle Scattering from Fractional Brownian Surfaces. *Symmetry*. 2021; 13(11):2042.
https://doi.org/10.3390/sym13112042

**Chicago/Turabian Style**

Anitas, Eugen Mircea.
2021. "Small-Angle Scattering from Fractional Brownian Surfaces" *Symmetry* 13, no. 11: 2042.
https://doi.org/10.3390/sym13112042