Deterministic fractals: extracting additional information from small-angle scattering data

Dense random packing with a power-law size distribution: The structure factor, mass-radius relation, and pair distribution function

We consider a dense random packing of disks with a power-law distribution of radii and investigate their correlation properties. We study the corresponding structure factor, mass-radius relation, and pair distribution function of the disk centers. A toy model of dense segments in one dimension (1D) is solved exactly. It is shown theoretically in 1D and numerically in 1D and 2D that such a packing exhibits fractal properties. It is found that the exponent of the power-law distribution and the fractal dimension coincide. An approximate relation for the structure factor in arbitrary dimensions is derived, which can be used as a fitting formula in small-angle scattering. These findings can be useful for understanding the microstructural properties of various systems such as ultra-high performance concrete, high-internal-phase-ratio emulsions, or biological systems.

Revised scattering exponents for a power-law distribution of surface and mass fractals

We consider scattering exponents arising in small-angle scattering from power-law polydisperse surface and mass fractals. It is shown that a set of fractals, whose sizes are distributed according to a power law, can change its fractal dimension when the power-law exponent is sufficiently big. As a result, the scattering exponent corresponding to this dimension appears due to the spatial correlations between positions of different fractals. For large values of the momentum transfer, the correlations do not play any role, and the resulting scattering intensity is given by a sum of intensities of all composing fractals. The restrictions imposed on the power-law exponents are found. The obtained results generalize Martin's formulas for the scattering exponents of the polydisperse fractals.

Fractal Analysis of DNA Sequences Using Frequency Chaos Game Representation and Small-Angle Scattering

The fractal characteristics of DNA sequences are studied using the frequency chaos game representation (FCGR) and small-angle scattering (SAS) technique. The FCGR allows representation of the frequencies of occurrence of k-mers (oligonucleotides of length k) in the form of images. The numerically encoded data are then used in a SAS analysis to enhance hidden features in DNA sequences. It is shown that the simulated SAS intensity allows us to obtain the fractal dimensions and scaling factors at various scales. These structural parameters can be used to distinguish unambiguously between the scaling properties of complex hierarchical DNA sequences. The validity of this approach is illustrated on several sequences from: Escherichia coli, Mouse mitochondrion, Homo sapiens mitochondrion and Human cosmid.

An efficient approach to study membrane nano-inclusions: from the complex biological world to a simple representation

Membrane nano-inclusions (NIs) are of great interest in biophysics, materials science, nanotechnology, and medicine. We hypothesized that the NIs within a biological membrane bilayer interact via a simple and efficient interaction potential, inspired by previous experimental and theoretical work. This interaction implicitly treats the membrane lipids but takes into account its effect on the NIs micro-arrangement. Thus, the study of the NIs is simplified to a two-dimensional colloidal system with implicit solvent. We calculated the structural properties from Molecular Dynamics simulations (MD), and we developed a Scaling Theory to discuss their behavior. We determined the thermal properties through potential energy per NI and pressure, and we discussed their variation as a function of the NIs number density. We performed a detailed study of the NIs dynamics using two approaches, MD simulations, and Dynamics Theory. We identified two characteristic values of number density, namely a critical number density n c = 3.67 × 10-3 Å-2 corresponded to the apparition of chain-like structures along with the liquid dispersed structure and the gelation number density n g = 8.40 × 10-3 Å-2 corresponded to the jamming state. We showed that the aggregation structure of NIs is of fractal dimension d F < 2. Also, we identified three diffusion regimes of membrane NIs, namely, normal for n < n c, subdiffusive for n c ≤ n < n g, and blocked for n ≥ n g. Thus, this paper proposes a simple and effective approach for studying the physical properties of membrane NIs. In particular, our results identify scaling exponents related to the microstructure and dynamics of membrane NIs.

Small-Angle Scattering and Multifractal Analysis of DNA Sequences

The arrangement of A, C, G and T nucleotides in large DNA sequences of many prokaryotic and eukaryotic cells exhibit long-range correlations with fractal properties. Chaos game representation (CGR) of such DNA sequences, followed by a multifractal analysis, is a useful way to analyze the corresponding scaling properties. This approach provides a powerful visualization method to characterize their spatial inhomogeneity, and allows discrimination between mono- and multifractal distributions. However, in some cases, two different arbitrary point distributions, may generate indistinguishable multifractal spectra. By using a new model based on multiplicative deterministic cascades, here it is shown that small-angle scattering (SAS) formalism can be used to address such issue, and to extract additional structural information. It is shown that the box-counting dimension given by multifractal spectra can be recovered from the scattering exponent of SAS intensity in the fractal region. This approach is illustrated for point distributions of CGR data corresponding to Escherichia coli, Phospholamban and Mouse mitochondrial DNA, and it is shown that for the latter two cases, SAS allows extraction of the fractal iteration number and the scaling factor corresponding to "ACGT" square, or to recover the number of bases. The results are compared with a model based on multiplicative deterministic cascades, and respectively with one which takes into account the existence of forbidden sequences in DNA. This allows a classification of the DNA sequences in terms of random and deterministic fractals structures emerging in CGR.

Structural Properties of Molecular Sierpiński Triangle Fractals

The structure of fractals at nano and micro scales is decisive for their physical properties. Generally, statistically self-similar (random) fractals occur in natural systems, and exactly self-similar (deterministic) fractals are artificially created. However, the existing fabrication methods of deterministic fractals are seldom defect-free. Here, are investigated the effects of deviations from an ideal deterministic structure, including small random displacements and different shapes and sizes of the basic units composing the fractal, on the structural properties of a common molecular fractal—the Sierpiński triangle (ST). To this aim, analytic expressions of small-angle scattering (SAS) intensities are derived, and it is shown that each type of deviation has its own unique imprint on the scattering curve. This allows the extraction of specific structural parameters, and thus the design and fabrication of artificial structures with pre-defined properties and functions. Moreover, the influence on the SAS intensity of various configurations induced in ST, can readily be extended to other 2D or 3D structures, allowing for exploration of structure-property relationships in various well-defined fractal geometries.

Small-Angle Scattering from Fractals: Differentiating between Various Types of Structures

Small-angle scattering (SAS; X-rays, neutrons, light) is being increasingly used to better understand the structure of fractal-based materials and to describe their interaction at nano- and micro-scales. To this aim, several minimalist yet specific theoretical models which exploit the fractal symmetry have been developed to extract additional information from SAS data. Although this problem can be solved exactly for many particular fractal structures, due to the intrinsic limitations of the SAS method, the inverse scattering problem, i.e., determination of the fractal structure from the intensity curve, is ill-posed. However, fractals can be divided into various classes, not necessarily disjointed, with the most common being random, deterministic, mass, surface, pore, fat and multifractals. Each class has its own imprint on the scattering intensity, and although one cannot uniquely identify the structure of a fractal based solely on SAS data, one can differentiate between various classes to which they belong. This has important practical applications in correlating their structural properties with physical ones. The article reviews SAS from several fractal models with an emphasis on describing which information can be extracted from each class, and how this can be performed experimentally. To illustrate this procedure and to validate the theoretical models, numerical simulations based on Monte Carlo methods are performed.

Structural Properties of Vicsek-like Deterministic Multifractals

Deterministic nano-fractal structures have recently emerged, displaying huge potential for the fabrication of complex materials with predefined physical properties and functionalities. Exploiting the structural properties of fractals, such as symmetry and self-similarity, could greatly extend the applicability of such materials. Analyses of small-angle scattering (SAS) curves from deterministic fractal models with a single scaling factor have allowed the obtaining of valuable fractal properties but they are insufficient to describe non-uniform structures with rich scaling properties such as fractals with multiple scaling factors. To extract additional information about this class of fractal structures we performed an analysis of multifractal spectra and SAS intensity of a representative fractal model with two scaling factors—termed Vicsek-like fractal. We observed that the box-counting fractal dimension in multifractal spectra coincide with the scattering exponent of SAS curves in mass-fractal regions. Our analyses further revealed transitions from heterogeneous to homogeneous structures accompanied by changes from short to long-range mass-fractal regions. These transitions are explained in terms of the relative values of the scaling factors.

The structure of deterministic mass and surface fractals: theory and methods of analyzing small-angle scattering data

Small-angle scattering (SAS) of X-rays, neutrons or light from ensembles of randomly oriented and placed deterministic fractal structures is studied theoretically.

Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake

The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering amplitude. The surface fractals can be decomposed into a sum of surface mass fractals for arbitrary fractal iteration, which enables various approximations for the scattering intensity. It is shown that for the Cantor fractal, one can neglect with good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, correlations can be built in the mass fractal amplitudes, which explains the decay of the scattering intensity I(q) ∼ q^D_s-4, with 1 < D_s < 2 being the fractal dimension of the perimeter. The curve I(q)q^4-D_s is found to be log-periodic in the fractal region with a period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of the sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of the Koch snowflake, which is solved in the limit of infinite iterations. The present analysis allows us to obtain additional information from SAS data, such as the edges of the fractal regions, the fractal iteration number and the scaling factor.

Small-Angle Scattering from Nanoscale Fat Fractals

Small-angle scattering (of neutrons, x-ray, or light; SAS) is considered to describe the structural characteristics of deterministic nanoscale fat fractals. We show that in the case of a polydisperse fractal system, with equal probability for any orientation, one obtains the fractal dimensions and scaling factors at each structural level. This is in agreement with general results deduced in the context of small-angle scattering analysis of a system of randomly oriented, non-interacting, nano-/micro-fractals. We apply our results to a two-dimensional fat Cantor-like fractal, calculating analytic expressions for the scattering intensities and structure factors. We explain how the structural properties can be computed from experimental data and show their correlation to the variation of the scaling factor with the iteration number. The model can be used to interpret recorded experimental SAS data in the framework of fat fractals and can reveal structural properties of materials characterized by a regular law of changing of the fractal dimensions. It can describe successions of power-law decays, with arbitrary decreasing values of the scattering exponents, and interleaved by regions of constant intensity.

Structural characterization of chaos game fractals using small-angle scattering analysis

Small-angle scattering (SAS) technique is applied to study the nano and microstructural properties of spatial patterns generated from chaos game representation (CGR). Using a simplified version of Debye formula, we calculate and analyze in momentum space, the monodisperse scattering structure factor from a system of randomly oriented and non-interacting 2D Sierpinski gaskets (SG). We show that within CGR approach, the main geometrical and fractal properties, such as the overall size, scaling factor, minimal distance between scattering units, fractal dimension and the number of units composing the SG, can be recovered. We confirm the numerical results, by developing a theoretical model which describes analytically the structure factor of SG. We apply our findings to scattering from single scale mass fractals, and respectively to a multiscale fractal representing DNA sequences, and for which an analytic description of the structure factor is not known a priori.

Scattering from surface fractals in terms of composing mass fractals

It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensityI(q) ∝ q^{D_{\rm s}-6}, where 2 <Ds< 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝r−τdr, withDs= τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.

Extended Vicsek fractals: Laplacian spectra and their applications

Extended Vicsek fractals (EVF) are the structures constructed by introducing linear spacers into traditional Vicsek fractals. Here we study the Laplacian spectra of the EVF. In particularly, the recurrence relations for the Laplacian spectra allow us to obtain an analytic expression for the sum of all inverse nonvanishing Laplacian eigenvalues. This quantity characterizes the large-scale properties, such as the gyration radius of the polymeric structures, or the global mean-first passage time for the random walk processes. Introduction of the linear spacers leads to local heterogeneities, which reveal themselves, for example, in the dynamics of EVF under external forces.