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Concept: Fractal


The time-evolutions of nanoparticle hydrodynamic radius and aggregate fractal dimension during the aggregation of fullerene (C(60)) nanoparticles (FNPs) were measured via simultaneous multiangle static and dynamic light scattering. The FNP aggregation behavior was determined as a function of monovalent (NaCl) and divalent (CaCl(2)) electrolyte concentration, and the impact of addition of dissolved natural organic matter (humic acid) to the solution was also investigated. In the absence of humic acid, the fractal dimension decreased over time with monovalent and divalent salts, suggesting that aggregates become slightly more open and less compact as they grow. Although the aggregates become slightly more open, the magnitude of the fractal dimension suggests intermediate aggregation between the diffusion- and reaction-limited regimes. We observed different aggregation behavior with monovalent and divalent salts upon the addition of humic acid to the solution. For NaCl-induced aggregation, the introduction of humic acid significantly suppressed the aggregation rate of FNPs at NaCl concentrations lower than 150mM. In this case, the aggregation was intermediate or reaction-limited even at NaCl concentrations as high as 500mM, giving rise to aggregates with a fractal dimension of 2.0. For CaCl(2)-induced aggregation, the introduction of humic acid enhanced the aggregation of FNPs at CaCl(2) concentrations greater than about 5mM due to calcium complexation and bridging effects. Humic acid also had an impact on the FNP aggregate structure in the presence of CaCl(2), resulting in a fractal dimension of 1.6 for the diffusion-limited aggregation regime. Our results with CaCl(2) indicate that in the presence of humic acid, FNP aggregates have a more open and loose structure than in the absence of humic acid. The aggregation results presented in this paper have important implications for the transport, chemical reactivity, and toxicity of engineered nanoparticles in aquatic environments.

Concepts: Chemistry, Soil, Aggregate, Aggregate data, Light scattering, Humus, Natural organic matter, Fractal


Understanding the role of kinetics in fiber network microstructure formation is of considerable importance in engineering gel materials to achieve their optimized performances/functionalities. In this work, we present a new approach for kinetic-structure analysis for fibrous gel materials. In this method, kinetic data is acquired using a rheology technique and is analyzed in terms of an extended Dickinson model in which the scaling behaviors of dynamic rheological properties in the gelation process are taken into account. It enables us to extract the structural parameter, i.e. the fractal dimension, of a fibrous gel from the dynamic rheological measurement of the gelation process, and to establish the kinetic-structure relationship suitable for both dilute and concentrated gelling systems. In comparison to the fractal analysis method reported in a previous study, our method is advantageous due to its general validity for a wide range of fractal structures of fibrous gels, from a highly compact network of the spherulitic domains to an open fibrous network structure. With such a kinetic-structure analysis, we can gain a quantitative understanding of the role of kinetic control in engineering the microstructure of the fiber network in gel materials.

Concepts: Structure, Colloid, Materials science, Gel, Gels, Aerogel, Fractal, Fractal dimension


Animal behaviour exhibits fractal structure in space and time. Fractal properties in animal space-use have been explored extensively under the Lévy flight foraging hypothesis, but studies of behaviour change itself through time are rarer, have typically used shorter sequences generated in the laboratory, and generally lack critical assessment of their results. We thus performed an in-depth analysis of fractal time in binary dive sequences collected via bio-logging from free-ranging little penguins (Eudyptula minor) across full-day foraging trips (2(16) data points; 4 orders of temporal magnitude). Results from 4 fractal methods show that dive sequences are long-range dependent and persistent across ca. 2 orders of magnitude. This fractal structure correlated with trip length and time spent underwater, but individual traits had little effect. Fractal time is a fundamental characteristic of penguin foraging behaviour, and its investigation is thus a promising avenue for research on interactions between animals and their environments.

Concepts: Space, Penguin, Penguins, Fractal, Scuba diving, Little Penguin, Eudyptula, White-flippered Penguin


Ordered and chaotic superlattices have been identified in Nature that give rise to a variety of colours reflected by the skin of various organisms. In particular, organisms such as silvery fish possess superlattices that reflect a broad range of light from the visible to the UV. Such superlattices have previously been identified as ‘chaotic’, but we propose that apparent ‘chaotic’ natural structures, which have been previously modelled as completely random structures, should have an underlying fractal geometry. Fractal geometry, often described as the geometry of Nature, can be used to mimic structures found in Nature, but deterministic fractals produce structures that are too ‘perfect’ to appear natural. Introducing variability into fractals produces structures that appear more natural. We suggest that the ‘chaotic’ (purely random) superlattices identified in Nature are more accurately modelled by multi-generator fractals. Furthermore, we introduce fractal random Cantor bars as a candidate for generating both ordered and ‘chaotic’ superlattices, such as the ones found in silvery fish. A genetic algorithm is used to evolve optimal fractal random Cantor bars with multiple generators targeting several desired optical functions in the mid-infrared and the near-infrared. We present optimized superlattices demonstrating broadband reflection as well as single and multiple pass bands in the near-infrared regime.

Concepts: Mathematics, Chaos theory, Topology, Reflection, Fractal, Cantor set, Hausdorff dimension, Space-filling curve


Globally, there are millions of small lakes, but a small number of large lakes. Most key ecosystem patterns and processes scale with lake size, thus this asymmetry between area and abundance is a fundamental constraint on broad-scale patterns in lake ecology. Nonetheless, descriptions of lake size-distributions are scarce and empirical distributions are rarely evaluated relative to theoretical predictions. Here we develop expectations for Earth’s lake area-distribution based on percolation theory and evaluate these expectations with data from a global lake census. Lake surface areas ≥8.5 km(2) are power-law distributed with a tail exponent (τ = 1.97) and fractal dimension (d = 1.38), similar to theoretical expectations (τ = 2.05; d = 4/3). Lakes <8.5 km(2) are not power-law distributed. An independently developed regional lake census exhibits a similar transition and consistency with theoretical predictions. Small lakes deviate from the power-law distribution because smaller lakes are more susceptible to dynamical change and topographic behavior at sub-kilometer scales is not self-similar. Our results provide a robust characterization and theoretical explanation for the lake size-abundance relationship, and form a fundamental basis for understanding and predicting patterns in lake ecology at broad scales.

Concepts: Scientific method, Lake, Hypothesis, Distribution, Theory, Explanation, Fractal, Scales


Stochastic growth processes give rise to diverse and intricate structures everywhere in nature, often referred to as fractals. In general, these complex structures reflect the non-trivial competition among the interactions that generate them. In particular, the paradigmatic Laplacian-growth model exhibits a characteristic fractal to non-fractal morphological transition as the non-linear effects of its growth dynamics increase. So far, a complete scaling theory for this type of transitions, as well as a general analytical description for their fractal dimensions have been lacking. In this work, we show that despite the enormous variety of shapes, these morphological transitions have clear universal scaling characteristics. Using a statistical approach to fundamental particle-cluster aggregation, we introduce two non-trivial fractal to non-fractal transitions that capture all the main features of fractal growth. By analyzing the respective clusters, in addition to constructing a dynamical model for their fractal dimension, we show that they are well described by a general dimensionality function regardless of their space symmetry-breaking mechanism, including the Laplacian case itself. Moreover, under the appropriate variable transformation this description is universal, i.e., independent of the transition dynamics, the initial cluster configuration, and the embedding Euclidean space.

Concepts: Mathematics, Chaos theory, Dimension, Vector space, Spacetime, Manifold, Point, Fractal


In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters' fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.

Concepts: Mathematics, Chaos theory, Chemistry, Thermodynamics, System, Standard Model, Fractal, Fractal dimension


Fractals, being “exactly the same at every scale or nearly the same at different scales” as defined by Benoit B. Mandelbrot, are complicated yet fascinating patterns that are important in aesthetics, mathematics, science and engineering. Extended molecular fractals formed by the self-assembly of small-molecule components have long been pursued but, to the best of our knowledge, not achieved. To tackle this challenge we designed and made two aromatic bromo compounds (4,4″-dibromo-1,1':3',1″-terphenyl and 4,4‴-dibromo-1,1':3',1″:4″,1‴-quaterphenyl) to serve as building blocks. The formation of synergistic halogen and hydrogen bonds between these molecules is the driving force to assemble successfully a whole series of defect-free molecular fractals, specifically Sierpiński triangles, on a Ag(111) surface below 80 K. Several critical points that govern the preparation of the molecular Sierpiński triangles were scrutinized experimentally and revealed explicitly. This new strategy may be applied to prepare and explore various planar molecular fractals at surfaces.

Concepts: Chaos theory, Chemistry, Science, Fractal, Hausdorff dimension, Benoît Mandelbrot, Mandelbrot set, Gaston Julia


Motor imagery is based on the volitional modulation of sensorimotor rhythms (SMRs); however, the sensorimotor processes in patients with amyotrophic lateral sclerosis (ALS) are impaired, leading to degenerated motor imagery ability. Thus, motor imagery classification in ALS patients has been considered challenging in the brain-computer interface (BCI) community. In this study, we address this critical issue by introducing the Grassberger-Procaccia and Higuchi’s methods to estimate the fractal dimensions (GPFD and HFD, respectively) of the electroencephalography (EEG) signals from ALS patients. Moreover, a Fisher’s criterion-based channel selection strategy is proposed to automatically determine the best patient-dependent channel configuration from 30 EEG recording sites. An EEG data collection paradigm is designed to collect the EEG signal of resting state and the imagination of three movements, including right hand grasping (RH), left hand grasping (LH), and left foot stepping (LF). Five late-stage ALS patients without receiving any SMR training participated in this study. Experimental results show that the proposed GPFD feature is not only superior to the previously-used SMR features (mu and beta band powers of EEG from sensorimotor cortex) but also better than HFD. The accuracies achieved by the SMR features are not satisfactory (all lower than 80%) in all binary classification tasks, including RH imagery vs. resting, LH imagery vs. resting, and LF imagery vs. resting. For the discrimination between RH imagery and resting, the average accuracies of GPFD in 30-channel (without channel selection) and top-five-channel configurations are 95.25% and 93.50%, respectively. When using only one channel (the best channel among the 30), a high accuracy of 91.00% can still be achieved by the GPFD feature and a linear discriminant analysis (LDA) classifier. The results also demonstrate that the proposed Fisher’s criterion-based channel selection is capable of removing a large amount of redundant and noisy EEG channels. The proposed GPFD feature extraction combined with the channel selection strategy can be used as the basis for further developing high-accuracy and high-usability motor imagery BCI systems from which the patients with ALS can really benefit.

Concepts: Neuroscience, Amyotrophic lateral sclerosis, Electroencephalography, Left-handedness, Electromyography, Lou Gehrig, Fractal, ALS Society of Canada


The primary objective of this study was to evaluate a range of calculation points on water retention curves (WRC) instead of the singularity point at air-entry suction in the pore-solid fractal (PSF) model, which additionally considered the hysteresis effect based on the PSF theory. The modified pore-solid fractal (M-PSF) model was tested using 26 soil samples from Yangling on the Loess Plateau in China and 54 soil samples from the Unsaturated Soil Hydraulic Database. The derivation results showed that the M-PSF model is user-friendly and flexible for a wide range of calculation point options. This model theoretically describes the primary differences between the soil moisture desorption and the adsorption processes by the fractal dimensions. The M-PSF model demonstrated good performance particularly at the calculation points corresponding to the suctions from 100 cm to 1000 cm. Furthermore, the M-PSF model, used the fractal dimension of the particle size distribution, exhibited an accepted performance of WRC predictions for different textured soils when the suction values were ≥100 cm. To fully understand the function of hysteresis in the PSF theory, the role of allowable and accessible pores must be examined.

Concepts: Mathematics, Soil, Dimension, Particle size distribution, Fractal, Technological singularity, Water content, Water retention curve