Fractal Regime Research Articles

Unsaturated hydraulic conductivity modeling for porous media with two fractal regimes

A reliable means to predict the saturation-dependence of the hydraulic conductivity would have important applications and implications across soil science. In our efforts to improve predictive capabilities we apply a bimodal pore size distribution to generate simultaneously the soil water retention curve (SWRC) and the unsaturated hydraulic conductivity K in porous media. Our specific pore size model incorporates two fractal regimes, which we treat within the pore–solid fractal approach. The calculation of the hydraulic conductivity employs critical path analysis from percolation theory, which has already been shown to perform the best overall among models commonly employed. To evaluate the developed piecewise functions, 8 soil samples with different textures, e.g., loam, silt loam, sandy loam and clay are selected. All soils show almost the same cross-over point on both water retention and hydraulic conductivity curves on semi-log plots. We find that the piecewise water retention and unsaturated hydraulic conductivity models fit well the measured data. However, the hydraulic conductivity curves predicted from the water retention data agree relatively well with the measured one just for the first regime and tend to underestimate K in the second. We also compare our results with those obtained from unimodal pore-size distribution reported by Ghanbarian-Alavijeh and Hunt (2012). Comparing the measured data with the unimodal and bimodal models indicates that the bimodal distribution provide somewhat more realistic predictions than the unimodal one. If prediction is sacrificed and we simply try to model K using our results, we find that we can generate a very accurate phenomenological description of K with only a slight change in the values of the fractal dimensionality. Reasons for this discrepancy are discussed.

海岸带地理特征对沉水植被丰度的影响

Submerged aquatic vegetation(SAV)consists of a taxonomically diverse group of plants that lives entirely beneath the water surface.SAV provides habitat and supplies food for aquatic life,absorbs excess nutrients,and helps purify the water.Since the 1960s,SAV coverage has experienced dramatic decline worldwide due to serious deterioration of water quality in coastal ecosystems. In recent years,landscape analyses have been used to predict direct or indirect effects of geographic characteristics on aquatic organisms.In this study,we analyzed the effects of coastal geographical characteristics(such as watershed land use types,sub-estuary mouth width,shoreline fractal dimension,ratio of watershed area/sub-estuary area,high resolution average wave height and estuary sediment,etc.) on the spatial variation of SAV abundance based on the long-term dataset(1984—2009) from 99 sub-estuaries and their linked coastal sub-watersheds of the Chesapeake Bay in the United States.Coastal watershed land use is an important factor influencing inputs of nutrient and sediment to its associated sub-estuary,and thus indirectly affects SAV abundance.Our results showed that sub-estuaries with different dominant land covers in each salinity regime had significantly different variation in SAV abundance.Abundance of different SAV species varies spatially among different salinity regimes in shallow water area.In each salinity regime,SAV abundance in sub-estuaries declined with their linked coastal dominant watershed land use type in the following order: mixed-land developed land or forested land agricultural land in fresh water(TF) regime;mixed-land or agricultural land developed land forested land in oligohaline(OH) regime;forested land or mixed-land developed land or agricultural land in mesohaline(MH) regime;mixed-land or forested land developed land agricultural land in polyhaline(PH) regime.Sub-estuaries with different salinity regimes for each dominant land cover also had significantly different levels of SAV abundance.In each dominant watershed land use type,SAV abundance in sub-estuaries declined with salinity regime in the following order: no significant difference among four salinity regimes for developed land;OH MH TF or PH for agricultural land;OH or PH TF or MH for mixed-land;MH or PH TF or OH for forested land.Relative weaker linear correlation was found between a single geographical characteristics and SAV abundance(-0.157≤r≤0.442),but SAV abundance had significantly positive correlations with a few estuary characteristics,such as sub-estuary mouth width(r=0.442,P0.001),shoreline fractal dimension(r=0.290,P=0.007) and average wave height(r=0.306,P=0.002).A wider sub-estuary mouth could lead more sea water to diluting pollutants in shorter time and thus favors SAV growth.A higher shoreline fractal dimension value represents more complex estuary structure and small tributaries in each sub-estuary that favors SAV growth as well.Higher wave height could positively dilute pollutants or negatively erode SAV beds in a sub-estuary.The integrated impact on SAV abundance was positive.Using the classification and regression tree(CART) model,we predicted that ratio of watershed area/sub-estuary area had the greatest impact on SAV abundance that appeared at the highest level of the tree,followed by shoreline fractal dimension,salinity regime,and average wave height.The four geographical variables explained 63% of the total spatial variation in SAV abundance across the Chesapeake Bay. Our findings imply that variation in the coastal geographical characteristics can indirectly affect spatial SAV abundance.Therefore,geographical characteristics in both sub-estuaries and their associated watersheds should be considered in making strategy of SAV recovery or protection policy at the regional scale in coastal area.

Structure of drying fronts in three-dimensional porous media

Evaporation in a three-dimensional (3D) porous medium, a sand column saturated by water, was studied using synchrotron x-ray tomography. Three-dimensional images of the medium with a resolution of 7 μm were obtained during the evaporation. The entire column was scanned seven times, resulting in nearly 10(4) 2D cross sections and illustrating the spatial distribution of air, liquid, and solid phases at the pore scale. The results were analyzed in order to gain new insights and better understanding of the characteristics of the drying front that was formed when the liquid-filled pores were invaded by air, as well as the structure of the liquid phase as it was dried. The analysis indicates that the liquid phase has a self-similar fractal structure, with its fractal dimension D(f) in all the cross sections being a function of the water content or saturation. In addition, D(f) for the 3D liquid structure, as well as its density correlation function, were computed using the 3D images. A crossover length scale ξ was identified that separates the fractal regime from the compact geometry. For length scales r>ξ, the density correlation function approaches asymptotically the water content of the porous medium. The drying front is shown to be rough and multi-affine, rather than self-affine. Its properties were also computed using the 3D images. The roughness characteristics agree with those for imbibition in porous media, but not with those of fracture surfaces and crack lines.

FRACTAL SELF-ORGANIZATION OF BACTERIA-INSPIRED AGENTS

We develop an agent-based model as a preliminary theoretical basis to guide the synthesis of a new class of materials with dynamic properties similar to bacterial colonies. Each agent in the model is representative of an individual bacterium capable of: the uptake of chemicals (nutrients), which are metabolized; active movement (part viscous, part diffusive), consuming metabolic energy; and cellular division, when agents have doubled in size. The agents grow in number and self-organize into fractal structures, depending on the rules that define the actions of the agents and the parameter values. The environment of the agents includes chemicals responsible for their growth and is described by a diffusion-reaction equation with Michaelis-Menten kinetics. These rules are modeled mathematically by a set of equations with five dimensionless groups that are functions of physical parameters. Simulations are performed for different parameter values. The resulting structures are characterized by their fractal scaling regime, box-counting and mass-radius dimensions, and lacunarity. Each parameter influences the overall structure in a unique way, generating a wide spectrum of structures. For certain combinations of parameter values, the model converges to a steady state, with a finite population of agents that no longer divide.

Gel formation and aging in weakly attractive nanocolloid suspensions at intermediate concentrations

We report x-ray photon correlation spectroscopy (XPCS) and rheometry experiments to study the temporal evolution of gel formation and aging in suspensions of silica nanocolloids possessing a tunable short-range attraction. The colloid volume fractions, φ = 0.20 and 0.43, are below the glass regime at high concentration and above the fractal regime at low concentration. Following a sudden initiation of the interparticle attraction, the suspensions display a protracted latency period in which they remain fluid before acquiring a measurable elastic shear modulus. The duration of the latency period and the subsequent rate of increase of the modulus vary strongly with the strength of the attraction. The XPCS results indicate dynamic heterogeneity among the colloids during this gel formation in which a growing fraction of the particles become localized. The temporal evolution of this localization correlates with that of the rheology. In particular, the time scale over which the fraction of localized particles increases tracks the duration of the latency period. Also, at φ = 0.20 the localization length characterizing the motion of the localized fraction scales onto the shear modulus with no free parameters as predicted by a self-consistent theory based on mode coupling [K. S. Schweizer and G. Yatsenko, J. Chem. Phys. 127, 164505 (2007)], while deviations from the predicted scaling at φ = 0.43 are observed near the gel point. The XPCS results also reveal slow, hyperdiffusive motion of the colloids in the newly formed gels that is attributed to strain from the relaxation of internal stress. While some features of this motion correlate with the evolving rheology, others appear decoupled from the macroscopic mechanical behavior.

Formation of fractal aggregates during green synthesis of silver nanoparticles

The aggregation behavior of silver nanoparticles (AgNPs) prepared by a green synthesis procedure using starch as the stabilizer was studied by the small angle X-ray scattering (SAXS) technique. The protecting ability of starch was affected by the presence of NaOH leading to different aggregation behaviors. In all the samples, mass as well as surface fractal regimes were observed. Assuming spherical form, the radii of nanoparticles were in the range of 11–17 nm.

Trapping and sedimentation of inertial particles in three-dimensional flows in a cylindrical container with exactly counter-rotating lids

Stirring and sedimentation of solid inertial particles in low-Reynolds-number flows has acquired great relevance in multiple environmental, industrial and microfluidic systems, but few detailed numerical studies have focused on chaotically advected experimentally realizable flows. We carry out one-way coupling simulations to study the dynamics of inertial particles in the steady three-dimensional flow in a cylindrical container with exactly counter-rotating lids, which was recently studied by Lackey & Sotiropoulos (Phys. Fluids, vol. 18, 2006, paper no. 053601). We elucidate the rich Lagrangian dynamics of the flow in the vicinity of toroidal invariant regions and show that depending on the Stokes number inertial particles could get trapped for long times in different equilibrium positions inside integrable islands. In the chaotically advected region of the flow the balance between inertia and gravity forces (represented by the settling velocity) can produce a striking fractal sedimentation regime, characterized by a sequence of discrete deposition events of seemingly random number of particles separated by hiatuses of random duration. The resulting staircase-like distribution of the time series of the number of particles in suspension is shown to be a devil's staircase whose fractal dimension is equal to the 0.87 value found in multiple dissipative dynamical systems in nature. Our work sheds new light on the complex mechanisms governing the stirring and deposition of inertial particles and provides new information about the parameters that are relevant in the characterization of particle dynamics in different regions of chaotically advected flows.

Fractal and chaotic patterns in animal morphology

In the review the spatial organization of cells, cell ensembles, tissues and metazoan body are considered using the concepts of fractal geometry, topology and dynamic chaos theory. We investigated both the scenario of transition from chaos into order during self-organization of cells in vitro and the reverse scenario of transition from order to chaos in the fractal morphogenesis of metazoan cell systems. Chaotic features in animal morphology were identified and quantified. Fractal morphogenesis was studied using epithelial branching channels of gastrovascular system in the scyphomeduse Aurelia aurita and tracheal gill system in the mayfly larvae Siphlonurus immanis and Parameletus chelifer, as well as structures of colonial interna in rhizocephalan crustaceans Peltogasterella gracilis and Polyascus polygenea. It was shown that completely identical fractal patterns do not occur even within a single animal body with radial or bilateral symmetric, functionally equivalent repetitive modules. Fractal dimension was used to quantify the spatial complexity of neuron morphology in central nervous system of the fishes Pholidapus dybowskii, Oncorhyhchus keta and Oncorhyhchus masou. During ontogenesis of Oncorhyhchus masou the values of fractal dimension and linear morphometric indicators were rising in studied neuron groups. Probably biological morphogenesis with chaotic fractal regime had an advantage in evolution, providing morphofunctional variability, plasticity and adaptability to unpredictable environmental changes.

Multifractal regime transition in a modified minority game model

The search for more realistic modeling of financial time series reveals several stylized facts of real markets. In this work we focus on the multifractal properties found in price and index signals. Although the usual minority game (MG) models do not exhibit multifractality, we study here one of its variants that does. We show that the nonsynchronous MG models in the nonergodic phase is multifractal and in this sense, together with other stylized facts, constitute a better modeling tool. Using the structure function (SF) approach we detected the stationary and the scaling range of the time series generated by the MG model and, from the linear (non-linear) behavior of the SF we identified the fractal (multifractal) regimes. Finally, using the wavelet transform modulus maxima (WTMM) technique we obtained its multifractal spectrum width for different dynamical regimes.

On the limits of fractal surface behaviour in silica. A virtual adsorbates simulation

A computer simulation technique, suited to replicate real adsorption experiments, was applied to pure simulated silica in order to gain insight into the fractal regime of its surface. The previously reported experimental fractal dimension was closely approached and the hitherto uncharted lower limit of fractal surface behaviour is reported herein.

Surface analysis of cryofixation-vacuum-freeze-dried polyaluminum chloride–humic acid (PACl–HA) flocs

The powder of polyaluminum chloride–humic acid (PACl–HA) flocs was prepared by cryofixation-vacuum-freeze-drying method. The FTIR spectra show that some characteristic functional groups in polyaluminum chloride (PACl), humic acid (HA), and kaolin still existed in the dried flocs. X-ray diffractometry (XRD) patterns indicate that these flocs are amorphous. Nitrogen adsorption–desorption isotherms were obtained for different samples of the dried PACl–HA flocs. The BET specific surface area, BJH cumulative absorbed volume and BJH desorption average pore diameter of them were determined. The peak values of 8.4–11.2 nm (pore diameter) for pore size distribution (PSD) curves indicate that the pores of the dried flocs are mostly mesopores. The surface fractal dimensions D s and the corresponding fractal scales determined from both SEM images and nitrogen adsorption–desorption data sets reveal the multi-scale surface fractal properties of the dried PACl–HA flocs, which exhibited two distinct fractal regimes: a regime of low fractal dimensions (2.07–2.26) at higher scales (23–387 nm), mainly belonging to exterior surface scales, and a higher fractal dimensions (2.24–2.37) at lower scales (0.80–7.81 nm), falling in pore surface scales. Both HA addition and kaolin reduction in dried floc can decrease the irregularity and roughness of external surface. However, for the irregularity and roughness of pore surface, the addition of HA or kaolin in dried floc can increase them. Furthermore, some difference was found between the pore surface fractal dimensions D s calculated from nitrogen adsorption and desorption data. The pore surface D s values calculated through thermodynamic model were much greater than three.

Synthesis of Titania Nanofibers for Photocatalytic Applications

Titania nanofibers were prepared using a colloidal solution of Titania nanoparticles and a templating method. The preparation comprises impregnation of cellulose fibers in the solution containing Titania nanoparticles followed by thermal removal of the cellulose template. It is evidenced by scanning electron microscopy (SEM) that the residue substance consists of micron size fibers with a microstructure of nanofibers. The pore size distribution of the substance is multi‐scale and fractal morphology was demonstrated with two fractal regimes with dimensions of 1.78 and 2.97 for sizes below and above 7.5 nm, respectively. The surface area of the Titania fibers was measured about 250 m2/g. The photocatalytic property of nanofibers was measured using methanol and cigarette smoke.

Conductivity fluctuation of YBa2Cu3O7 - delta/Sr2YSbO6/SrTiO3 thin films

Thermal fluctuations in the electric conductivity of YBa2Cu3O7 - delta superconducting thin films grown on Sr2YSbO6 novel substrate materials in the film form were experimentally studied. YBa2Cu3O7 - delta films were grown by using a dc-technique on Sr2YSbO6 substrates, which were produced by rf magnetron sputtering. X-ray diffraction analysis evidenced that the Sr2YSbO6 films grow on conventional SrTiO3 substrates in a preferential orientation along the (100) planes direction with lattice parameter a=4,43() Å. The YBa2Cu3O7 - delta thin films, grown on Sr2YSbO6 films, exhibit an oriented growth in the (001) crystallographic direction, with lattice constant c=11,65(9) Å. Morphological characterizations were performed by means atomic force microscopy. Experiments of electrical resistivity show that the YBa2Cu3O7 - delta films present a normal-superconducting transition with critical temperature Tc=82,33 K. Fluctuation analysis for the YBa2Cu3O7 - delta thin films were performed by utilizing the concept of logarithmic derivative of the conductivity excess. Above the critical temperature Tc we experimentally determine the occurrence of Gaussian 3D, 2D and fractal fluctuation regimes. A genuinely critical region identified by the exponent lambdaCR=0,35 were obtained close to Tc. This critical regime is effectively described by the 3D-XY model.

From Homogeneous to Fractal Normal and Tumorous Microvascular Networks in the Brain

We studied normal and tumorous three-dimensional (3D) microvascular networks in primate and rat brain. Tissues were prepared following a new preparation technique intended for high-resolution synchrotron tomography of microvascular networks. The resulting 3D images with a spatial resolution of less than the minimum capillary diameter permit a complete description of the entire vascular network for volumes as large as tens of cubic millimeters. The structural properties of the vascular networks were investigated by several multiscale methods such as fractal and power-spectrum analysis. These investigations gave a new coherent picture of normal and pathological complex vascular structures. They showed that normal cortical vascular networks have scale-invariant fractal properties on a small scale from 1.4 mum up to 40 to 65 mum. Above this threshold, vascular networks can be considered as homogeneous. Tumor vascular networks show similar characteristics, but the validity range of the fractal regime extend to much larger spatial dimensions. These 3D results shed new light on previous two dimensional analyses giving for the first time a direct measurement of vascular modules associated with vessel-tissue surface exchange.

Highly porous TiO2 nanofibres with a fractal structure

TiO2 nanofibres were prepared using a templating method with tetraisopropylorthotitanate(TiPT) as a precursor. The preparation comprises liquid phase deposition oncellulose fibres followed by thermal removal of the cellulose template. The obtainedTiO2 fibrous substance consists of micron-size fibres with a microstructure of nanofibres. It wasdemonstrated that nanofibres are basically formed through the aggregation ofTiO2 nanoparticles and nanorods into chain structures during thethermal treatment process. The measured surface area of theTiO2 fibreswas about 250 m2 g−1. It was shown that the pore size distribution is multi-scale and a fractal morphologywas demonstrated with two fractal regimes with dimensions of 1.78 and 2.97for sizes below and above 7.5 nm, respectively. The crystalline phase of theTiO2 nanofibres, as well as the nanoparticles in the solution, could be controlled by the pH of thesolution. A pH of 1.2 resulted in rutile phase, while a pH of 1.8 resulted in anatase phase.

Relation between Aggregation Kinetics and the Structure of Kaolinite Aggregates

Dynamic and static light scattering were applied to the determination of the stability ratio and fractal dimension of kaolinite (KGa-2) at different kaolinite or/and electrolyte concentrations at pH 9.5. Dynamic light scattering was used to measure the kinetics of early stage aggregation to determine the stability ratio, W, as well as the cluster sizes which determine the fractal regime. Static light scattering was used to measure the fractal dimension, D(f). Results show that the two classes of "universality" (Lin et al. Nature 1989, 339, 360) characterizing the diffusion- and reaction-limited regimes of cluster-cluster aggregation do apply to colloidal kaolinite as limit cases when W approximately 1 or W > 100, respectively. In the intermediate regime where 5 < W < 100, the growth of the aggregate radius showed a power-law behavior similar to diffusion-limited cluster aggregation. For the intermediate aggregation regime, a scaling relation between fractal dimension and stability ratio, reflecting a continuous increase in particle packing density in the aggregate as the sticking probability of particles was reduced, was demonstrated.

(Non-)fractality on Large Scales

The debate about the possible smoothness of the Universe on large scales as opposed to an unbounded fractal hierarchy has been the subject of increasing interest in recent years. The controversy arises as a consequence of different statistical analyses performed on surveys of galaxy redshifts. I review the observational evidence supporting the idea that a gradual transition occurs in the galaxy distribution: from a fractal regime at small scales to large scale homogeneity.

Colloidal Aggregation Induced by Long Range Attractions

The structure of colloidal clusters formed by long-range attractive interactions under diluted conditions is studied by means of Monte Carlo simulations. For a not-too-long attraction range, clusters show self-similar internal structure with lower density than that typical for diffusive aggregation. For long-range interactions, low kappa, nonfractal clusters are formed (dense at short scales but open at long ones). The dependence on the volume fraction shows that more-compact clusters are grown the higher the colloidal density for diffusive aggregation and attraction-driven aggregation in the fractal regime. The whole trend is explained in terms of the interpenetration among aggregates. In attraction-driven aggregations, the interpenetration of clusters competes with aggregation in the tips of the clusters, causing low-density clusters.

Fractal aircraft trajectories and nonclassical turbulent exponents

The dimension (D) of aircraft trajectories is fundamental in interpreting airborne data. To estimate D, we studied data from 18 trajectories of stratospheric aircraft flights 1600 km long taken during a "Mach cruise" (near constant Mach number) autopilot flight mode of the ER-2 research aircraft. Mach cruise implies correlated temperature and wind fluctuations so that DeltaZ approximately Deltax (H(z) ) where Z is the (fluctuating) vertical and x the horizontal coordinate of the aircraft. Over the range approximately 3-300 km , we found H(z) approximately 0.58+/-0.02 close to the theoretical 5/9=0.56 and implying D=1+ H(z) =14/9 , i.e., the trajectories are fractal. For distances <3 km aircraft inertia smooths the trajectories, for distances >300 km , D=1 again because of a rise of 1 m/km due to fuel consumption. In the fractal regime, the horizontal velocity and temperature exponents are close to the nonclassical value 1/2 (rather than 1/3 ). We discuss implications for aircraft measurements as well as for the structure of the atmosphere.

Modelling soil water retention scaling. Comparison of a classical fractal model with a piecewise approach

Soil water retention curves are very important for modeling and predicting geohydrological processes. Fractal-based models seem to be appropriate theoretical frameworks for investigating this soil characteristics. Since soil water retention is sensitive to both structure and soil texture, it is difficult to see a unique value of fractal dimension as a physical descriptor of both soil conditions. The objectives of this work were to (i) present a piecewise fractal approach to approximate the soil water retention data, (ii) test the model with previously published and unpublished data sets, and (iii) compare its performance with a traditional surface fractal model. The goodness-of-fit of the piecewise model was excellent ( R 2>0.95). Almost all the soil water retention data (21 data sets in total) showed two fractal scaling regimes. The cutoff ranged from a minimum h c=10.889 kPa (Ariana silty clay loam soil) to a maximum h c=2951 kPa (Walla-Walla silt loam soil). The fractal dimension corresponding to the first domain ranged from D 1p=2.59 to D 1p=2.85, while those values corresponding to the second regime varied between D 2p=2.72 and D 2p=2.95. The fit of a classical surface fractal model rendered poor results in terms of goodness-of-fit parameters with a large dispersion of predicted water content values at low tensions. The presented piecewise approach could be consistent to some extent with the bimodal pore-size distributions usually observed in experimental studies.